Motion Mountain Physics Textbook Volume 2 - Relativity and Cosmology

Authors Christoph Schiller

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Christoph Schiller

MOTION MOUNTAIN
the adventure of physics – vol.ii
relativity and cosmology

www.motionmountain.net
Christoph Schiller

Motion Mountain

The Adventure of Physics
Volume II

Relativity and Cosmology

Edition 31, available as free pdf
with films at www.motionmountain.net
Editio trigesima prima.

Proprietas scriptoris © Chrestophori Schiller
primo anno Olympiadis trigesimae secundae.

Omnia proprietatis iura reservantur et vindicantur.
Imitatio prohibita sine auctoris permissione.
Non licet pecuniam expetere pro aliqua, quae
partem horum verborum continet; liber
pro omnibus semper gratuitus erat et manet.

Thirty-first edition.

Copyright © 1990–2023 by Christoph Schiller,
from the third year of the 24th Olympiad
to the first year of the 32nd Olympiad.

This pdf file is licensed under the Creative Commons
Attribution-Noncommercial-No Derivative Works 3.0 Germany
Licence, whose full text can be found on the website
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with the additional restriction that reproduction, distribution and use,
in whole or in part, in any product or service, be it
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the copyright owner. The pdf file was and remains free for everybody
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electronically, but only in unmodified form and only at no charge.
To Britta, Esther and Justus Aaron

τῷ ἐμοὶ δαὶμονι
Die Menschen stärken, die Sachen klären.
PR E FAC E

“ ”
Primum movere, deinde docere.*
Antiquity

T
his book series is for anybody who is curious about motion in nature. How do
hings, people, animals, images and empty space move? The answer leads
o many adventures. This volume presents the best ones about extremely fast,

Motion Mountain – The Adventure of Physics
powerful and distant motion. In the exploration of motion – physics – special and
general relativity make up two important stages, as shown in Figure 1.
Special relativity is the exploration of nature’s speed limit 𝑐. General relativity is the
exploration of the force limit 𝑐4 /4𝐺. The text shows that in both domains, all results
follow from these two limit values. In particular, cosmology is the exploration of motion
near nature’s distance limit 1/√Λ . This simple, intuitive and unusual way of learning
relativity should reward the curiosity of every reader – whether student or professional.
The present volume is the second of a six-volume overview of physics that arose from
a threefold aim that I have pursued since 1990: to present motion in a way that is simple,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
up to date and captivating.
In order to be simple, the text focuses on concepts, while keeping mathematics to the
necessary minimum. Understanding the concepts of physics is given precedence over
using formulae in calculations. The whole text is within the reach of an undergraduate.
In order to be up to date, the text is enriched by the many gems – both theoretical and
empirical – that are scattered throughout the scientific literature.
In order to be captivating, the text tries to startle the reader as much as possible. Read-
ing a book on general physics should be like going to a magic show. We watch, we are
astonished, we do not believe our eyes, we think, and finally we understand the trick.
When we look at nature, we often have the same experience. Indeed, every page presents
at least one surprise or provocation for the reader to think about.
The motto of the text, die Menschen stärken, die Sachen klären, a famous statement
on pedagogy, translates as: ‘To fortify people, to clarify things.’ Clarifying things – and
adhering only to the truth – requires courage, as changing the habits of thought produces
fear, often hidden by anger. But by overcoming our fears we grow in strength. And we
experience intense and beautiful emotions. All great adventures in life allow this, and
exploring motion is one of them. Enjoy it.

Christoph Schiller
* ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is called motiv-
ating; both terms go back to the same Latin root.
8 preface

Final, unified description of motion: upper limit c4/4Ghbar
Adventures: describing precisely all motion, understanding
the origin of colours, space -time and particles, enjoying
extreme thinking, calculating masses and couplings,
catching a further, tiny glimpse of bliss (vol. VI).

PHYSICS: An arrow indicates an
Describing motion with precision, increase in precision by
i.e., using the least action principle. adding a motion limit.

upper limit: Quantum theory
General relativity: 1/4G hbar with classical gravity Quantum field theory
upper limit c4/4G Adventures: bouncing (the ‘standard model’):
Adventures: the neutrons, under- upper limit c/hbar
night sky, measu- standing tree Adventures: building

Motion Mountain – The Adventure of Physics
ring curved and growth (vol. V). accelerators, under-
wobbling space, standing quarks, stars,
exploring black bombs and the basis of
holes and the life, matter & radiation
universe, space (vol. V).
and time (vol. II).

Classical gravity: upper limit: c Special relativity Quantum theory:
upper limit 1/4G Adventures: light, upper limit 1/hbar
Adventures: magnetism, length Adventures: biology,
climbing, skiing, c contraction, time birth, love, death,
space travel, limits dilation and chemistry, evolution,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the wonders of G fast E0 = mc2 h, e, k enjoying colours, art,
astronomy and limits motion (vol. II). limit paradoxes, medicine
geology (vol. I). uniform tiny and high-tech business
motion motion (vol. IV and vol. V).

Galilean physics, heat and electricity: no limits
The world of everyday motion: human scale, slow and weak.
Adventures: sport, music, sailing, cooking, describing
beauty and understanding its origin (vol. I);
using electricity, light and computers,
understanding the brain and people (vol. III).
F I G U R E 1 A complete map of physics, the science of motion, as ﬁrst proposed by Matvei Bronshtein
(b. 1907 Vinnytsia, d. 1938 Leningrad). The Bronshtein cube starts at the bottom with everyday motion,
and shows the connections to the ﬁelds of modern physics. Each connection increases the precision of
the description and is due to a limit to motion that is taken into account. The limits are given for
uniform motion by the gravitational constant G, for fast motion by the speed of light c, and for tiny
motion by the Planck constant h, the elementary charge e and the Boltzmann constant k.
preface 9

Using this b o ok
Marginal notes refer to bibliographic references, to other pages or to challenge solutions.
In the colour edition, marginal notes, pointers to footnotes and links to websites are
typeset in green. Over time, links on the internet tend to disappear. Most links can be
recovered via www.archive.org, which keeps a copy of old internet pages. In the free
pdf edition of this book, available at www.motionmountain.net, all green pointers and
links are clickable. The pdf edition also contains all films; they can be watched directly
in Adobe Reader.
Solutions and hints for challenges are given in the appendix. Challenges are classified
as easy (e), standard student level (s), difficult (d) and research level (r). Challenges for
which no solution has yet been included in the book are marked (ny).

Advice for learners
Learning allows us to discover what kind of person we can be. Learning widens know-
ledge, improves intelligence and provides a sense of achievement. Therefore, learning

Motion Mountain – The Adventure of Physics
from a book, especially one about nature, should be efficient and enjoyable. Avoid bad
learning methods like the plague! Do not use a marker, a pen or a pencil to highlight
or underline text on paper. It is a waste of time, provides false comfort and makes the
text unreadable. Add notes and comments instead! And do not learn from a screen. In
particular, do not learn from videos, from games or from a smartphone. All games and
almost all videos are drugs for the brain. Smartphones are drug dispensers that make
people addicted and prevent learning. Learn from paper – at your speed, and allow your
mind to wander! Nobody marking paper or looking at a screen is learning efficiently.
In my experience as a pupil and teacher, one learning method never failed to trans-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
form unsuccessful pupils into successful ones: if you read a text for study, summarize
every section you read, in your own words and images, aloud. If you are unable to do
so, read the section again. Repeat this until you can clearly summarize what you read in
your own words and images, aloud. And enjoy the telling aloud! You can do this alone
or with friends, in a room or while walking. If you do this with everything you read, you
will reduce your learning and reading time significantly; you will enjoy learning from
good texts much more and hate bad texts much less. Masters of the method can use it
even while listening to a lecture, in a low voice, thus avoiding to ever take notes.

Advice for teachers
A teacher likes pupils and likes to lead them into exploring the field he or she chose. His
or her enthusiasm is the key to job satisfaction. If you are a teacher, before the start of a
lesson, picture, feel and tell yourself how you enjoy the topic of the lesson; then picture,
feel and tell yourself how you will lead each of your pupils into enjoying that topic as
much as you do. Do this exercise consciously, every day. You will minimize trouble in
your class and maximize your teaching success.
This book is not written with exams in mind; it is written to make teachers and stu-
dents understand and enjoy physics, the science of motion.
10 preface

Feedback
The latest pdf edition of this text is and will remain free to download from the internet.
I would be delighted to receive an email from you at fb@motionmountain.net, especially
on the following issues:
Challenge 1 s — What was unclear and should be improved?
— What story, topic, riddle, picture or film did you miss?
Also help on the specific points listed on the www.motionmountain.net/help.html web
page is welcome. All feedback will be used to improve the next edition. You are welcome
to send feedback by mail or by sending in a pdf with added yellow notes, to provide
illustrations or photographs, or to contribute to the errata wiki on the website. If you
would like to translate a chapter of the book in your language, please let me know.
On behalf of all readers, thank you in advance for your input. For a particularly useful
contribution you will be mentioned – if you want – in the acknowledgements, receive a
reward, or both.

Motion Mountain – The Adventure of Physics
Support
Your donation to the charitable, tax-exempt non-profit organisation that produces, trans-
lates and publishes this book series is welcome. For details, see the web page www.
motionmountain.net/donation.html. The German tax office checks the proper use of
your donation. If you want, your name will be included in the sponsor list. Thank you in
advance for your help, on behalf of all readers across the world.
The paper edition of this book is available, either in colour or in black and white,
from www.amazon.com, in English and in certain other languages. And now, enjoy the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
reading.
C ON T E N T S

7 Preface
Using this book 9 • Advice for learners 9 • Advice for teachers 9 • Feedback 10 •
Support 10
11 Contents
15 1 Maximum speed, observers at rest and motion of light
Aberration and the speed of rain 17 • The speed of light 19 • Can one play tennis

Motion Mountain – The Adventure of Physics
using a laser pulse as the ball and mirrors as rackets? 22 • Albert Einstein 25 •
An invariant limit speed and its consequences 26 • Special relativity with a few
lines 28 • Acceleration of light and the Doppler effect 31 • The difference between
light and sound 36 • Can one shoot faster than one’s shadow? 37 • The compos-
ition of velocities 39 • Observers and the principle of special relativity 40 • What
is space-time? 45 • Can we travel to the past? – Time and causality 46 • Curi-
osities about special relativity 48 • Faster than light: how far can we travel? 48 •
Synchronization and time travel – can a mother stay younger than her own daugh-
ter? 49 • Length contraction 52 • Relativistic films – aberration and Doppler ef-
fect 54 • Which is the best seat in a bus? 55 • How fast can one walk? 58 • Is the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
speed of shadow greater than the speed of light? 58 • Parallel to parallel is not
parallel – Thomas precession 62 • A never-ending story – temperature and relativ-
ity 62 • A curiosity: what is the one-way speed of light? 63 • Summary 63
65 2 Relativistic mechanics
Mass in relativity 65 • Why relativistic snooker is more difficult 67 • Mass and
energy are equivalent 68 • Weighing light 71 • Collisions, virtual objects and
tachyons 72 • Systems of particles – no centre of mass 74 • Why is most mo-
tion so slow? 75 • The history of the mass–energy equivalence formula 76 •
4-vectors 76 • 4-velocity 78 • 4-acceleration and proper acceleration 79 • 4-
momentum or energy–momentum or momenergy 81 • 4-force – and the nature
of mechanics 83 • Rotation in relativity 84 • Wave motion 86 • The action of a
free particle – how do things move? 87 • Conformal transformations 89 • Ac-
celerating observers 91 • Accelerating frames of reference 93 • Constant accele-
ration 94 • Event horizons 96 • The importance of horizons 98 • Acceleration
changes colours 99 • Can light move faster than 𝑐? 100 • The composition of ac-
celerations 100 • Limits on the length of solid bodies 101
104 3 Special relativity in four sentences
Could the speed of light vary? 104 • Where does special relativity break down? 105
107 4 Simple general relativity: gravitation, maximum speed and max-
imum force
Maximum force – general relativity in one statement 108 • The meaning of the
12 contents

force and power limits 110 • The experimental evidence 112 • Deducing general
relativity 113 • Gravity, space-time curvature, horizons and maximum force 117 •
Conditions of validity for the force and power limits 119 • Gedanken experiments
and paradoxes about the force limit 120 • Gedanken experiments with the
power and the mass flow limits 125 • Why maximum force has remained undis-
covered for so long 128 • An intuitive understanding of general relativity 129 •
An intuitive understanding of cosmology 132 • Experimental challenges for the
third millennium 133 • A summary of general relativity – and minimum force 134
136 5 How maximum speed changes space, time and gravity
Rest and free fall 136 • What clocks tell us about gravity 137 • What tides tell us
about gravity 141 • Bent space and mattresses 143 • Curved space-time 145 •
The speed of light and the gravitational constant 147 • Why does a stone thrown
into the air fall back to Earth? – Geodesics 149 • Can light fall? 151 • Curiosities
and fun challenges about gravitation 152 • What is weight? 157 • Why do
apples fall? 158 • A summary: the implications of the invariant speed of light on
gravitation 159

Motion Mountain – The Adventure of Physics
160 6 Open orbits, bent light and wobbling vacuum
Weak fields 160 • Bending of light and radio waves 161 • Time delay 163 • Re-
lativistic effects on orbits 164 • The geodesic effect 166 • The Thirring effects 169 •
Gravitomagnetism 170 • Gravitational waves 174 • Production and detection of
gravitational waves 179 • Curiosities and fun challenges about weak fields 183 •
A summary on orbits and waves 184
185 7 From curvature to motion
How to measure curvature in two dimensions 185 • Three dimensions: curvature
of space 188 • Curvature in space-time 190 • Average curvature and motion in
general relativity 192 • Universal gravity 193 • The Schwarzschild metric 193 •

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Curiosities and fun challenges about curvature 194 • Three-dimensional
curvature: the Ricci tensor 194 • Average curvature: the Ricci scalar 195 •
The Einstein tensor 195 • The description of momentum, mass and energy 196 •
Einstein’s field equations 198 • Universal gravitation – again 199 • Understand-
ing the field equations 200 • Hilbert’s action – how does space bend? 201 • The
symmetries of general relativity 202 • Mass in general relativity 203 • The force
limit and the cosmological constant 203 • Is gravity an interaction? 204 • How
to calculate the shape of geodesics 205 • Riemann gymnastics 206 • Curiosities
and fun challenges about general relativity 208 • A simple summary of the field
equations 210
211 8 Why can we see the stars? – Motion in the universe
Which stars do we see? 211 • How do we watch the stars? 214 • What do we
see at night? 216 • What is the universe? 223 • The colour and the motion of the
stars 226 • Do stars shine every night? 228 • A short history of the universe 230
• The history of space-time 234 • Why is the sky dark at night? 239 • The col-
our variations of the night sky 242 • Is the universe open, closed or marginal? 243
• Why is the universe transparent? 245 • The big bang and its consequences 246
• Was the big bang a big bang? 247 • Was the big bang an event? 247 • Was the
big bang a beginning? 247 • Does the big bang imply creation? 248 • Why can
we see the Sun? 249 • Why do the colours of the stars differ? 250 • Are there dark
stars? 252 • Are all stars different? – Gravitational lenses 252 • What is the shape
of the universe? 254 • What is behind the horizon? 255 • Why are there stars
contents 13

all over the place? – Inflation 256 • Why are there so few stars? – The energy and
entropy content of the universe 256 • Why is matter lumped? 257 • Why are stars
so small compared with the universe? 258 • Are stars and galaxies moving apart
or is the universe expanding? 258 • Is there more than one universe? 258 • Why
are the stars fixed? – Arms, stars and Mach’s principle 258 • At rest in the
universe 260 • Does light attract light? 260 • Does light decay? 261 • Summary
on cosmology 261
262 9 Black holes – falling forever
Why explore black holes? 262 • Mass concentration and horizons 262 • Black hole
horizons as limit surfaces 266 • Orbits around black holes 267 • Black holes have
no hair 269 • Black holes as energy sources 271 • Formation of and search for
black holes 273 • Singularities 274 • Curiosities and fun challenges about black
holes 275 • Summary on black holes 278 • A quiz – is the universe a black
hole? 279
280 10 Does space differ from time?
Can space and time be measured? 282 • Are space and time necessary? 283 • Do

Motion Mountain – The Adventure of Physics
closed time-like curves exist? 283 • Is general relativity local? – The hole argu-
ment 284 • Is the Earth hollow? 285 • A summary: are space, time and mass
independent? 286
287 11 General relativity in a nutshell – a summary for the layman
The accuracy of the description 289 • Research in general relativity and cosmo-
logy 290 • Could general relativity be different? 292 • The limitations of general
relativity 293
295 a Units, measurements and constants
SI units 295 • The meaning of measurement 298 • Curiosities and fun challenges
about units 298 • Precision and accuracy of measurements 300 • Limits to preci-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
sion 301 • Physical constants 302 • Useful numbers 309
310 Challenge hints and solutions
321 Bibliography
350 Credits
Acknowledgements 350 • Film credits 351 • Image credits 351
353 Name index
361 Subject index
R elativity

In our quest to learn how things move,
the experience of hiking and seeing leads us to discover
that there is a maximum energy speed in nature,
that two events that occur at the same time for one observer
may not for another, and
that acceleration limits observation distance by a horizon.
We discover that empty space can bend, wobble and move,
we experience the fascination of black holes,
we find that there is a maximum force in nature,
we perceive why we can see the stars
and we understand why the sky is dark at night.
Chapter 1

M A X I M UM SPE E D, OB SE RV E R S AT
R E ST A N D MOT ION OF L IG H T

“ ”
Fama nihil est celerius.**
Antiquity

L
ight is indispensable for a precise description of motion. To check whether a
ine or a path of motion is straight, we must look along it. In other words, we use

Motion Mountain – The Adventure of Physics
ight to define straightness. How do we decide whether a plane is flat? We look across
it,*** again using light. How do we observe motion? With light. How do we measure
length to high precision? With light. How do we measure time to high precision? With
light: once it was light from the Sun that was used; nowadays it is light from caesium
Page 295 atoms.
In short, light is important because

⊳ Light is the standard for ideal, undisturbed motion.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Physics would have evolved much more rapidly if, at some earlier time, light propagation
had been recognized as the ideal example of motion.
But is light really a phenomenon of motion? Yes. This was already known in ancient
Greece, from a simple daily phenomenon, the shadow. Shadows prove that light is a mov-
ing entity, emanating from the light source, and moving in straight lines.**** The Greek
Ref. 1 thinker Empedocles (c. 490 to c. 430 b ce) drew the logical conclusion that light takes
a certain amount of time to travel from the source to the surface showing the shadow.
Empedocles thus stated that

** ‘Nothing is faster than rumour.’ This common sentence is a simplified version of Virgil’s phrase: fama,
malum qua non aliud velocius ullum. ‘Rumour, the evil faster than all.’ From the Aeneid, book IV, verses
173 and 174.
*** Note that looking along the plane from all sides is not sufficient for this check: a surface that a light
beam touches right along its length in all directions does not need to be flat. Can you give an example? One
Challenge 2 s needs other methods to check flatness with light. Can you specify one?
**** Whenever a source produces shadows, the emitted entities are called rays or radiation. Apart from
light, other examples of radiation discovered through shadows were infrared rays and ultraviolet rays, which
emanate from most light sources together with visible light, and cathode rays, which were found to be to the
motion of a new particle, the electron. Shadows also led to the discovery of X-rays, which again turned out
to be a version of light, with high frequency. Channel rays were also discovered via their shadows; they turn
out to be travelling ionized atoms. The three types of radioactivity, namely α-rays (helium nuclei), β-rays
(again electrons), and γ-rays (high-energy X-rays) also produce shadows. All these discoveries were made
between 1890 and 1910: those were the ‘ray days’ of physics.
16 1 maximum speed, observers at rest and

F I G U R E 2 How do you check whether the lines

Motion Mountain – The Adventure of Physics
are curved or straight?

⊳ The speed of light is finite.

We can confirm this result with a different, equally simple, but subtle argument. Speed
can be measured. And measurement is comparison with a standard. Therefore the per-
fect or ideal speed, which is used as the implicit measurement standard, must have a fi-
Challenge 3 s nite value. An infinite velocity standard would not allow measurements at all. (Why?) In
nature, lighter bodies tend to move with higher speed. Light, which is indeed extremely

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
light, is an obvious candidate for motion with perfect but finite speed. We will confirm
this in a minute.
A finite speed of light means that whatever we see is a message from the past. When
we see the stars,* the Sun or a person we love, we always see an image of the past. In a
sense, nature prevents us from enjoying the present – but teaches us to learn to enjoy the
past.
The speed of light is high; therefore it was not measured until the years 1668 to 1676,
even though many, including Isaac Beeckman in 1629 and Galileo in 1638, had tried to
Ref. 3 do so earlier. ** The first measurement method was realized and published by the Danish
astronomer Ole Rømer*** when he was studying the orbits of Io and the other Galilean

* The photograph of the night sky and the Milky Way, on page 14 is copyright Anthony Ayiomamitis and is
found on his splendid website www.perseus.gr.
** During his whole life, and still in 1638, René Descartes argued publicly that the speed of light was infin-
ite for reasons of principle. But in 1637, he had assumed a finite value in his explanation of Snell’s ‘law’.
Ref. 2 This shows how confused philosophers can be. In fact, Descartes wrote to Beeckman in 1634 that if one
could prove that the speed of light is finite, he would be ready to admit directly that he ‘knew nothing of
philosophy.’ We should take him by his word.
*** Ole (Olaf) Rømer (b. 1644 Aarhus, d. 1710 Copenhagen), important astronomer. He was the teacher of
the Dauphin in Paris, at the time of Louis XIV. The idea of measuring the speed of light in this way was due
to the astronomer Giovanni Cassini, whose assistant Rømer had been. Rømer continued his measurements
until 1681, when Rømer had to leave France, like all protestants (such as Christiaan Huygens), so that his
motion of light 17

Jupiter and Io
(second measurement)

Earth (second
measurement)

Sun Earth (first Jupiter and Io
measurement) (first measurement)

F I G U R E 3 Rømer’s method of measuring the speed of light.

Motion Mountain – The Adventure of Physics
Vol. I, page 210 satellites of Jupiter. He did not obtain any specific value for the speed of light because
he had no reliable value for the satellite’s distance from Earth and because his timing
measurements were imprecise. The lack of a numerical result was quickly corrected by
Ref. 4 his peers, mainly Christiaan Huygens and Edmund Halley. (You might try to deduce
Challenge 4 s Rømer’s method from Figure 3.) Since Rømer’s time it has been known that light takes a
bit more than 8 minutes to travel from the Sun to the Earth. This result was confirmed in a
beautiful way fifty years later, in the 1720s, independently, by the astronomers Eustachio
Manfredi (b. 1674 Bologna , d. 1739 Bologna) and James Bradley (b. 1693 Sherborne ,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Vol. I, page 152 d. 1762 Chalford). Their measurements allowed the ‘rain method’ to measure the speed
Ref. 5 of light.

Aberration and the speed of rain
How can we measure the speed of falling rain? We walk rapidly with an umbrella, meas-
ure the angle 𝛼 at which the rain appears to fall, and then measure our own velocity 𝑣.
(We can clearly see the angle while walking if we look at the rain to our left or right, if
possible against a dark background.) As shown in Figure 4, the speed 𝑐r of the rain is
then given by
𝑐r = 𝑣/ tan 𝛼 . (1)

In the same way we can measure the speed of wind when on a surfboard or on a ship.
The same method can be applied to the speed of light. Figure 4 shows that we just need
to measure the angle between the motion of the Earth and the light coming from a star
above Earth’s orbit. Because the Earth is moving relative to the Sun and thus to the star,

work was interrupted. Back in Denmark, a fire destroyed all his measurement notes. As a result, he was not
able to continue improving the precision of his method. Later he became an important administrator and
reformer of the Danish state.
18 1 maximum speed, observers at rest and

Motion Mountain – The Adventure of Physics

F I G U R E 4 The rainwalker’s or windsurfer’s method of measuring the speed of light.

the angle is not 90°. For the speed of light 𝑐, we get

Challenge 5 s (Why is the expression for light different?) The deviation from the geometrically ex-
Ref. 7 pected angle was called the aberration of light by Eustachio Manfredi. The aberration
is determined by comparing measurements over the course of a year, in particular, six
months apart. The explanation of aberration was also found by James Bradley, who in-
dependently, made similar measurements.* The measured value of the aberration angle

* Umbrellas were not common in Europe in 1719 or 1726; they became fashionable later. The umbrella part
of the story is made up. It is said that Bradley understood aberration while sailing on the Thames, when
he noted that on a moving ship the apparent wind, showed by an on-board flag, has a direction that de-
pends on the sailing direction and thus differs from that on land. For many years, independently, Manfredi
and Bradley had observed numerous stars, notably Gamma Draconis, and during that time they had been
puzzled by the sign of the aberration, which was opposite to the effect they were looking for, namely that
of the star parallax. Both the parallax and the aberration for a star above the ecliptic make them describe
a small ellipse in the course of an Earth year, though the ellipses differ by their orientation and their rota-
Challenge 6 s tion sign. Can you see why? Today we know that the largest known parallax for a star is 0.77 󸀠󸀠 , whereas the
major axis of the aberration ellipse is 20.5 󸀠󸀠 for all stars. The discovery by Bradley and Manfredi convinced
even church officials that the Earth moves around the Sun, and Galileo’s books were eventually taken from
the index of forbidden books. Since the church delayed the publication of Manfredi’s discovery, Bradley
is often named as the sole discoverer of aberration. But the name of the effect recalls Manfredi’s priority.
Because of the discovery, Manfredi became member of the Académie des Sciences and the Royal Society.
motion of light 19

for a star exactly above the ecliptic is 20.49552(1) 󸀠󸀠 ≈ 0.1 mrad – a really small angle. It
is called the aberration constant. Its existence clearly shows that the Earth orbits the Sun,
when observed by a distant observer. Yes, the Earth moves.
Using the aberration angle, we can deduce the speed of light if we know the speed of
the Earth when travelling around the Sun. For this, we first have to determine its dis-
tance from the Sun. The simplest method is the one by the Greek thinker Aristarchus of
Samos (c. 310 to c. 230 b ce). We measure the angle between the Moon and the Sun at
the moment when the Moon is precisely half full. The cosine of that angle gives the ratio
Vol. I, page 179 between the distance to the Moon (determined as explained earlier on) and the distance
Challenge 7 s to the Sun. The explanation is left as a puzzle for the reader.
The angle of Aristarchus * is almost a right angle (which would yield an infinite dis-
Ref. 6 tance), and good instruments are needed to measure it with precision, as Hipparchus
noted in an extensive discussion of the problem around 130 b ce. Precise measurement
of the angle became possible only in the late seventeenth century, when it was found to be
89.86°, giving a Sun–Moon distance ratio of about 400. Today, thanks to radar distance
Page 308 measurements of planets, the average distance to the Sun is known with the incredible

Motion Mountain – The Adventure of Physics
precision of 30 metres;** its value is 149 597 870.691(30) km, or roughly 150 million kilo-
metres.

The speed of light
Using the distance between the Earth and the Sun, the Earth’s orbital speed is 𝑣 =
2π𝑅/𝑇 = 29.7 km/s. Therefore, the aberration angle gives us the following result

⊳ The speed of light (in vacuum) is 𝑐 = 0.300 Gm/s, or 0.3 m/ns, or
0.3 mm/ps, or 1080 million km/h.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
This is an astonishing speed value, especially when compared with the highest speed ever
achieved by a man-made object, namely the Helios II satellite, which travelled around the
Sun at 253 Mm/h = 70.2 km/s, with the growth of children, about 3 nm/s, or with the
growth of stalagmites in caves, about 0.3 pm/s. We begin to realize why measurement of
the speed of light is a science in its own right.
The first precise measurement of the speed of light was made in 1849 by Hippolyte
Fizeau (b. 1819 Paris, d. 1896 Venteuil). His value was only 5 % greater than the modern
one. He sent a beam of light towards a distant mirror and measured the time the light
took to come back. How did Fizeau measure the time without any electric device? In fact,
Vol. I, page 61 he used the same ideas that are used to measure bullet speeds; part of the answer is given
Challenge 9 s in Figure 5. (How far away does the mirror have to be?) A modern reconstruction of his
Ref. 9 experiment by Jan Frercks has even achieved a precision of 2 %. Today, the measurement
Ref. 8 * Aristarchus also determined the radius of the Sun and of the Moon as multiples of those of the Earth.
Aristarchus was a remarkable thinker: he was the first to propose the heliocentric system, and perhaps the
first to propose that stars were other, faraway suns. For these ideas, several of his contemporaries proposed
that he should be condemned to death for impiety. When the monk and astronomer Nicolaus Copernicus
(b. 1473 Thorn, d. 1543 Frauenburg) reproposed the heliocentric system two thousand years later, he did not
mention Aristarchus, even though he got the idea from him.
** Moon distance variations can even be measured to the nearest centimetre; can you guess how this is
Challenge 8 s achieved?
20 1 maximum speed, observers at rest and

half-silvered
mirror
large distance

mirror light
source

Motion Mountain – The Adventure of Physics
F I G U R E 5 Fizeau’s set-up to measure the speed of light (photo © AG Didaktik und Geschichte der
Physik, Universität Oldenburg).

red
shutter
switch
beam

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
light path of light pulse
pulse

10 mm

F I G U R E 6 The ﬁrst photograph of a green light pulse moving from right to left through a bottle with
milky water, marked in millimetres (photograph © Tom Mattick).

is much simpler; in the chapters on electrodynamics we will discover how to measure the
speed of light using two standard Unix or Linux computers connected by a cable, using
Vol. III, page 32 the ‘ping’ command.
The speed of light is so high that in everyday life it is even difficult to prove that it is
finite. Perhaps the most beautiful way to prove this is to photograph a light pulse flying
across one’s field of view, in the same way as one can photograph a car driving by or a
Ref. 10 bullet flying through the air. Figure 6 shows the first such photograph, produced in 1971
with a standard off-the-shelf reflex camera, a very fast shutter invented by the photo-
graphers, and, most noteworthy, not a single piece of electronic equipment. (How fast
Challenge 10 s does such a shutter have to be? How would you build such a shutter? And how would
you make sure it opened at the right instant?)
A finite speed of light also implies that a rapidly rotating light beam bends, as shown
motion of light 21

F I G U R E 7 A consequence of the ﬁniteness
of the speed of light. Watch out for the
tricky details – light does travel straight from
the source, it does not move along the
drawn curved line; the same occurs for
water emitted by a rotating water sprinkler.

Motion Mountain – The Adventure of Physics
F I G U R E 8 A ﬁlm
taken with a special
ultrafast camera
showing a short

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
light pulse that
bounces off a
mirror (QuickTime
ﬁlm © Wang Lihong
and Washington
University at St.
Louis).

as in Figure 7. In everyday life, the high speed of light and the slow rotation of lighthouses
make the effect unnoticeable. But maybe, one day, ...
Finally, in the twenty-first century, films of moving light pulses started to appear. A
beautiful example is shown in Figure 8. Such films again confirm that light has a finite
speed.
In summary, light moves extremely rapidly, but with a finite speed. For example, light is
Challenge 11 s much faster than lightning, as you might like to check yourself. A century of increasingly
precise measurements of the speed of light in all its forms have culminated in the modern
value
𝑐 = 299 792 458 m/s. (3)

In fact, this value has now been fixed exactly, by definition, and the metre has been
22 1 maximum speed, observers at rest and

TA B L E 1 Properties of the motion of light.

O b s e r va t i o n s a b o u t l i g h t

Light can move through vacuum.
Light transports energy.
Light has momentum: it can hit bodies.
Light has angular momentum: it can rotate bodies.
Light moves across other light undisturbed.
In vacuum, the speed of light is 𝑐 = 299 792 458 m/s, or roughly 30 cm/ns – always and every-
where.
Light in vacuum always moves faster than any material body does.
The proper speed of light is infinite. Page 48
The speed of light pulses, their true signal speed, is the forerunner speed, not the group velocity.
In vacuum, the forerunner speed is always and everywhere 𝑐. Vol. III, page 135
Light beams are approximations when the wavelength is neglected.

Motion Mountain – The Adventure of Physics
Light beams move in a straight line when far from matter.
Shadows can move without any speed limit.
Normal and high-intensity light is a wave. Light of extremely low intensity is a stream of particles.
In matter, both the forerunner speed and the energy speed of light are at most 𝑐.
In matter, the group velocity of light pulses can be negative, zero, positive or infinite.

defined in terms of the speed of light 𝑐 since 1983. The good approximate values 0.3 Gm/s
or 0.3 μm/fs are obviously easier to remember. A summary of what is known today about
the motion of light is given in Table 1. Two of the most surprising properties of light mo-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
tion were discovered in the late nineteenth century. They form the basis of what is called
Ref. 11 the theory of special relativity.

C an one play tennis using a laser pulse as the ball and mirrors
as rackets?

“ ”
Et nihil est celerius annis.*
Ovid, Metamorphoses.

All experiments ever performed show: the speed of electromagnetic radiation in vacuum
does not depend on the frequency of the radiation, nor on its polarization, nor on its
intensity.
Ref. 12 For example, electromagnetic pulses from the Crab nebula pulsar have been shown
to have the same speed over 13 decades of frequencies, from radio waves to 𝛾-rays. The
speed value is the same to a precision of 14 digits. Observations using 𝛾-ray bursts have
improved this precision to 20 digits. After starting together and travelling together for
Ref. 13 thousands of millions of years across the universe, light pulses with different frequencies
and polarizations still arrive side by side.
Comparisons between the speed of 𝛾-rays and the speed of visible light have also been

* ‘Nothing is faster than the years.’ Book X, verse 520.
motion of light 23

performed in accelerators. Also the speed of radio waves of different frequencies when
Ref. 14 travelling around the Earth can be compared. All such experiments found no detectable
change of the speed of light with frequency. Additional experiments show that the speed
Ref. 15 of light is the same in all directions of space, to at least 21 digits of precision.
Light from the most powerful lasers, light from the weakest pocket lamps and light
from the most distant stars has the same speed. In the same way, linearly polarized, cir-
cularly polarized and elliptically polarized light, but also thermal, i.e., unpolarized light
has the same speed.
In summary,

⊳ Nature provides no way to accelerate or decelerate the motion of light in
vacuum.

Watching pulsating stars in the sky proves it. The speed of light in vacuum is always the
same: it is invariant. But this invariance is puzzling.
We all know that in order to throw a stone as fast and as far as possible, we run as

Motion Mountain – The Adventure of Physics
we throw it; we know instinctively that in that case the stone’s speed with respect to the
ground is higher than if we do not run. We also know that hitting a tennis ball more
rapidly makes it faster.
However, to the initial astonishment of everybody, experiments show that light emit-
ted from a moving lamp has the same speed as light emitted from a resting one. The
simplest way to prove this is to look at the sky. The sky shows many examples of double
stars: these are two stars that rotate around each other along ellipses. In some of these
systems, we see the ellipses (almost) edge-on, so that each star periodically moves to-
wards and away from us. If the speed of light would vary with the speed of the source,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
we would see bizarre effects, because the light emitted from some positions would catch
up the light emitted from other positions. In particular, we would not be able to observe
the elliptical shape of the orbits. However, such bizarre effects are not seen, and perfect
Ref. 16 ellipses are observed. Willem de Sitter gave this beautiful argument already in 1913; he
confirmed its validity with a large number of double stars.
In other words, light in vacuum is never faster than light:

⊳ All light beams in vacuum have the same speed.

Ref. 13, Ref. 17 Many specially designed experiments have confirmed this result to high precision. The
speed of light can be measured with a precision of better than 1 m/s; but even for lamp
speeds of more than 290 000 000 m/s the speed of the emitted light does not change. (Can
Challenge 12 s you guess what lamps were used?)
In everyday life, we also know that a stone or a tennis ball arrives more rapidly if we
run towards it than in the case that we stand still or even run away from it. But aston-
ishingly again, for light in a vacuum, no such effect exists! All experiments clearly show
that if we run towards a lamp, we measure the same speed of light as in the case that we
stand still or even run away from it. Also these experiments have been performed to the
Ref. 18 highest precision possible. Even for the highest observer speeds, the speed of the arriving
light remains the same.
Both sets of experiments, those with moving lamps and those with moving observ-
24 1 maximum speed, observers at rest and

F I G U R E 9 All devices based on electric motors prove that the speed of light is invariant (© Miele,
EasyGlide).

Motion Mountain – The Adventure of Physics
ers, thus show that the velocity of light has exactly the same magnitude for everybody,
everywhere and always – even if observers are moving with respect to each other or with
respect to the light source.

⊳ The speed of light in vacuum is invariant.

The speed of light in vacuum is indeed the ideal, perfect measurement standard for speed.
By the way, an equivalent alternative term for ‘speed of light’ is ‘radar speed’ or ‘radio

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Vol. III, page 108 speed’; we will see in the part on electrodynamics why this is the case.
The speed of light is also not far from the speed of neutrinos. This was shown most
spectacularly by the observation of a supernova in 1987, when the light flash and the
neutrino pulse arrived on Earth only 12 seconds apart. (The difference is probably due to
a tiny speed difference and to a different starting point of the two flashes.) What would
be the first digit for which the two speed values could differ, knowing that the supernova
Challenge 13 s was 1.7 ⋅ 105 light years away, and assuming the same starting point?
Ref. 19 There is also a further set of experimental evidence for the invariance of the speed of
light. Every electromagnetic device, such as an electric vacuum cleaner, shows that the
Vol. III, page 53 speed of light is invariant. We will discover that magnetic fields would not result from
electric currents, as they do every day in every electric motor and in every loudspeaker,
if the speed of light were not invariant. This was actually how the invariance was first
deduced, by several researchers. Only after these results did Albert Einstein show that
the invariance of the speed of light is also in agreement with the observed motion of
Ref. 20 bodies. We will check this agreement in this chapter. The connection between relativity
and electric vacuum cleaners, as well as other machines, will be explored in the chapters
Vol. III, page 53 on electrodynamics.
The motion of light and the motion of bodies are deeply connected. If the speed of
light were not invariant, observers would be able to move at the speed of light. Why?
Since light is a wave, an observer moving almost as fast as such a light wave would see a
light wave moving slowly. And an observer moving at the same speed as the wave would
motion of light 25

F I G U R E 10 Albert Einstein (1879–1955).

see a frozen wave. However, experiment and the properties of electromagnetism prevent
Vol. III, page 53 both observations; observers and bodies cannot reach the speed of light.

⊳ The speed of light in vacuum is a limit speed.

Motion Mountain – The Adventure of Physics
Observers and bodies thus always move slower than light.
In summary, the speed of light in vacuum is an invariant limit speed. Therefore, there
is no way to accelerate a light pulse. And, in contrast to a tennis ball, there is no way
to see a light pulse before it actually arrives. Thus, playing tennis with light is neither
Challenge 14 d possible nor is it fun – at least in vacuum. But what about other situations?

Albert Einstein
Albert Einstein (b. 1879 Ulm, d. 1955 Princeton) was one of the greatest physicists ever.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
(By the way, the ‘s’ in his family name is pronounced ‘sh’ and the two instances of ‘ei’ are
pronounced like ‘eye’, so that the full pronunciation is ["albErt "aInStaIn].) In 1905, he
published three important papers: one about Brownian motion, one about special relativ-
ity and one about the idea of light quanta. The first paper showed definitely that matter is
made of molecules and atoms; the second showed the invariance of the speed of light; and
the third paper was one of the starting points of quantum theory. Each paper was worth a
Nobel Prize, but he was awarded the prize only for the last one. In 1906, he published the
Page 76 proof of the famous formula 𝐸 = 𝑐2 𝑚, after a few others also had proposed it. Although
Einstein was one of the founders of quantum theory, he later turned against it. His famous
discussions with his friend Niels Bohr nevertheless helped to clarify quantum theory in
its most counter-intuitive aspects. Later, he explained the Einstein–de Haas effect which
proves that magnetism is due to motion inside materials. After many other discoveries, in
1915 and 1916 Einstein published his highest achievement: the general theory of relativity,
Page 136 one of the most beautiful and remarkable works of science. In the remaining forty years
of his life, he searched for the unified theory of motion, without success.
Being Jewish and famous, Einstein was a favourite target of attacks and discrimination
by the National Socialist movement; therefore, in 1933 he emigrated from Germany to
the USA; since that time, he stopped contact with Germans, except for a few friends,
among them Max Planck. Another of his enemies was the philosopher Henri Bergson.
An influential figure of the time, he somehow achieved, with his confused thinking, to
prevent that Einstein received the Nobel Prize in Physics. Until his death, Einstein kept
26 1 maximum speed, observers at rest and

TA B L E 2 How to convince yourself and others that there is a maximum
energy speed 𝑐 in nature. Compare this table with the table about
maximum force, on page 109 below, and with the table about a smallest
action, on page 19 in volume IV.

S tat e m e n t Te s t

The maximum energy speed value Check all observations.
𝑐 is observer-invariant.
Local energy speed values > 𝑐 are Check all observations.
not observed.
Local energy speed values > 𝑐 Check all attempts.
cannot be produced.
Local energy speed values > 𝑐 Solve all paradoxes.
cannot even be imagined.
The maximum local energy speed Deduce the theory of
value 𝑐 is a principle of nature. special relativity from it.

Motion Mountain – The Adventure of Physics
Check that all
consequences, however
weird, are confirmed by
observation.

his Swiss passport in his bedroom. He was not only a great physicist, but also a great
Ref. 21 thinker; his collection of thoughts about topics outside physics are well worth reading.
However, his family life was disastrous, and he made each of his family members deeply

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
unhappy.
Anyone interested in emulating Einstein should know first of all that he published
many papers.* He was both ambitious and hard-working. Moreover, many of his papers
Ref. 22 were wrong; he would then correct them in subsequent papers, and then do so again. This
happened so frequently that he made fun of himself about it. Einstein indeed realized the
well-known definition of a genius as a person who makes the largest possible number of
mistakes in the shortest possible time.

An invariant limit speed and its consequences
Experiments and theory show that observers cannot reach the speed of light. Equival-
ently, no object can reach the speed of light. In other words, not only is the speed of light
the standard of speed; it is also the maximum speed in nature. More precisely, the velo-
city 𝑣 of any physical system in nature – i.e., of any localized mass or energy – is bound
by
𝑣⩽𝑐. (4)

This relation is the basis of special relativity; in fact, the complete theory of special re-
lativity is contained in it.

* All his papers and letters are now freely available online, at einsteinpapers.press.princeton.edu.
motion of light 27

The existence of an invariant limit speed 𝑐 is not as surprising at we might think: we
Page 104 need such an invariant value in order to be able to measure speeds. Nevertheless, an in-
variant maximum speed implies many fascinating results: it leads to observer-varying
time and length intervals, to an intimate relation between mass and energy, to the exist-
ence of event horizons and to the existence of antimatter, as we will see.
Already in 1895, Henri Poincaré * called the discussion of viewpoint invariance the
theory of relativity, and the name was common in 1905. Einstein regretted that the theory
was called this way; he would have preferred the name ‘Invarianztheorie’, i.e., ‘theory of
Ref. 23 invariance’, but was not able to change the name any more. Thus Einstein called the
Ref. 19 description of motion without gravity the theory of special relativity, and the description
of motion with gravity the theory of general relativity. Both fields are full of fascinating
and counter-intuitive results, as we will find out.**
Can an invariant limit speed really exist in nature? Table 2 shows that we need to
explore three points to accept the idea. We need to show that first, no higher speed is
observed, secondly, that no higher energy speed can ever be observed, and thirdly, that
all consequences of the invariance of the speed of light, however weird they may be, apply

Motion Mountain – The Adventure of Physics
to nature. In fact, this programme defines the theory of special relativity; thus it is all we
do in this and the next chapter.
The invariance of the speed of light is in complete contrast with Galilean mechanics,
which describes the behaviour of stones, and proves that Galilean mechanics is wrong at
high velocities. At low velocities the Galilean description remains good, because the error
is small. But if we want a description valid at all velocities, we have to discard Galilean
mechanics. For example, when we play tennis, by hitting the ball in the right way, we
can increase or decrease its speed. But with light this is impossible. Even if we mount a
mirror on an aeroplane and reflect a light beam with it, the light still moves away with

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the same speed, both for the pilot and for an observer on Earth. All experiments confirm
this weird behaviour of light.
If we accelerate a bus that we are driving, the cars on the other side of the road pass by
with higher and higher speeds. For light, experiment shows that this is not so: light always
passes by with the same speed. Even with the current measurement precision of 2 ⋅ 10−13 ,
Ref. 15 we cannot discern any changes of the speed of light for different speeds of the observer.
Light does not behave like cars or any other matter object. Again, all experiments confirm
this weird behaviour.
Why exactly is the invariance of the speed of light almost unbelievable, even though
the measurements show it unambiguously? Take two observers O and Ω (pronounced
Vol. I, page 441 ‘omega’) moving with relative velocity 𝑣, such as two cars on opposite sides of the street.
Imagine that at the moment they pass each other, a light flash is emitted by a lamp in O.
The light flash moves through positions 𝑥(𝑡) for observer O and through positions 𝜉(𝜏)
(pronounced ‘xi of tau’) for Ω. Since the speed of light is measured to be the same for

* Henri Poincaré (1854 Nancy–1912 Paris), important mathematician and physicist. Poincaré was one of the
most productive scientists of his time, advancing relativity, quantum theory and many parts of mathematics.
Ref. 24 ** Among the most beautiful introductions to relativity are still those given by Albert Einstein himself. It
has taken almost a century for books almost as beautiful to appear, such as the texts by Schwinger or by
Ref. 25, Ref. 26 Taylor and Wheeler.
28 1 maximum speed, observers at rest and

both, we have
𝑥 𝜉
=𝑐= . (5)
𝑡 𝜏

However, in the situation described, we obviously have 𝑥 ≠ 𝜉. In other words, the invari-
ance of the speed of light implies that 𝑡 ≠ 𝜏, i.e., that

⊳ Time is different for observers moving relative to each other.

Challenge 15 e Time is thus not unique. This surprising result, which has been confirmed by many
Ref. 27 experiments, was first stated clearly in 1905 by Albert Einstein. Every observer has its own
time. Two observers’ times agree only if they do not move against each other. Though
many others knew about the invariance of 𝑐, only the young Einstein had the courage to
say that time is observer-dependent, and to explore and face the consequences. Let us do
so as well.
One remark is in order. The speed of light 𝑐 is a limit speed. What is meant with this

Motion Mountain – The Adventure of Physics
statement is that

⊳ The speed of light in vacuum is a limit speed.

Indeed, particles can move faster than the speed of light in matter, as long as they move
slower than the speed of light in vacuum. This situation is regularly observed.
In solid or liquid matter, the speed of light is regularly two or three times lower than
the speed of light in vacuum. For special materials, the speed of light can be even lower: in
Ref. 28 the centre of the Sun, the speed of light is estimated to be around 30 km/year = 1 mm/s,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
and even in the laboratory, for some materials, the speed of light has been measured to
Ref. 29 be as low as 0.3 m/s.
Vol. I, page 327 When an aeroplane moves faster than the speed of sound in air, it creates a cone-
shaped shock wave behind it. When a charged particle moves faster than the speed
of light in matter, it emits a cone of radiation, so-called Vavilov–Čerenkov radiation.
Vavilov–Čerenkov radiation is regularly observed; for example, it is the cause of the blue
glow of the water in nuclear reactors and it appears in transparent plastic crossed by fast
particles, a connection used in detectors for accelerator experiments.
In this and the following chapters, when we use the term ‘speed of light’, we mean the
speed of light in vacuum. In air, the speed of light is smaller than that in vacuum only by
a fraction of one per cent, so that in most cases, the difference between air and vacuum
can be neglected.

Special relativity with a few lines
The speed of light is invariant and constant for all observers. We can thus deduce all
Ref. 30 relations between what two different observers measure with the help of Figure 11. It
shows two observers moving with constant speed against each other, drawn in space-
time. The first is sending a light flash to the second, from where it is reflected back to the
first. Since the speed of light is invariant, light is the only way to compare time and space
coordinates for two distant observers. Also two distant clocks (like two distant metre
motion of light 29

first
𝑡 observer second
or clock observer
or clock

𝑘2 𝑇
light flash

𝑡1 = (𝑘2 + 1)𝑇/2 𝑡2 = 𝑘𝑇

light flash
𝑇

𝑂
𝑥
F I G U R E 11 A drawing containing most of special

Motion Mountain – The Adventure of Physics
relativity, including the expressions for time dilation
and for the Lorentz transformation.

bars) can only be compared, or synchronized, using light or radio flashes. Since light
speed is invariant, all light paths in the same direction are parallel in such diagrams.
A constant relative speed between two observers implies that a constant factor 𝑘
Challenge 16 s relates the time coordinates of events. (Why is the relation linear?) If a flash starts at
a time 𝑇 as measured for the first observer, it arrives at the second at time 𝑘𝑇, and then

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 17 s back again at the first at time 𝑘2 𝑇. The drawing shows that

𝑐+𝑣 𝑣 𝑘2 − 1
𝑘=√ or = . (6)
𝑐−𝑣 𝑐 𝑘2 + 1

Page 31 This factor will appear again in the Doppler effect.*
Figure 11 also shows that the first observer measures a time 𝑡1 for the event when the
light is reflected; however, the second observer measures a different time 𝑡2 for the same
event. Time is indeed different for two observers in relative motion. This effect is called
time dilation. In other terms, time is relative. Figure 12 shows a way to illustrate the result.
The time dilation factor between the two observers is found from Figure 11 by com-
paring the values 𝑡1 and 𝑡2 ; it is given by

𝑡1 1
= = 𝛾(𝑣) . (7)
𝑡2 √ 𝑣2
1− 𝑐2

Time intervals for a moving observer are shorter by this factor 𝛾; the time dilation factor
is always larger than 1. In other words,

* The explanation of relativity using the factor 𝑘 is sometimes called k-calculus.
30 1 maximum speed, observers at rest and

one moving watch

first second
time time

F I G U R E 12 Moving clocks
two fixed watches
go slow: moving clocks mark
time more slowly than do
stationary clocks.

Motion Mountain – The Adventure of Physics
F I G U R E 13 Moving clocks go slow: moving lithium atoms in a storage ring (left) read out with lasers
(right) conﬁrm the prediction to highest precision (© TSR relativity team at the Max Planck Gesellschaft).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
⊳ Moving clocks go slower.

Challenge 18 e For everyday speeds the effect is tiny. That is why we do not detect time differences in
everyday life. Nevertheless, Galilean physics is not correct for speeds near that of light;
Ref. 31 the correct expression (7) has been tested to a precision better than one part in 10 million,
with an experiment shown in Figure 13. The same factor 𝛾 also appears in the formula
𝐸 = 𝑐2 𝛾𝑚 for the equivalence of mass and energy, which we will deduce below. Expres-
sions (6) or (7) are the only pieces of mathematics needed in special relativity: all other
results derive from it.
If a light flash is sent forward starting from the second observer to the first and re-
flected back, the second observer will make a similar statement: for him, the first clock
is moving, and also for him, the moving clock marks time more slowly.

⊳ Each of the observers observes that the other clock marks time more slowly.

The situation is similar to that of two men comparing the number of steps between two
identical ladders that are not parallel, as shown in Figure 14. A man on either ladder will
always observe that the steps of the other ladder are shorter. There is nothing deeper than
Page 52 this observation at the basis of time dilation and length contraction.
motion of light 31

first
ladder
𝑦 second
(first
ladder
observer)
(second
observer)

𝑥

Motion Mountain – The Adventure of Physics
F I G U R E 14 The observers on both ladders claim
that the other ladder is shorter.

Naturally, many people have tried to find arguments to avoid the strange conclusion
that time differs from observer to observer. But none have succeeded, and all experi-
mental results confirm that conclusion: time is relative. Let us have a look at some of
these experiments.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Acceleration of light and the Doppler effect
Can light in vacuum be accelerated? It depends on what you mean. Most physicists are
snobbish and say that every mirror accelerates light, because it changes its direction. We
will see in the chapter on electromagnetism that matter also has the power to bend light,
Vol. III, page 157 and thus to accelerate it. However, it will turn out that all these methods only change the
direction of propagation; none has the power to change the speed of light in a vacuum. In
particular, light is an example of a motion that cannot be stopped. There are only a few
Challenge 19 s other such examples. Can you name one?
What would happen if we could accelerate light to higher speeds? For this to be pos-
sible, light would have to be made of massive particles. If light had mass, it would be
necessary to distinguish the ‘massless energy speed’ 𝑐 from the speed of light 𝑐L , which
would be lower and would depend on the kinetic energy of those massive light particles.
The speed of light would not be invariant, but the massless energy speed would still be so.
Such massive light particles could be captured, stopped and stored in a box. Such boxes
would make electric illumination unnecessary; it would be sufficient to store some day-
light in them and release the light, slowly, during the following night, maybe after giving
it a push to speed it up.*

* Incidentally, massive light would also have longitudinal polarization modes. This is in contrast to obser-
vations, which show that light is polarized exclusively transversally to the propagation direction.
32 1 maximum speed, observers at rest and

Redshifts of quasar spectra

Lyman α Hγ Hβ Hα
almost static reference:
Vega
v = 13.6 km/s at 27 al

redshift redshift

quasar 3C273 in Virgo
v = 44 Mm/s at 2 Gal

quasar APM 08279-5255
redshift in Lynx
v = 276 Mm/s at 12 Gal

Motion Mountain – The Adventure of Physics
Leo

F I G U R E 15 Top: the Doppler effect for light from two quasars. Below: the – magniﬁed, false colour –
Doppler effect for the almost black colour of the night sky – the cosmic background radiation – due to
the Earth travelling through space. In the latter case, the Doppler shift implies a tiny change of the
effective temperature of the night sky (© Maurice Gavin, NASA).

Physicists have tested the possibility of massive light in quite some detail. Observa-
Ref. 32, Ref. 18 tions now put any possible mass of light particles, or photons, at less than 1.3 ⋅ 10−52 kg
from terrestrial experiments, and at less than 4 ⋅ 10−62 kg from astrophysical arguments
(which are slightly less compelling). In other words, light is not heavy, light is light.
But what happens when light hits a moving mirror? The situation is akin to that of
a light source moving with respect to the receiver: the receiver will observe a different
colour from that observed by the sender. This frequency shift is called the Doppler effect.
motion of light 33

If this red text appears blue,
you are too fast.

Motion Mountain – The Adventure of Physics
F I G U R E 16 The Doppler sonar system of dolphins, the Doppler effect system in a sliding door opener,
the Doppler effect as a speed warning and Doppler sonography to detect blood ﬂow (coloured) in the
umbilical cord of a foetus (© Wikimedia, Hörmann AG, Medison).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Christian Doppler* was the first to study the frequency shift in the case of sound waves.
We all know the change in whistle tone between approaching and departing trains: that
is the Doppler effect for sound. We can determine the speed of the train in this way. Bats,
dolphins and wales use the acoustical Doppler effect to measure the speed of prey, and
the effect is used to measure blood flow and heart beat of unborn babies in ultrasound
Vol. I, page 313 systems (despite being extremely loud for the babies), as shown in Figure 16.
Doppler was also the first person to extend the concept of frequency shift to the case
Vol. III, page 106 of light waves. As we will see, light is (also) a wave, and its colour is determined by its
frequency, or equivalently, by its wavelength 𝜆. Like the tone change for moving trains,
Doppler realized that a moving light source produces a colour at the receiver that differs
from the colour at the source. Simple geometry, and the conservation of the number of
Challenge 20 e maxima and minima, leads to the result

𝜆r 1 𝑣 𝑣
= (1 − cos 𝜃r ) = 𝛾 (1 − cos 𝜃r ) . (8)
𝜆s √ 𝑐 𝑐
1 − 𝑣2 /𝑐2

* Christian Andreas Doppler (b. 1803 Salzburg, d. 1853 Venezia), important physicist. Doppler studied the
effect named after him for sound and light. Already in 1842 he predicted (correctly) that one day we would
be able to use the effect to measure the motion of distant stars by looking at their colours. For his discovery
Ref. 33 of the effect – and despite its experimental confirmation in 1845 and 1846 – Doppler was expelled from the
Imperial Academy of Science in 1852. His health degraded and he died shortly afterwards.
34 1 maximum speed, observers at rest and

sender
at rest

receiver

moving
red-shifted signal sender blue-shifted signal

𝑣 receiver

Motion Mountain – The Adventure of Physics
𝑦

𝑦
𝜃r
light 𝑥
signal receiver

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
any
sender 𝑧
𝜃s
𝑥
𝑣

𝑧

F I G U R E 17 The set-up for the observation of the Doppler effect in one and three dimensions: waves
emitted by an approaching source arrive with higher frequency and shorter wavelength, in contrast to
waves emitted by a departing source (wave graph © Pbroks13).

The variables 𝑣 and 𝜃r in this expression are defined in Figure 17. Light from an approach-
ing source is thus blue-shifted, whereas light from a departing source is red-shifted.
The first observation of the Doppler effect for light, also called the colour shift, was
made by Johannes Stark* in 1905, who studied the light emitted by moving atoms. All

* Johannes Stark (b. 1874 Schickenhof, d. 1957 Eppenstatt), discovered in 1905 the optical Doppler effect
in channel rays, and in 1913 the splitting of spectral lines in electrical fields, nowadays called the Stark
effect. For these two discoveries he received the 1919 Nobel Prize in Physics. He left his professorship in
1922 and later turned into a full-blown National Socialist. A member of the National Socialist party from
1930 onwards, he became known for aggressively criticizing other people’s statements about nature purely
motion of light 35

subsequent experiments confirmed the calculated colour shift within measurement er-
Ref. 34 rors; the latest checks have found agreement to within two parts per million.
In contrast to sound waves, a colour change is also found when the motion is trans-
verse to the light signal. Thus, a yellow rod in rapid motion across the field of view will
have a blue leading edge and a red trailing edge prior to the closest approach to the ob-
server. The colours result from a combination of the longitudinal (first-order) Doppler
shift and the transverse (second-order) Doppler shift. At a particular angle 𝜃unshifted the
colour will stay the same. (How does the wavelength change in the purely transverse
Challenge 21 s case? What is the expression for 𝜃unshifted in terms of the speed 𝑣?)
The colour or frequency shift explored by Doppler is used in many applications. Al-
most all solid bodies are mirrors for radio waves. Many buildings have doors that open
automatically when one approaches. A little sensor above the door detects the approach-
ing person. It usually does this by measuring the Doppler effect of radio waves emitted by
the sensor and reflected by the approaching person. (We will see later that radio waves
Vol. III, page 108 and light are manifestations of the same phenomenon.) So the doors open whenever
something moves towards them. Police radar also uses the Doppler effect, this time to

Motion Mountain – The Adventure of Physics
measure the speed of cars.*
As predicted by Doppler himself, the Doppler effect is regularly used to measure the
speed of distant stars, as shown in Figure 15. In these cases, the Doppler shift is often
characterized by the red-shift number 𝑧, defined with the help of wavelength 𝜆 or fre-
quency 𝑓 by
Δ𝜆 𝑓S 𝑐+𝑣
𝑧= = −1 = √ −1 . (9)
𝜆 𝑓R 𝑐−𝑣

Can you imagine how the number 𝑧 is determined? Typical values for 𝑧 for light sources

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 23 s
in the sky range from −0.1 to 3.5, but higher values, up to more than 10, have also been
Challenge 24 s found. Can you determine the corresponding speeds? How can they be so high?
Because of the rotation of the Sun and the Doppler effect, one edge of the Sun is blue-
Ref. 35 shifted, and the other is red-shifted. It is possible to determine the rotation speed of the
Sun in this way. The time of a rotation lies between 27 and 33 days, depending of the
latitude. The Doppler effect also showed that the surface of the Sun oscillates with periods
of the order of 5 minutes.
Even the rotation of our galaxy was discovered using the Doppler effect of its stars.
Astronomers thus discovered that the Sun takes about 220 million years for a rotation
around the centre of the Milky Way.
What happens if one really tries to play tennis with light, using a racket that moves
at really high, thus relativistic speed? Such passionate tennis players actually exist; the
fastest rackets built so far had a speed over 80 % per cent of the speed of light. They
Ref. 36 were produced in 2013 by shooting extremely powerful and short laser pulses, with a
power of 0.6 ZW and a duration of 50 fs, onto a 10 nm thin diamond-like carbon foil.
Such pulses eject a flat and rapid electron cloud into the vacuum; for a short time, this
cloud acted as a relativistic mirror. When a second laser beam was reflected from this

for ideological reasons; he became rightly despised by the academic community all over the world, already
during his lifetime.
Challenge 22 s * At what speed does a red traffic light appear green?
36 1 maximum speed, observers at rest and

relativistic racket, the light speed remained unchanged, but its frequency was increased
by a factor of about 14, changing the beam colour from the near infrared to the extreme
ultraviolet. This relativistic electron mirror had a reflectivity far less than 1 %, though, its
lifetime was only a few picoseconds, and its size only about 2 μm; therefore calling it a
racket is a slight exaggeration.
In summary, whenever we try to change the vacuum speed of light, we only manage
to change its colour. That is the Doppler effect. In other terms, attempts to accelerate or
decelerate light only lead to colour change. And a colour change does not change the
Page 22 speed of light at all, as shown above.
Modern Doppler measurements are extremely precise. Our Sun moves with up to
9 cm/s with respect to the Earth, due to the planets that orbit it. Nowadays, the Doppler
shift due to this speed value is measured routinely, using a special laser type called a fre-
quency comb. This device allows to measure light frequencies within fractions of 1 Hz.
Frequency combs allow the detection of even smaller speed values through the induced
Ref. 37 Doppler shifts. This method is used on a regular basis to detect exoplanets orbiting dis-
tant stars.

Motion Mountain – The Adventure of Physics
The connection between colour change and light acceleration attempts leads to a
Vol. I, page 201 puzzle: we know from classical physics that when light passes a large mass, such as a
Challenge 25 s star, it is deflected. Does this deflection lead to a Doppler shift?

The difference bet ween light and sound
The Doppler effect for light is much more fundamental than the Doppler effect for sound.
Even if the speed of light were not yet known to be invariant, the Doppler effect alone
would prove that time is different for observers moving relative to each other. Why?
Time is what we read from our watch. In order to determine whether another watch

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
is synchronized with our own one, we look at both watches. In short, we need to use light
Ref. 38 signals to synchronize clocks. Now, any change in the colour of light moving from one
observer to another necessarily implies that their watches run differently, and thus that
time is different for the two of them. To see this, note that also a light source is a clock
– ‘ticking’ very rapidly. So if two observers see different colours from the same source,
they measure different numbers of oscillations for the same clock. In other words, time
Page 29 is different for observers moving against each other. Indeed, equation (6) for the Dop-
pler effect implies the whole of special relativity, including the invariance of the speed
of light. (Can you confirm that the connection between observer-dependent frequencies
Challenge 26 s and observer-dependent time breaks down in the case of the Doppler effect for sound?)
Why does the behaviour of light imply special relativity, while that of sound in air does
not? The answer is that light is a limit for the motion of energy. Experience shows that
there are supersonic aeroplanes, but there are no superluminal rockets. In other words,
the limit 𝑣 ⩽ 𝑐 is valid only if 𝑐 is the speed of light, not if 𝑐 is the speed of sound in air.
However, there is at least one system in nature where the speed of sound is indeed
a limit speed for energy: the speed of sound is the limit speed for the motion of dislo-
Vol. V, page 298 cations in crystalline solids. (We discuss this motion in detail later on.) As a result, the
theory of special relativity is also valid for dislocations, provided that the speed of light is
replaced everywhere by the speed of sound! Indeed, dislocations obey the Lorentz trans-
Ref. 39 formations, show length contraction, and obey the famous energy formula 𝐸 = 𝑐2 𝛾𝑚. In
motion of light 37

F I G U R E 18 Lucky Luke.

Motion Mountain – The Adventure of Physics
all these effects the speed of sound 𝑐 plays the same role for dislocations as the speed of
light plays for general physical systems.
Given special relativity is based on the statement that nothing can move faster than
light, we need to check this statement carefully.

C an one sho ot faster than one ’ s shad ow?
For Lucky Luke to achieve the feat shown in Figure 18, his bullet has to move faster than
Challenge 27 e the speed of light. (What about his hand?) In order to emulate Lucky Luke, we could

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
take the largest practical amount of energy available, taking it directly from an electrical
power station, and accelerate the lightest ‘bullets’ that can be handled, namely electrons.
This experiment is carried out daily in particle accelerators; an example was the Large
Electron Positron ring, the LEP, of 27 km circumference, located partly in France and
partly in Switzerland, near Geneva. There, 40 MW of electrical power (the same amount
used by a small city) were used to accelerate electrons and positrons to record energies
Ref. 40 of over 16 nJ (104.5 GeV) each, and their speed was measured. The result is shown in
Figure 19: even with these impressive means it is impossible to make electrons move
more rapidly than light. (Can you imagine a way to measure kinetic energy and speed
Challenge 28 e separately?)
The speed–energy relation of Figure 19 is a consequence of the maximum speed, and
Page 69 its precise details are deduced below. These and many similar observations thus show
that there is a limit to the velocity of objects and radiation. Bodies and radiation cannot
move at velocities higher that the speed of light.* The accuracy of Galilean mechanics was

* There are still people who refuse to accept this result, as well as the ensuing theory of relativity. Every
reader should enjoy the experience, at least once in his life, of conversing with one of these men. (Strangely,
no woman has yet been reported as belonging to this group of people. Despite this conspicuous effect,
Ref. 41 studying the influences of sex on physics is almost a complete waste of time.)
Ref. 42 Crackpots can be found, for example, via the internet, in the sci.physics.relativity newsgroup. See also
the www.crank.net website. Crackpots are sometimes interesting, mainly because they demonstrate the
importance of precision in language and in reasoning, which they all, without exception, neglect.
38 1 maximum speed, observers at rest and

𝑣 𝑝 = 𝑚𝑣

𝑐
𝑚𝑣
𝑝=
√1−𝑣2 /𝑐2

𝑝

𝑣2 𝑇 = 12 𝑚𝑣2

𝑐2 F I G U R E 19 Experimental values (black
2 1 dots) for the electron velocity 𝑣 as
𝑇 = 𝑐 𝑚( − 1) function of their momentum 𝑝 and as
√1−𝑣2 /𝑐2
function of their kinetic energy 𝑇. The
predictions of Galilean physics (blue)

Motion Mountain – The Adventure of Physics
𝑇 and the predictions of special relativity
(red) are also shown.

taken for granted for more than two centuries, so that nobody ever thought of checking
Ref. 43 it; but when this was finally done, as in Figure 19, it was found to be wrong.
The same result appears when we consider momentum instead of energy. Particle ac-
celerators show that momentum is not proportional to speed: at high speeds, doubling

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the momentum does not lead to a doubling of speed. In short, experiments show that
neither increasing the energy nor increasing the momentum of even the lightest particles
allows reaching the speed of light.
The people most unhappy with this speed limit are computer engineers: if the speed
limit were higher, it would be possible to build faster microprocessors and thus faster
computers; this would allow, for example, more rapid progress towards the construction
of computers that understand and use language.
The existence of a limit speed runs counter to Galilean mechanics. In fact, it means
that for velocities near that of light, say about 15 000 km/s or more, the expression 𝑚𝑣2 /2
is not equal to the kinetic energy 𝑇 of the particle. In fact, such high speeds are rather
common: many families have an example in their home. Just calculate the speed of elec-
trons inside a cathode ray tube inside an old colour television, given that the transformer
Challenge 29 s inside produces 30 kV.
The speed of light is a limit speed for objects. This property is easily seen to be a con-
sequence of its invariance. Bodies that can be at rest in one frame of reference obviously
move more slowly than light in that frame. Now, if something moves more slowly than
something else for one observer, it does so for all other observers as well. (Trying to
Challenge 30 d imagine a world in which this would not be so is interesting: bizarre phenomena would
occur, such as things interpenetrating each other.) Since the speed of light is the same
for all observers, no object can move faster than light, for every observer.
We conclude that
motion of light 39

time 𝑡

first second
observer observer third
(e.g. Earth) (e.g. train) observer
(e.g. stone)
𝑘se 𝑇

𝑘te 𝑇

𝑇

𝑂

Motion Mountain – The Adventure of Physics
space 𝑥
F I G U R E 20 How to deduce the composition of
velocities.

⊳ The maximum speed is the speed of massless entities.

Electromagnetic waves, including light, and gravitational waves are the only known en-
tities that travel at the maximum speed. Though the speed of neutrinos cannot be distin-
guished experimentally from the maximum speed, recent experiments showed that they

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 44 do have a tiny mass.
Conversely, if a phenomenon exists whose speed is the limit speed for one observer,
Challenge 31 e then this limit speed must necessarily be the same for all observers. Is the connection
Challenge 32 r between limit property and observer invariance generally valid in nature?

The composition of velo cities
If the speed of light is a limit, no attempt to exceed it can succeed. This implies that when
two velocities are composed, as when one throws a stone while running or travelling, the
values cannot simply be added. Imagine a train that is travelling at velocity 𝑣te relative to
the Earth, and a passenger throws a stone inside it, in the same direction, with velocity 𝑣st
relative to the train. It is usually assumed as evident that the velocity of the stone relative
to the Earth is given by 𝑣se = 𝑣st + 𝑣te . In fact, both reasoning and measurement show a
different result.
Page 26 The existence of a maximum speed, together with Figure 20, implies that the 𝑘-factors
must satisfy 𝑘se = 𝑘st 𝑘te .* Then we only need to insert the relation (6) between each 𝑘-

* By taking the (natural) logarithm of this equation, one can define a quantity, the rapidity, that quantifies
the speed and is additive.
40 1 maximum speed, observers at rest and

Challenge 33 e factor and the respective speed to get

𝑣st + 𝑣te
𝑣se = . (10)
1 + 𝑣st 𝑣te /𝑐2

Challenge 34 e This is called the velocity composition formula. The result is never larger than 𝑐 and is
always smaller than the naive sum of the velocities.* Expression (10) has been confirmed
Page 68 by each of the millions of cases for which it has been checked. You may check that it
Ref. 18 simplifies with high precision to the naive sum for everyday life speed values.

Observers and the principle of special relativit y
Special relativity is built on a simple principle:

⊳ The maximum local speed of energy transport is the same for all observers.

Motion Mountain – The Adventure of Physics
Ref. 46 Or, as Hendrik Lorentz** liked to say, the equivalent:

⊳ The speed 𝑣 of a physical system is bound by

𝑣⩽𝑐 (11)

for all observers, where 𝑐 is the speed of light.

This invariance of the speed of light was known since the 1850s, because the expression

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Vol. III, page 106 𝑐 = 1/√𝜀0 𝜇0 , known to people in the field of electricity, does not depend on the speed of
the observer or of the light source, nor on their orientation or position. The invariance
of 𝑐, including its speed independence, was found by optical experiments that used mov-
ing prisms, moving water, moving bodies with double refraction, interfering light beams
travelling in different directions, interfering circulating light beams or light from moving
stars. The invariance was also found by electromagnetic experiments that used moving
insulators in electric and magnetic fields.*** All experiments show without exception
that the speed of light in vacuum is invariant, whether they were performed before or
Ref. 45 * One can also deduce the Lorentz transformation directly from this expression.
** Hendrik Antoon Lorentz (b. 1853 Arnhem, d. 1928 Haarlem) was, together with Boltzmann and Kelvin,
one of the most important physicists of his time. He deduced the so-called Lorentz transformation and
the Lorentz contraction from Maxwell’s equations for the electromagnetic field. He was the first to un-
derstand, long before quantum theory confirmed the idea, that Maxwell’s equations for the vacuum also
describe matter and all its properties, as long as moving charged point particles – the electrons – are in-
cluded. He showed this in particular for the dispersion of light, for the Zeeman effect, for the Hall effect
and for the Faraday effect. He also gave the correct description of the Lorentz force. In 1902, he received the
physics Nobel Prize together with Pieter Zeeman. Outside physics, he was active in the internationalization
of scientific collaborations. He was also instrumental in the creation of the largest human-made structures
on Earth: the polders of the Zuiderzee.
*** All these experiments, which Einstein did not bother to cite in his 1905 paper, were performed by the
Ref. 47 complete who’s who of 19th century physics, such as Wilhelm Röntgen, Alexander Eichenwald, François
Ref. 48 Arago, Augustin Fresnel, Hippolyte Fizeau, Martin Hoek, Harold Wilson, Albert Michelson, (the first
motion of light 41

half-

Motion Mountain – The Adventure of Physics
transparent
mirror mirror mirror

light intereference
source detector

F I G U R E 21 Testing the invariance of the speed of light on the motion of the observer: the
reconstructed set-up of the ﬁrst experiment by Albert Michelson in Potsdam, performed in 1881, and a
modern high-precision, laser-based set-up that keeps the mirror distances constant to less than a

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
proton radius and constantly rotates the whole experiment around a vertical axis (© Astrophysikalisches
Institut Potsdam, Stephan Schiller).

after special relativity was formulated. The experiment performed by Albert Michelson,
and the high-precision version to date, by Stephan Schiller and his team, are illustrated
in Figure 21. All such experiments found no change of the speed of light with the motion
Ref. 49 of the Earth within measurement precision, which is around 2 parts in 10−17 at present.
You can also confirm the invariance of the speed of light yourself at home; the way to do
Vol. III, page 53 this is explained in the section on electrodynamics.
The existence of an invariant limit speed has several important consequences. To ex-
plore them, let us keep the remaining of Galilean physics intact.* The limit property and
the invariance of the speed of light imply:
US-American to receive, in 1907, the Nobel Prize in Physics) Edward Morley, Oliver Lodge, John Strutt
Rayleigh, Dewitt Brace, Georges Sagnac and Willem de Sitter among others.
Vol. I, page 156 * This point is essential. For example, Galilean physics states that only relative motion is observable. Galilean
physics also excludes various mathematically possible ways to realize an invariant light speed that would
contradict everyday life.
Einstein’s original 1905 paper starts from two principles: the invariance of the speed of light and the
equivalence, or relativity, of all inertial observers. The latter principle had already been stated in 1632 by
Galileo; only the invariance of the speed of light was new. Despite this fact, the new theory was named – by
Ref. 23 Poincaré – after the old principle, instead of calling it ‘invariance theory’, as Einstein would have preferred.
42 1 maximum speed, observers at rest and

⊳ In a closed free-floating (‘inertial’) room, there is no way to tell the speed of the
room. Or, as Galileo writes in his Dialogo: il moto [ ...] niente opera ed è come s’ e’
non fusse. ‘Motion [ ...] has no effect and behaves as if it did not exist’. Sometimes
this statement is shortened to: motion is like nothing.
⊳ There is no notion of absolute rest: rest is an observer-dependent, or relative concept.*
⊳ Length and space depend on the observer; length and space are not absolute, but
relative.
⊳ Time depends on the observer; time is not absolute, but relative.
⊳ Mass and energy are equivalent.
We can draw more specific conclusions when two additional conditions are realised.
First, we study situations where gravitation can be neglected. (If this not the case, we
need general relativity to describe the system.) Secondly, we also assume that the data
about the bodies under study – their speed, their position, etc. – can be gathered without
disturbing them. (If this not the case, we need quantum theory to describe the system.)
How exactly differ the time intervals and lengths measured by two observers? To an-

Motion Mountain – The Adventure of Physics
swer, we only need a pencil and a ruler. To start, we explore situations where no inter-
action plays a role. In other words, we start with relativistic kinematics: all bodies move
without disturbance.
If an undisturbed body is observed to travel along a straight line with a constant ve-
locity (or to stay at rest), one calls the observer inertial, and the coordinates used by the
observer an inertial frame of reference. Every inertial observer is itself in undisturbed
motion. Examples of inertial observers (or frames) thus include – in two dimensions –
those moving on a frictionless ice surface or on the floor inside a smoothly running train
or ship. For a full example – in all three spatial dimensions – we can take a cosmonaut
travelling in a space-ship as long as the engine is switched off or a person falling in va-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
cuum. Inertial observers in three dimensions can also be called free-floating observers,
where ‘free’ stands again for ‘undisturbed’. Inertial observers are thus much rarer than
Challenge 36 e non-inertial observers. Can you confirm this? Nevertheless, inertial observers are the
most simple ones, and they form a special set:
⊳ Any two inertial observers move with constant velocity relative to each other (as long
as gravity and interactions play no role, as assumed above).
⊳ All inertial observers are equivalent: they describe the world with the same equations.
This statement, due to Galileo, was called the principle of relativity by Henri Poincaré.
To see how exactly the measured length and space intervals change from one inertial
observer to the other, we assume a Roman one, using space and time coordinates 𝑥, 𝑦,
𝑧 and 𝑡, and a Greek one, using coordinates 𝜉, 𝜐, 𝜁 and 𝜏,** that move with constant
velocity 𝑣 relative to each other, as shown in Figure 22. The invariance of the speed of
light in any direction for any two observers means that the coordinate differences found
Challenge 37 e by two observers are related by

(𝑐d𝑡)2 − (d𝑥)2 − (d𝑦)2 − (d𝑧)2 = (𝑐d𝜏)2 − (d𝜉)2 − (d𝜐)2 − (d𝜁)2 . (12)
Challenge 35 s * Can you give the precise argument leading to this deduction?
** They are read as ‘xi’, ‘upsilon’, ‘zeta’ and ‘tau’. The names, correspondences and pronunciations of all
Greek letters are explained in Appendix A in the first volume.
motion of light 43

𝑣 = const
observer (greek)

light 𝑐

observer (roman) F I G U R E 22 Two inertial
𝑣=0 observers and a beam of light.
Both measure the same speed
of light 𝑐.

Galilean physics special relativity

𝑡 𝜏

Motion Mountain – The Adventure of Physics
L
no consistent L
graph possible

F I G U R E 23 The
𝜉 space-time diagram
for light seen from
O, Ω O, Ω 𝑥 two inertial observers,
using coordinates
(𝑡, 𝑥) and (𝜏, 𝜉).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
We now chose the axes in such a way that the velocity points in the 𝑥 and 𝜉-direction.
Then we have
(𝑐d𝑡)2 − (d𝑥)2 = (𝑐d𝜏)2 − (d𝜉)2 . (13)

Assume that a flash lamp is at rest at the origin for the Greek observer, thus with 𝜉 =
0, and produces two flashes separated by a time interval d𝜏. For the Roman observer,
the flash lamp moves with speed 𝑣, so that d𝑥 = 𝑣d𝑡. Inserting this into the previous
Challenge 38 e expression, we deduce
d𝜏
d𝑡 = = 𝛾d𝜏 . (14)
√1 − 𝑣2 /𝑐2

This expression thus relates clock intervals measured by one observer to the clock inter-
vals measured by another. At relative speeds 𝑣 that are small compared to the velocity
of light 𝑐, such as occur in everyday life, the stretch factor, relativistic correction, Lorentz
factor or relativistic contraction 𝛾 is equal to 1 for all practical purposes. In these cases,
the time intervals found by the two observers are essentially equal: time is then the same
for all. However, for velocities near that of light the value of 𝛾 increases. The largest value
humans have ever achieved is about 2 ⋅ 105 ; the largest observed value in nature is about
Challenge 39 s 1012 . Can you imagine where they occur?
44 1 maximum speed, observers at rest and

For a relativistic correction 𝛾 larger than 1 – thus in principle for any relative speed
different from zero – the time measurements of the two observers give different values.
Because time differs from one observer to another, moving observers observe time dila-
tion.
But that is not all. Once we know how clocks behave, we can easily deduce how co-
ordinates change. Figures 22 and 23 show that the 𝑥 coordinate of an event L is the sum
of two intervals: the 𝜉 coordinate plus any distance between the two origins. In other
words, we have
𝜉 = 𝛾(𝑥 − 𝑣𝑡) . (15)

Using the invariance of the space-time interval, we get

𝜏 = 𝛾(𝑡 − 𝑥𝑣/𝑐2 ) . (16)

Henri Poincaré called these two relations the Lorentz transformations of space and time
after their discoverer, the Dutch physicist Hendrik Antoon Lorentz.* In one of the most

Motion Mountain – The Adventure of Physics
Ref. 50 beautiful discoveries of physics, in 1892 and 1904, Lorentz deduced these relations from
Vol. III, page 76 the equations of electrodynamics, where they had been lying, waiting to be discovered,
since 1865.** In that year James Clerk Maxwell had published the equations that describe
everything electric, magnetic and optical. However, it was Einstein who first understood
that 𝑡 and 𝜏, as well as 𝑥 and 𝜉, are equally valid descriptions of space and time.
The Lorentz transformation describes the change of viewpoint from one inertial frame
to a second, moving one. This change of viewpoint is called a (Lorentz) boost. The for-
mulae (15) and (16) for the boost are central to the theories of relativity, both special and
general. In fact, the mathematics of special relativity will not get more difficult than that:

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
if you know what a square root is, you can study special relativity in all its beauty.
The Lorentz transformations (15) and (16) contain many curious results. Again they
Challenge 40 e show that time depends on the observer. They also show that length depends on the
Page 52 observer: in fact, moving observers observe length contraction. Space and time are thus
indeed relative.
The Lorentz transformations (15) and (16) are also strange in another respect. When
two observers look at each other, each of them claims to measure shorter intervals than
Challenge 41 s the other. In other words, special relativity shows that the grass on the other side of the
fence is always shorter – if we ride along beside the fence on a bicycle and if the grass is
Page 52 inclined. We explore this bizarre result in more detail shortly.
Many alternative formulae for Lorentz boosts have been explored, such as expressions
in which the relative acceleration of the two observers is included, as well as the relative
Ref. 51 velocity. However, all alternatives had to be discarded after comparing their predictions
with experimental results. Before we have a look at such experiments, we continue with
a few logical deductions from the boost relations.

* For information about Hendrik Antoon Lorentz, see page 40.
** The same discovery had been published first in 1887 by Woldemar Voigt (b. 1850 Leipzig,
d. 1919 Göttingen); Voigt – pronounced ‘Fohgt’ – was also the discoverer of the Voigt effect and the Voigt
tensor. Later, in 1889, George Fitzgerald (b. 1851 Dublin, d. 1901 Dublin) also found the result.
motion of light 45

What is space-time?

“
Von Stund’ an sollen Raum für sich und Zeit
für sich völlig zu Schatten herabsinken und nur
noch eine Art Union der beiden soll

”
Selbstständigkeit bewahren.*
Hermann Minkowski.

The Lorentz transformations tell us something important: space and time are two aspects
of the same basic entity. They mix in different ways for different observers. The mixing
is commonly expressed by stating that time is the fourth dimension. This makes sense
because the common basic entity – called space-time – can be defined as the set of all
events, events being described by four coordinates in time and space, and because the
Challenge 42 s set of all events has the properties of a manifold.** (Can you confirm this?) Complete
space-time is observer-invariant and absolute; space-time remains unchanged by boosts.
Only its split into time and space depends on the viewpoint.
In other words, the existence of a maximum speed in nature forces us to introduce

Motion Mountain – The Adventure of Physics
the invariant space-time manifold, made of all possible events, for the description of
nature. In the absence of gravitation, i.e., in the theory of special relativity, the space-
time manifold is characterized by a simple property: the space-time interval d𝑖 between
Ref. 52 two events, defined as

𝑣2
d𝑖2 = 𝑐2 d𝑡2 − d𝑥2 − d𝑦2 − d𝑧2 = 𝑐2 d𝑡2 (1 − ) , (17)
𝑐2

is independent of the (inertial) observer: it is an invariant. Space-time is also called
Minkowski space-time, after Hermann Minkowski,*** the teacher of Albert Einstein; he

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
was the first, in 1904, to define the concept of space-time and to understand its useful-
ness and importance. We will discover later that when gravitation is present, the whole of
space-time bends; such bent space-times, called Riemannian space-times, will be essential
in general relativity.
The space-time interval d𝑖 of equation (17) has a simple physical meaning. It is the
time measured by an observer moving from event (𝑡, 𝑥) to event (𝑡 + d𝑡, 𝑥 + d𝑥), the so-
called proper time, multiplied by 𝑐. If we neglect the factor 𝑐, we can also call the interval
the wristwatch time.
In short, we can say that we live in space-time. Space-time exists independently of
all things; it is a container, a background for everything that happens. And even though
coordinate systems differ from observer to observer, the underlying entity, space-time, is
the same and unique, even though space and time by themselves are not. (All this applies
also in the presence of gravitation, in general relativity.)

* ‘Henceforth space by itself and time by itself shall completely fade into shadows and only a kind of union
of the two shall preserve autonomy.’ This famous statement was the starting sentence of Minkowski’s 1908
talk at the meeting of the Gesellschaft für Naturforscher und Ärzte.
Vol. V, page 365 ** The term ‘manifold’ is defined in all mathematical details later in our walk.
*** Hermann Minkowski (b. 1864 Aleksotas, d. 1909 Göttingen) was mainly a mathematician. He had de-
veloped, independently, similar ideas to Einstein, but the latter was faster. Minkowski then developed the
concept of space-time. Unfortunately, Minkowski died suddenly at the age of 44.
46 1 maximum speed, observers at rest and

How does Minkowski space-time differ from Galilean space-time, the combination of
everyday space and time? Both space-times are manifolds, i.e., continuum sets of points,
both have one temporal and three spatial dimensions, and both manifolds have the topo-
Challenge 43 s logy of the punctured sphere. (Can you confirm this?) Both manifolds are flat, i.e., free of
curvature. In both cases, space is what is measured with a metre rule or with a light ray,
and time is what is read from a clock. In both cases, space-time is fundamental, unique
and absolute; it is and remains the background and the container of things and events.
The central difference, in fact the only one, is that Minkowski space-time, in contrast
to the Galilean case, mixes space and time. The mixing is different for observers with
different speeds, as shown in Figure 23. The mixing is the reason that time and space are
observer-dependent, or relative, concepts.
Mathematically, time is a fourth dimension; it expands space to space-time. Calling
time the fourth dimension is thus only a statement on how relativity calculates – we will
do that below – and has no deeper meaning.
The maximum speed in nature thus forces us to describe motion with space-time.
That is interesting, because in space-time, speaking in tabloid terms, motion does not

Motion Mountain – The Adventure of Physics
exist. Motion exists only in space. In space-time, nothing moves. For each point particle,
space-time contains a world-line. (See Figure 24.) In other words, instead of asking why
motion exists, we can equivalently ask why space-time is criss-crossed by world-lines.
But at this point of our adventure we are still far from answering either question. What
we can do is to explore how motion takes place.

C an we travel to the past? – Time and causalit y
We know that time is different for different observers. Does time nevertheless order
events in sequences? The answer given by relativity is a clear ‘yes and no’. Certain sets of

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
events are not naturally ordered by time; others sets are. This is best seen in a space-time
diagram, such as Figure 24.
Clearly, two events can be placed in a time sequence only if one event is or could be
the cause of the other. But this connection can only apply if the first event could send en-
ergy, e.g. through a signal, to the second. In other words, a temporal sequence between
two events implies that the signal speed connecting the two events must not be larger
than the speed of light. Figure 24 shows that event E at the origin of the coordinate sys-
tem can only be influenced by events in quadrant IV (the past light cone, when all space
dimensions are included), and can itself influence only events in quadrant II, the future
light cone. Events in quadrants I and III neither influence nor are influenced by event E:
signal speed above that of light would be necessary to achieve that. Thus the full light
cone defines the boundary between events that can be ordered with respect to event E
– namely those inside the cone – and those that cannot – those outside the cone, which
happen elsewhere for all observers. (Some authors sloppily call all the events happening
elsewhere the present.)
The past light cone gives the complete set of events that can influence what happens at
E, the coordinate origin. One says that E is causally connected to events in the past light
cone. Note that causal connection is an invariant concept: all observers agree on whether
Challenge 44 s or not it applies to two given events. Can you confirm this?
In short, time orders events only partially. In particular, for two events that are not
motion of light 47

time t
time

ne
II future T

lig
lig
T

co
ht
ht future

ht
co
pa

lig

lig
ne
th
III I
elsewhere E elsewhere space E elsewhere y

IV x
past past

F I G U R E 24 A space-time diagram for a moving object T seen from an inertial observer O in the case of

Motion Mountain – The Adventure of Physics
one and two spatial dimensions; the slope of the world-line at a point is the speed at that point, and
thus is never steeper than that of light.

Challenge 45 e causally connected, their temporal order (or their simultaneity) depends on the observer!
A vector inside the light cone is called time-like; one on the light cone is called light-
like or null; and one outside the cone is called space-like. For example, the world-line of
an observer, i.e., the set of all events that make up its past and future history, consists of
time-like events only.
Special relativity thus teaches us that causality and time can be defined only because

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
light cones exist. If transport of energy at speeds faster than that of light did exist, time
could not be defined. Causality, i.e., the possibility of (partially) ordering events for all
observers, is due to the existence of a maximal speed.
If the speed of light could be surpassed, we could always win the lottery. Can you see
Challenge 46 e why? In other words, if the speed of light could be surpassed in some way, the future
Challenge 47 s could influence the past. Can you confirm this? In such situations, one would observe
acausal effects. However, there is an everyday phenomenon which tells that the speed of
light is indeed maximal: our memory. If the future could influence the past, we would
also be able to remember the future. To put it in another way, if the future could influ-
ence the past, the second principle of thermodynamics would not be valid.* No known
data from everyday life or from experiments provide any evidence that the future can
influence the past. In other words,

⊳ Time travel to the past is impossible.

How the situation changes in quantum theory will be revealed later on. Interestingly,

* Another related result is slowly becoming common knowledge. Even if space-time had a non-trivial shape,
such as a cylindrical topology with closed time-like curves, one still would not be able to travel into the past,
in contrast to what many science fiction novels suggest. The impossibility of this type of time travel is made
Ref. 53 clear by Steven Blau in a recent pedagogical paper.
48 1 maximum speed, observers at rest and

time travel to the future is possible, as we will see shortly.

Curiosities ab ou t special relativit y
Special relativity is full of curious effects. Let us start with a puzzle that helps to sharpen
our thinking. Seen by an observer on an island, two lightning strokes hit simultaneously:
one hits the island, and another, many kilometres away, the open sea. A second observer
is a pilot in a relativistic aeroplane and happens to be just above the island when the
Challenge 48 e lightning hits the island. Which lightning hits first for the pilot?
For the pilot, the distant lightning, hitting the sea, hits first. But this is a trick question:
despite being the one that hits first, the distant lightning is observed by the pilot to hit
after the one on the island, because light from the distant hit needs time to reach him.
However, the pilot can compensate for the propagation time and can deduce that the
Challenge 49 e distant lightning hit first.
When you wave your hand in front of a mirror, your image waves with the same fre-
Challenge 50 e quency. What happens if the mirror moves away with relativistic speed?

Motion Mountain – The Adventure of Physics
We will discover in the section on quantum theory that the yellow colour of gold is a
Vol. IV, page 195 relativistic effect; also the liquid state of mercury at room temperature is a consequence
of relativity. Both effects are due to the high speed of the outer electrons of these atoms.
Let us explore a few additional consequences of special relativity.

Faster than light : how far can we travel?
How far away from Earth can we travel, given that the trip should not last more than
a lifetime, say 80 years, and given that we are allowed to use a rocket whose speed can
approach the speed of light as closely as desired? Given the time 𝑡 we are prepared to

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
spend in a rocket, given the speed 𝑣 of the rocket, and assuming optimistically that it
can accelerate and decelerate in a negligible amount of time, the distance 𝑑 we can move
Challenge 51 e away is given by
𝑣𝑡
𝑑= . (18)
√1 − 𝑣2 /𝑐2

The distance 𝑑 is larger than 𝑐𝑡 already for 𝑣 > 0.72𝑐, and, if 𝑣 is chosen large enough,
it increases beyond all bounds! In other words, light speed does not limit the distance
we can travel in a lifetime or in any other time interval. We could, in principle, roam the
Page 51 entire universe in less than a second. (The fuel issue is discussed below.)
For rocket trips it makes sense to introduce the concept of proper velocity 𝑤, defined
as
𝑑 𝑣
𝑤= = =𝛾𝑣. (19)
𝑡 √ 2 2
1 − 𝑣 /𝑐

As we have just seen, proper velocity is not limited by the speed of light; in fact the proper
velocity of light itself is infinite.*

* Using proper velocity, the relation given in equation (10) for the composition of two velocities wa = 𝛾a va
motion of light 49

first
twin

trip of
Earth second twin
time time
comparison
and
first change of
twin rocket

Motion Mountain – The Adventure of Physics
F I G U R E 25 The twin paradox.

Synchronization and time travel – can a mother stay younger
than her own daughter?
The maximum speed in nature implies that time is different for different observers mov-
ing relative to each other. So we have to be careful about how we synchronize clocks that
are far apart, even if they are at rest with respect to each other in an inertial reference
frame. For example, if we have two similar watches showing the same time, and if we

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
carry one of them for a walk and back, they will show different times afterwards. This
Ref. 55, Ref. 56 experiment has actually been performed several times and has fully confirmed the pre-
diction of special relativity. The time difference for a person or a watch in an aeroplane
travelling around the Earth once, at about 900 km/h, is of the order of 100 ns – not very
noticeable in everyday life. This is sometimes called the clock paradox. In fact, the delay
is easily calculated from the expression

𝑡
=𝛾. (21)
𝑡󸀠
Also human bodies are clocks; they show the elapsed time, usually called age, by vari-
ous changes in their shape, weight, hair colour, etc. If a person goes on a long and fast
trip, on her return she will have aged less and thus stayed younger than a second person
who stayed at her (inertial) home. In short, the invariance of 𝑐 implies: Travellers remain
younger.
The most extreme illustration of this is the famous twin paradox. An adventurous

Challenge 52 e and wb = 𝛾b vb simplifies to
𝑤s‖ = 𝛾a 𝛾b (𝑣a + 𝑣b‖ ) and 𝑤s⊥ = 𝑤b⊥ , (20)
where the signs ‖ and ⊥ designate the component in the direction of and the component perpendicular to
Ref. 54 va , respectively. One can in fact express all of special relativity in terms of ‘proper’ quantities.
50 1 maximum speed, observers at rest and

higher atmosphere

high
counter

decays

low
counter
F I G U R E 26 More muons than expected arrive at

Motion Mountain – The Adventure of Physics
the ground because fast travel keeps them young.

twin jumps on a relativistic rocket that leaves Earth and travels for many years. Far from
Earth, he jumps on another relativistic rocket going the other way and returns to Earth.
The trip is illustrated in Figure 25. At his arrival, he notes that his twin brother on Earth
is much older than himself. This result has also been confirmed in many experiments
Ref. 57 – though not with real twins yet. Can you explain the result, especially the asymmetry
Challenge 53 s between the two twins?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Special relativity thus confirms, in a surprising fashion, the well-known observation
that those who travel a lot remain younger. On the other hand, the human traveller with
the largest measured youth effect so far was the cosmonaut Sergei Krikalyov, who has
spent 803 days in orbit, and nevertheless aged only a few milliseconds less than people
on Earth.
The twin paradox is also the confirmation of the possibility of time travel to the future.
With the help of a fast rocket that comes back to its starting point, we can arrive at local
times that we would never have reached within our lifetime by staying home. Alas, we
can never return to the past to talk about it.*
One of the simplest experiments confirming the prolonged youth of really fast trav-
ellers involves the counting of muons. Muons are particles that are continuously formed
Vol. V, page 162 in the upper atmosphere by cosmic radiation and then fly to the ground. Muons at rest
(with respect to the measuring clock) have a finite half-life of 2.2 μs (or, at the speed of
light, 660 m). After this amount of time, half of the muons have decayed. This half-life
can be measured using simple muon counters. In addition, there exist more special coun-
ters that only count muons travelling within a certain speed range, say from 0.9950𝑐 to
0.9954𝑐. One can put one of these special counters on top of a mountain and put another

Ref. 58 * There are even special books on time travel, such as the well-researched text by Nahin. Note that the
concept of time travel has to be clearly defined; otherwise one has no answer to the clerk who calls his office
chair a time machine, as sitting on it allows him to get to the future.
motion of light 51

in the valley below, as shown in Figure 26. The first time this experiment was performed,
Ref. 59 the height difference was 1.9 km. Flying 1.9 km through the atmosphere at the mentioned
speed takes about 6.4 μs. With the half-life just given, a naive calculation finds that only
about 13 % of the muons observed at the top should arrive at the lower site in the val-
Challenge 54 s ley. However, it is observed that about 82 % of the muons arrive below. The reason for
this result is the relativistic time dilation. Indeed, at the mentioned speed, muons exper-
ience a proper time difference of only 0.62 μs during the travel from the mountain top
to the valley. This time is much shorter than that observed by the human observers. The
shortened muon time yields a much lower number of lost muons than would be the case
without time dilation; moreover, the measured percentage confirms the value of the pre-
Challenge 55 s dicted time dilation factor 𝛾 within experimental errors, as you may want to check. The
same effect is observed when relativistic muons are made to run in circles at high speed
Ref. 60 inside a so-called storage ring. The faster the muons turn, the longer they live.
Half-life dilation has also been found for many other decaying systems, such as pi-
ons, hydrogen atoms, neon atoms and various nuclei, always confirming the predictions
of special relativity. The effect is so common that for fast particles one speaks of the ap-

Motion Mountain – The Adventure of Physics
parent lifetime 𝜏𝑎𝑝𝑝 through the relation 𝜏𝑎𝑝𝑝 = 𝛾𝜏. Since all bodies in nature are made
of particles, the ‘youth effect’ of high speeds – usually called time dilation – applies to
bodies of all sizes; indeed, it has not only been observed for particles, but also for lasers,
Ref. 18 radio transmitters and clocks.
If motion leads to time dilation, a clock on the Equator, constantly running around
the Earth, should go slower than one at the poles. However, this prediction, which was
Ref. 61 made by Einstein himself, is incorrect. The centrifugal acceleration leads to a reduction
in gravitational acceleration whose time dilation exactly cancels that due to the rotation
velocity. This story serves as a reminder to be careful when applying special relativity in

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
situations involving gravity: pure special relativity is only applicable when space-time is
flat, i.e., when gravity is not present.
In summary, a mother can stay younger than her daughter. The mother’s wish to
remain younger than her daughter is not easy to fulfil, however. Let us imagine that a
mother is accelerated in a spaceship away from Earth at 10 m/s2 for ten years, then de-
celerates at 10 m/s2 for another ten years, then accelerates for ten additional years to-
wards the Earth, and finally decelerates for ten final years in order to land safely back on
our planet. The mother has taken 40 years for the trip. She got as far as 22 000 light years
from Earth. At her return on Earth, 44 000 years have passed. All this seems fine, until we
realize that the necessary amount of fuel, even for the most efficient engine imaginable,
is so large that the mass returning from the trip is only one part in 2 ⋅ 1019 of the mass
Challenge 56 e that started. The necessary amount of fuel does not exist on Earth. The same problem
Ref. 62 appears for shorter trips.
We also found that we cannot (simply) synchronize clocks at rest with respect to each
other simply by walking, clock in hand, from one place to another. The correct way to
Challenge 57 s do so is to exchange light signals. Can you describe how? The precise definition of syn-
chronization is necessary, because we often need to call two distant events simultaneous,
for example when we define coordinates. Obviously, a maximum speed implies that sim-
ultaneity depends on the observer. Indeed, this dependence has been confirmed by all
experiments.
52 1 maximum speed, observers at rest and

observations
observations
by the pilot
by the farmer

pilot
time
farmer
time

plane ends
barn ends

Motion Mountain – The Adventure of Physics
F I G U R E 27 The observations of the pilot and the barn owner.

Length contraction
The length of an object measured by an observer attached to the object is called its proper
length. The length measured by an inertial observer passing by is always smaller than the
Challenge 58 e proper length. This result follows directly from the Lorentz transformations.
For a Ferrari driving at 300 km/h or 83 m/s, the length is contracted by 0.15 pm: less

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
than the diameter of a proton. Seen from the Sun, the Earth moves at 30 km/s; this gives
a length contraction of 6 cm. Neither of these effects has ever been measured.* But larger
effects could be. Let us explore the possibilities.
Imagine a pilot flying with his plane through a barn with two doors, one at each end.
The plane is slightly longer than the barn, but moves so rapidly that its relativistically
contracted length is shorter than the length of the barn. Can the farmer close the barn
(at least for a short time) with the plane completely inside? The answer is positive. But
why can the pilot not say the following: relative to him, the barn is contracted; therefore
the plane does not fit inside the barn? The answer is shown in Figure 27. For the farmer,
the doors close (and reopen) at the same time. For the pilot, they do not. For the farmer,
the pilot is in the dark for a short time; for the pilot, the barn is never dark. (That is not
Challenge 60 s completely true: can you work out the details?) For obvious reasons, this experiment has
never been realized.
Let us explore some different length contraction experiments. Can a rapid snow-
boarder fall into a hole that is a bit shorter than his board? Imagine him boarding so
(unrealistically) fast that the length contraction factor 𝛾 is 4. For an observer on the
ground, the snowboard is four times shorter, and when it passes over the hole, it will fall
into it. However, for the boarder, it is the hole which is four times shorter; it seems that
the snowboard cannot fall into it.

Challenge 59 s * Is the Earth contraction value measurable at all?
motion of light 53

ski or snowboard ski or snowboard
height
h
trap trap

F I G U R E 28 The observations of the trap digger (left) and of the snowboarder (right), as often
(misleadingly) published in the literature.

𝑑

rails
B rope F
glider
𝑣
𝑣(𝑡) 𝑣(𝑡)

Motion Mountain – The Adventure of Physics
𝑙<𝑑
F I G U R E 29 Does the conducting glider keep the lamp F I G U R E 30 What happens to the
lit at large speeds? rope?

Ref. 63 A first careful analysis shows that, in contrast to the observation of the hole digger, the
snowboarder does not experience the board’s shape as fixed: while passing over the hole,
the boarder observes that the board takes on a parabolic shape and falls into the hole,
Challenge 61 e as shown in Figure 28. Can you confirm this? In other words, shape is not an observer-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
invariant concept. (However, rigidity is observer-invariant, if defined properly; can you
Challenge 62 s confirm this?)
The snowboard explanation and figure however, though published, are not correct,
Ref. 64 as Harald van Lintel and Christian Gruber have pointed out. We should not forget to
estimate the size of the effect. At relativistic speeds the time required for the hole to
affect the full thickness of the board cannot be neglected. The snowboarder only sees
his board take on a parabolic shape if it is extremely thin and flexible. For usual boards
moving at relativistic speeds, the snowboard has no time to fall any appreciable height ℎ
Challenge 63 e or to bend into the hole before passing it. Figure 28 is so exaggerated that it is incorrect.
The snowboarder would simply speed over the hole.
In fact, we can simplify the discussion of such examples of length contraction by ex-
ploring what happens when a rod moves on an inclined path towards a slot, without any
Ref. 65 gravity. A careful exploration shows that if the slot and the rod are parallel for the rod
observer, they are not parallel for the slot observer, and vice versa. The concept of parallel
is relative!
The paradoxes around length contraction become even more interesting in the case of
Ref. 66 a conductive glider that makes electrical contact between two rails, as shown in Figure 29.
The two rails are parallel, but one rail has a gap that is longer than the glider. Can you
work out whether a lamp connected in series stays lit when the glider moves along the
Challenge 64 s rails with relativistic speed? (Make the simplifying and not fully realistic assumption that
electrical current flows as long and as soon as the glider touches the rails.) Do you get
54 1 maximum speed, observers at rest and

the same result for all observers? And what happens when the glider is longer than the
detour? Or when it approaches the lamp from the other side of the detour? Be warned:
this problem gives rise to heated debates! What is unrealistic in this experiment?
Ref. 67 Another example of length contraction appears when two objects, say two cars, are
connected over a distance 𝑑 by a straight rope, as shown in Figure 30. Imagine that both
are at rest at time 𝑡 = 0 and are accelerated together in exactly the same way. The observer
at rest will maintain that the two cars always remain the same distance apart. On the other
hand, the rope needs to span a distance 𝑑󸀠 = 𝑑/√1 − 𝑣2 /𝑐2 , and thus has to expand when
the two cars are accelerating. In other words, the rope will break. Who is right? You can
check by yourself that this prediction is confirmed by all observers, in the cars and on
Challenge 65 s Earth.
A funny – but again unrealistic – example of length contraction is that of a submar-
Ref. 68 ine moving horizontally. Imagine that before moving, the resting submarine has tuned
its weight to float in water without any tendency to sink or to rise. Now the submar-
ine moves in horizontal direction. The captain observes the water outside to be Lorentz
contracted; thus the water is denser and he concludes that the submarine will rise. A

Motion Mountain – The Adventure of Physics
nearby fish sees the submarine to be contracted, thus denser than water, and concludes
that the submarine will sink. Who is wrong, and what is the correct buoyancy force?
Challenge 66 s Alternatively, answer the following question: why is it impossible for a submarine to
Challenge 67 s move at relativistic speed?
In summary, for macroscopic bodies, length contraction is interesting but will prob-
ably never be observed. However, length contraction does play an important role for
images.

R elativistic films – aberration and Doppler effect

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
In our adventure so far, we have encountered several ways in which the observed sur-
roundings change when we move at relativistic speed. We now put them all together.
First of all, Lorentz contraction and aberration lead to distorted images. Secondly, aber-
ration increases the viewing angle beyond the roughly 180 degrees that we are used to
in everyday life. At relativistic speeds, when we look in the direction of motion, we see
light that is invisible for an observer at rest, because for the latter, it comes from behind.
Thirdly, the Doppler effect produces colour-shifted images. Fourthly, our rapid motion
changes the brightness and contrast of the image: the so-called searchlight effect. Each of
these changes depends on the direction of sight; they are shown in Figure 31.
Modern computers enable us to simulate the observations made by rapid observers
with photographic quality, and even to produce simulated films and computer games.*
The images of Figure 32 are particularly helpful in allowing us to understand image dis-
tortion. They show the viewing angle, the circle which distinguish objects in front of
the observer from those behind the observer, the coordinates of the observer’s feet and

* See for example the many excellent images and films at www.anu.edu.au/Physics/Searle by Anthony
Searle and www.anu.edu.au/Physics/vrproject by Craig Savage and his team; you can even do interactive
motion steering with the free program downloadable at realtimerelativity.org. There is also beautiful ma-
terial at www.tat.physik.uni-tuebingen.de/~weiskopf/gallery/index.html by Daniel Weiskopf, at www.itp.
uni-hannover.de/~dragon/stonehenge/stone1.htm by Norbert Dragon and Nicolai Mokros, and at www.
tempolimit-lichtgeschwindigkeit.de by Ute Kraus, once at Hanns Ruder’s group.
motion of light 55

F I G U R E 31 Flying through three straight and vertical columns with 0.9 times the speed of light as
visualized by Daniel Weiskopf: on the left with the original colours; in the middle including the Doppler
effect; and on the right including brightness effects, thus showing what an observer would actually see
(© Daniel Weiskopf ).

the point on the horizon toward which the observer is moving. Adding these markers
in your head when watching other pictures or films may help you to understand more

Motion Mountain – The Adventure of Physics
clearly what they show.
We note that the image seen by a moving observer is a distorted version of that seen
by one at rest at the same point. Figure 33 shows this clearly. But a moving observer
never sees different things than a resting one at the same point. Indeed, light cones are
independent of observer motion.
Studying the images with care shows another effect. Even though the Lorentz con-
traction is measurable, it cannot be photographed. This surprising result was discovered
Ref. 69 only in 1959. Measuring implies simultaneity at the object’s position; in contrast, photo-
graphing implies simultaneity at the observer’s position. On a photograph or in a film,
the Lorentz contraction is modified by the effects due to different light travel times from

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the different parts of an object; the result is a change in shape that is reminiscent of, but
not exactly the same as, a rotation. This is shown in Figure 34. The total deformation is
the result of the angle-dependent aberration. We discussed the aberration of star pos-
Page 18 itions at the beginning of this chapter. In complete images, aberration transforms circles
into circles: such transformations are called conformal. As a result, a sphere is seen to
have a circular outline even at relativistic speeds – though its thickness changes.
Aberration leads to the pearl necklace paradox. If the relativistic motion keeps intact
the circular shape of spheres, but transforms rods into shorter rods, what happens to a
Challenge 68 s pearl necklace moving along its own long axis? Does it get shorter or not?
A further puzzle: imagine that a sphere moves and rotates at high speed. Can all the
Challenge 69 r mentioned effects lead to an apparent, observer-dependent sense of rotation?

Which is the best seat in a bus?
Ref. 67 Let us explore another surprise of special relativity. Imagine two twins inside two identic-
ally accelerated cars, one in front of the other, starting from standstill at time 𝑡 = 0, as
described by an observer at rest with respect to both of them. (There is no connecting
rope now.) Both cars contain the same amount of fuel. We easily deduce that the accele-
ration of the two twins stops, when the fuel runs out, at the same time in the frame of the
Challenge 70 e outside observer. In addition, the distance between the cars has remained the same all
along for the outside observer, and the two cars continue rolling with an identical con-
56 1 maximum speed, observers at rest and

Views for an observer at rest

Views for an observer at relativistic speed

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 32 Flying through twelve vertical columns (shown in the two uppermost images) with 0.9
times the speed of light as visualized by Nicolai Mokros and Norbert Dragon, showing the effect of
speed and position on distortions (© Nicolai Mokros).
motion of light 57

Motion Mountain – The Adventure of Physics
F I G U R E 33 What a researcher standing and one running rapidly through a corridor observe (ignoring
colour and brightness effects) (© Daniel Weiskopf ).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 34 A stationary row of
dice (below), and the same row,
ﬂying above it at relativistic
speed towards the observer,
though with Doppler and
brightness effects switched off.
(Mpg ﬁlm © Ute Kraus at www.
tempolimit-lichtgeschwindigkeit.
de).

stant velocity 𝑣, as long as friction is negligible. If we call the events at which the front
car and back car engines switch off f and b, their time coordinates in the outside frame at
rest are related simply by 𝑡f = 𝑡b . By using the Lorentz transformations you can deduce
58 1 maximum speed, observers at rest and

Challenge 71 e for the frame of the freely rolling twins the relation

𝑡󸀠b = 𝛾Δ𝑥 𝑣/𝑐2 + 𝑡󸀠f , (22)

which means that the front twin has aged more than the back twin! Thus, in accelerated
systems, ageing is position-dependent.
For choosing a seat in a bus, though, this result does not help. It is true that the best
seat in an accelerating bus is the back one, but in a decelerating bus it is the front one. At
the end of a trip, the choice of seat does not matter.
Is it correct to deduce from the above that people on high mountains age faster than
Challenge 72 s people in valleys, so that living in a valley helps postponing grey hair?

How fast can one walk?
In contrast to running, walking means to move the feet in such a way that at least one
of them is on the ground at any time. This is one of the rules athletes have to follow in

Motion Mountain – The Adventure of Physics
Olympic walking competitions; they are disqualified if they break it. A student athlete
was thinking about the theoretical maximum speed he could achieve in the Olympic
Games. The ideal would be that each foot accelerates instantly to (almost) the speed of
light. The highest walking speed is then achieved by taking the second foot off the ground
at exactly the same instant at which the first is put down. By ‘same instant’, the student
originally meant ‘as seen by a competition judge at rest with respect to Earth’. The mo-
tion of the feet is shown in the left diagram of Figure 35; it gives a limit speed for walking
of half the speed of light.
But then the student noticed that a moving judge will regularly see both feet off the
ground and thus disqualify the athlete for running. To avoid disqualification by any

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 70
judge, the rising foot has to wait for a light signal from the lowered one. The limit speed
for Olympic walking then turns out to be only one third of the speed of light.

Is the speed of shad ow greater than the speed of light?

“ ”
Quid celerius umbra?*
Antiquity

Actually, motion faster than light does exist and is even rather common. Nature only
constrains the motion of mass and energy. However, non-material points or non-energy-
transporting features and images can move faster than light. There are several simple
Page 48 examples. To be clear, we are not talking about proper velocity, which in these cases can-
Challenge 73 s not be defined anyway. (Why?) The following examples show speeds that are genuinely
higher than the speed of light in vacuum.
As first example, consider the point at which scissors cut paper, marked X in Fig-
ure 36. If the scissors are closed rapidly enough, the point moves faster than light. Similar
examples can also be found in every window frame, and in fact in any device that has
twisting parts.
Another example of superluminal motion is a music record – an old-fashioned LP –

* ‘What is faster than the shadow?’ A motto often found on sundials.
motion of light 59

average athlete average athlete
speed: 𝑐/2 speed: 𝑐/3
feet of
fee𝑡 athlete
time 𝑡 time 𝑡
󸀠 of
𝑡 athlete

𝑡󸀠
moving F I G U R E 35 For
judge the athlete on
the left, the
J light signal competition
󸀠
𝑥 judge moving in
the opposite
J direction sees
light signal 𝑥󸀠
moving both feet off the
judge ground at

Motion Mountain – The Adventure of Physics
certain times,
but not for the
space 𝑥 space 𝑥 athlete on the
right.

J.S. Bach

F I G U R E 36 A simple example of motion that can F I G U R E 37 Another example of
be faster than light. faster-than-light motion.

disappearing into its sleeve, as shown in Figure 37. The point where the border of the
record meets the border of the sleeve can travel faster than light.
Another example suggests itself when we remember that we live on a spherical planet.
Imagine you lie on the floor and stand up. Can you show that the initial speed with which
Challenge 74 s the horizon moves away from you can be larger than that of light?
A further standard example is the motion of a spot of light produced by shining a laser
60 1 maximum speed, observers at rest and

Ref. 71 beam onto the Moon. If the laser beam is deflected, the spot can easily move faster than
light. The same applies to the light spot on the screen of an oscilloscope when a signal
of sufficiently high frequency is fed to the input. In fact, when a beam is swept across an
inclined surface, the spot can move backwards, split and recombine. Researchers are still
looking for such events both in the universe and in the laboratory.
Finally, here is the simplest example of all. Imagine to switch on a light bulb in front
Ref. 71 of a wall. During the switch-on process, the boundary between the illuminated surface
and the surface that is still dark moves with a speed higher than the speed of light. Light
bulbs produce superluminal speeds.
All these are typical examples of the speed of shadows, sometimes also called the speed
of darkness. Both shadows and darkness can indeed move faster than light. In fact, there
Challenge 75 s is no limit to their speed. Can you find another example?
In addition, there is an ever-increasing number of experimental set-ups in which the
phase velocity or even the group velocity of light is higher than 𝑐. They regularly make
headlines in the newspapers, usually along the lines of ‘light moves faster than light’.
Vol. III, page 133 We will discuss this surprising phenomenon in more detail later on. In fact, these cases

Motion Mountain – The Adventure of Physics
can also be seen – with some abstraction – as special cases of the ‘speed of shadow’
phenomenon.
For a different example, imagine that we are standing at the exit of a straight tunnel of
length 𝑙. We see a car, whose speed we know to be 𝑣, entering the other end of the tunnel
and driving towards us. We know that it entered the tunnel because the car is no longer
in the Sun or because its headlights were switched on at that moment. At what time 𝑡,
after we see it entering the tunnel, does it drive past us? Simple reasoning shows that 𝑡 is
given by
𝑡 = 𝑙/𝑣 − 𝑙/𝑐 . (23)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
In other words, the approaching car seems to have a velocity 𝑣appr of

𝑙 𝑣𝑐
𝑣appr = = , (24)
𝑡 𝑐−𝑣

which is higher than 𝑐 for any car velocity 𝑣 higher than 𝑐/2. For cars this does not happen
too often, but astronomers know a type of bright object in the sky called a quasar (a
contraction of ‘quasi-stellar object’), which sometimes emits high-speed gas jets. If the
emission is in or near the direction of the Earth, its apparent speed – even the purely
transverse component – is higher than 𝑐. Such situations are now regularly observed
Ref. 72 with telescopes.
Note that to a second observer at the entrance of the tunnel, the apparent speed of the
car moving away is given by
𝑣𝑐
𝑣leav = , (25)
𝑐+𝑣

which is never higher than 𝑐/2. In other words, objects are never seen departing with
more than half the speed of light.
The story has a final twist. We have just seen that motion faster than light can be
observed in several ways. But could an object moving faster than light be observed at
motion of light 61

time
observer
emitted or reflected light

tachyon

light cone

Motion Mountain – The Adventure of Physics
space F I G U R E 38 Hypothetical space-time
diagram for tachyon observation.

all? Surprisingly, it could be observed only in rather unusual ways. First of all, since such
an imaginary object, usually called a tachyon, moves faster than light, we can never see
Ref. 73 it approaching. If it can be seen at all, a tachyon can only be seen departing. Seeing a
tachyon would be similar to hearing a supersonic jet. Only after a tachyon has passed

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
nearby, assuming that it is visible in daylight, could we notice it. We would first see a
flash of light, corresponding to the bang of a plane passing with supersonic speed. Then
we would see two images of the tachyon, appearing somewhere in space and departing in
opposite directions, as can be deduced from Figure 38. One image would be red-shifted,
the other blue-shifted. Even if one of the two images were approaching us, it would be
getting fainter and smaller. This is, to say the least, rather unusual behaviour. Moreover,
if you wanted to look at a tachyon at night, illuminating it with a torch, you would have
to turn your head in the direction opposite to the arm with the torch! This requirement
Challenge 76 e also follows from the space-time diagram: can you see why? Nobody has ever seen such
phenomena.
Ref. 74 Tachyons, if they existed, would be strange objects: they would accelerate when they
Page 73 lose energy, a zero-energy tachyon would be the fastest of all, with infinite speed, and the
direction of motion of a tachyon depends on the motion of the observer. No object with
these properties has ever been observed. Worse, as we just saw, tachyons would seem
to appear from nothing, defying laws of conservation; and note that, just as tachyons
cannot be seen in the usual sense, they cannot be touched either, since both processes
are due to electromagnetic interactions, as we will see later in our adventure. Tachyons
therefore cannot be objects in the usual sense. In the quantum part of our adventure
we will show that quantum theory actually rules out the existence of (real) tachyons.
However, quantum theory also requires the existence of ‘virtual’ tachyons, as we will
discover.
62 1 maximum speed, observers at rest and

R 𝑣 G

𝑢
𝑤
O

F I G U R E 39 If O’s stick is parallel to
R’s and R’s is parallel to G’s, then
O’s stick and G’s stick are not.

Motion Mountain – The Adventure of Physics
Parallel to parallel is not parallel – Thomas precession
The limit speed has many strange consequences. Any two observers can keep a stick
parallel to the other’s, even if they are in motion with respect to each other. But strangely,
given a chain of three or more sticks for which any two adjacent ones are parallel, the
first and the last sticks will not generally be parallel. In particular, they never will be if
the motions of the various observers are in different directions, as is the case when the
velocity vectors form a loop.
The simplest set-up is shown in Figure 39. In special relativity, a general concatenation
Ref. 75 of pure boosts does not give a pure boost, but a boost plus a rotation. As a result, the first

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
and last stick in a chain of parallel sticks are usually not parallel.
An example of this effect appears in rotating motion. Imagine that we walk in a circle
with relativistic speed and hold a stick. We always keep the stick parallel to the direction
it had just before. At the end of the turn, the stick will have an angle with respect to
the direction at the start. Similarly, the axis of a rotating body circling a second body
will not be pointing in the same direction after one turn. This effect is called Thomas
precession, after Llewellyn Thomas, who discovered it in 1925, a full 20 years after the
birth of special relativity. The effect had escaped the attention of dozens of other famous
physicists. Thomas precession is important for the orbit of electrons inside atoms, where
the stick is the spin axis of the rapidly orbiting electron. All these surprising phenomena
are purely relativistic, and are thus measurable only in the case of speeds comparable to
that of light.

A never-ending story – temperature and relativit y
What temperature is measured by an observer who moves with respect to a heat bath?
The literature on the topic is confusing. Max Planck, Albert Einstein and Wolfgang Pauli
agreed on the following result: the temperature 𝑇 seen by an observer moving with speed
𝑣 is related to the temperature 𝑇0 measured by the observer at rest with respect to the heat
bath via
𝑇 = 𝑇0 √1 − 𝑣2 /𝑐2 . (26)
motion of light 63

A moving observer thus always measures lower temperature values than a resting one.
In 1908, Max Planck used this expression, together with the corresponding transform-
ation for thermal energy, to deduce that the entropy is invariant under Lorentz trans-
formations. Being the discoverer of the Boltzmann constant 𝑘, Planck proved in this way
Ref. 76 that the Boltzmann constant is a relativistic invariant.
Not all researchers agree on the expression for the transformation of energy, however.
(They do agree on the invariance of 𝑘, though.) Others maintain that 𝑇 and 𝑇0 should
be interchanged in the formula. Also, powers other than the simple square root have
Ref. 77 been proposed. The origin of these discrepancies is simple: temperature is only defined
for equilibrium situations, i.e., for baths. But a bath for one observer is not a bath for
the other. For low speeds, a moving observer sees a situation that is almost a heat bath;
but at higher speeds the issue becomes tricky. Temperature is deduced from the speed
of matter particles, such as atoms or molecules. For rapidly moving observers, there is
no good way to measure temperature, because the distribution is not in equilibrium.
Any naively measured temperature value for a moving observer depends on the energy
range of matter particles that is used! In short, thermal equilibrium is not an observer-

Motion Mountain – The Adventure of Physics
invariant concept. Therefore, no temperature transformation formula is correct for high
speeds. (Only with certain additional assumptions, Planck’s expression holds. And sim-
ilar issues appear for the relativistic transformation of entropy.) In fact, there are not even
any experimental observations that would allow such a formula to be checked. Realizing
such a measurement is a challenge for future experimenters – but not for relativity itself.

A curiosit y: what is the one-way speed of light?
We have seen that the speed of light, as usually defined, is given by 𝑐 only if either the
observer is inertial or the observer measures the speed of light passing nearby (rather

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
than light passing at a distance). In short, the speed of light has to be measured locally.
But this condition does not eliminate one last subtlety.
Usually, length is measured by the time it takes light to travel. In this case the speed
of light will obviously be invariant. So how can we check the invariance? We need to
eliminate length measurements. The simplest way to do this is to reflect light from a
mirror, as shown in Figure 40. The invariance of the speed of light implies that if light
goes up and down a short straight line, then the clocks at the two ends measure times
given by
𝑡3 − 𝑡1 = 2 (𝑡2 − 𝑡1 ) . (27)

Here it is assumed that the clocks have been synchronised according to the prescription
on page 51. If the factor were not exactly two, the speed of light would not be invariant.
In fact, all experiments so far have yielded a factor of two, within measurement errors.
Ref. 78, Ref. 79 But these experiments instil us with a doubt: it seems that the one-way velocity of light
Challenge 77 s cannot be measured. Do you agree? Is the issue important?

Summary
For all physical systems, the locally measured energy speed, the forerunner speed and the
measured signal speed are limited by 𝑐 = 299 782 458 m/s, the speed of light in vacuum.
As a result, time, age, distance, length, colour, spatial orientation, angles and temperature
64 1 maximum speed, observers at rest and

time
clock 1 clock 2

𝑡3

𝑡2

𝑡1

space

Motion Mountain – The Adventure of Physics
F I G U R E 40 Clocks and the measurement of the speed
of light as two-way velocity.

– as long as it can be defined – depend on the observer. In contrast, the speed of light in
vacuum 𝑐 is invariant.

R E L AT I V I ST IC M E C HA N IC S

T
he speed of light is an invariant quantity and a limit value. Therefore, we need
o rethink all observables that we defined with the help of velocity – thus all of
hem! The most basic observables are mass, momentum and energy. In other
words, we need to recreate mechanics based on the invariant limit speed: we need to
build relativistic mechanics.

Motion Mountain – The Adventure of Physics
The exploration of relativistic mechanics will first lead us to the equivalence of mass
and energy, a deep relation that is the basis of the understanding of motion. Relativistic
mechanics will also lead us to the concept of horizon, a concept that we will need later
to grasp the details of black holes, the night sky and the universe as whole.

Mass in relativit y
Vol. I, page 100 In Galilean physics, the mass ratio between two bodies was defined using collisions. More
precisely, mass was given by the negative inverse of the velocity change ratio

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑚2 Δ𝑣
=− 1 . (28)
𝑚1 Δ𝑣2

However, experiments show that this expression is wrong for speeds near that of light; it
must be changed. In fact, experiments are not needed: thinking alone can show that it is
Challenge 78 s wrong. Can you do so?
There is only one solution to the problem of mass definition. Indeed, experiments
confirm that the two Galilean conservation theorems for momentum and for mass have
Ref. 80 to be changed into
∑ 𝛾𝑖 𝑚𝑖 𝑣𝑖 = const (29)
𝑖

and
∑ 𝛾𝑖 𝑚𝑖 = const . (30)
𝑖

These expressions are the (relativistic) conservation of momentum and the (relativistic)
conservation of mass–energy. They will remain valid throughout the rest of our adventure.
The conservation of momentum and energy implies, among other things, that tele-
Challenge 79 s portation is not possible in nature, in contrast to science fiction. Can you confirm this?
Obviously, in order to recover Galilean physics, the relativistic correction factors 𝛾𝑖
66 2 relativistic

Observer A
𝑚 𝑚
before: 𝑣
after:
𝑀 𝑉

Observer B

before:
𝑚 𝑉 𝑉 𝑚
after: F I G U R E 41 An inelastic collision of two identical
𝑀 particles seen from two different inertial frames of
reference.

have to be almost equal to 1 for everyday velocities, that is, for velocities nowhere near the

Motion Mountain – The Adventure of Physics
speed of light. That is indeed the case. In fact, even if we did not know the expression of
the relativistic correction factor, we can deduce it from the collision shown in Figure 41.
In the first frame of reference (A) we have 𝛾𝑣 𝑚𝑣 = 𝛾𝑉 𝑀𝑉 and 𝛾𝑣 𝑚 + 𝑚 = 𝛾𝑉 𝑀. From
the observations of the second frame of reference (B) we deduce that 𝑉 composed with
Challenge 80 e 𝑉 gives 𝑣, in other words, that
2𝑉
𝑣= . (31)
1 + 𝑉2 /𝑐2

When these equations are combined, the relativistic correction 𝛾 is found to depend on

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the magnitude of the velocity 𝑣 through

1
𝛾𝑣 = . (32)
√1 − 𝑣2 /𝑐2

With this expression the mass ratio between two colliding particles is defined as the ratio

𝑚1 Δ(𝛾2𝑣2 )
=− . (33)
𝑚2 Δ(𝛾1𝑣1 )

This is the generalization of the definition of mass ratio from Galilean physics. The cor-
rection factors 𝛾𝑖 ensure that the mass defined by this equation is the same as the one
defined in Galilean mechanics, and that it is the same for all types of collision a body
may have.* In this way, mass remains a quantity characterizing the difficulty of accel-
erating a body, and it can still be used for systems of bodies as well. (In the chapter on
Vol. I, page 103 Galilean mechanics we also used a second, generalized mass definition based on acce-
leration ratios. We do not explore its relativistic generalization because it contains some
subtleties which we will encounter shortly.)

Challenge 81 e * The results below also show that 𝛾 = 1 + 𝑇/𝑐2 𝑚, where 𝑇 is the kinetic energy of a particle.
mechanics 67

before

A 𝑝A B

non-relativistic pool
rule: 𝜑 + 𝜃 = 90° A

after
𝜃
𝑝A 𝜑

F I G U R E 42 A useful rule for playing non-relativistic
B snooker – and to predict non-relativistic elastic
collisions.

Motion Mountain – The Adventure of Physics
Following the example of Galilean physics, we call the quantity

𝑝 = 𝛾𝑚𝑣 (34)

the (linear) relativistic (three-) momentum of a particle. Total momentum is a conserved
quantity for any system not subjected to external influences, and this conservation is a
direct consequence of the way mass is defined.
For low speeds, or 𝛾 ≈ 1, relativistic momentum is the same as Galilean momentum,
and is then proportional to velocity. But for high speeds, momentum increases faster
than velocity, tending to infinity when approaching light speed. The result is confirmed

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Page 38 by experimental data, as was shown in Figure 19.
Now that we have the correct definitions of mass and momentum, we can explore
collisions in more detail.

Why relativistic sno oker is more difficult
A well-known property of collisions between a moving sphere or particle and a resting
one of the same mass is important when playing snooker, pool or billiards. After such
Challenge 82 e a collision, the two spheres will depart at a right angle from each other. As shown in
Figure 42, the two angles 𝜑 and 𝜃 add up to a right angle. (The only exception to this rule
is the case that the collision is exactly head on; in that case the first sphere simply stops.)
However, experiments show that the right-angle rule does not apply to relativistic
collisions. Indeed, using the conservation of momentum and a bit of dexterity you can
Challenge 83 e calculate that
2
tan 𝜃 tan 𝜑 = , (35)
𝛾+1

where the angles are defined in Figure 43. It follows that the sum 𝜑 + 𝜃 is smaller than
a right angle in the relativistic case. Relativistic speeds thus completely change the game
of snooker. Indeed, every accelerator physicist knows this: for electrons or protons, these
angles can easily be deduced from photographs taken in cloud or bubble chambers,
68 2 relativistic

accelerator beam target detectors

relativistic pool rule: 𝜑 + 𝜃 < 90°
𝜃
𝜑

Motion Mountain – The Adventure of Physics
F I G U R E 43 The dimensions of detectors for particle accelerators with single beams are based on the
relativistic snooker angle rule – as an example, the HARP experiment at CERN (© CERN).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
which show the tracks left by particles when they move through them, as shown in Fig-
Ref. 18 ure 44. All such photographs confirm the relativistic expression. In fact, the shapes of
detectors are chosen according to expression (35), as sketched in Figure 43. If the for-
mula – and relativity – were wrong, most of these detectors would not work, as they
would miss most of the particles after the collision. If relativity were wrong, such de-
tectors would have to be much larger. In fact, these particle experiments also prove the
Challenge 84 e formula for the composition of velocities. Can you show this?

Mass and energy are equivalent
Page 66 Let us go back to the collinear and inelastic collision of Figure 41. What is the mass 𝑀 of
Challenge 85 s the final system? Calculation shows that

𝑀/𝑚 = √2(1 + 𝛾𝑣 ) > 2 . (36)

In other words, the mass of the final system is larger than the sum 2𝑚 of the two original
masses. In contrast to Galilean mechanics, the sum of all masses in a system is not a
conserved quantity. Only the sum ∑𝑖 𝛾𝑖 𝑚𝑖 of the corrected masses is conserved.
Relativity provides the solution to this puzzle. Everything falls into place if, for the
mechanics 69

F I G U R E 44 The ‘Big European Bubble Chamber’ and an example of tracks of relativistic particles it
produced, with the momentum values deduced from the photograph (© CERN).

Motion Mountain – The Adventure of Physics
energy 𝐸 of an object of mass 𝑚 and velocity 𝑣, we use the expression

𝑐2 𝑚
𝐸 = 𝑐2 𝛾𝑚 = , (37)
√1 − 𝑣2 /𝑐2

applying it both to the total system and to each component. The conservation of the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
corrected mass can then be read as the conservation of energy, simply without the factor
𝑐2 . In the example of the two identical masses sticking to each other, the two parts are
thus each described by mass and energy, and the resulting system has an energy 𝐸 given
by the sum of the energies of the two parts. (We recall that the uncorrected masses do
not add up.) In particular, it follows that the energy 𝐸0 of a body at rest and its mass 𝑚
are related by
𝐸0 = 𝑐2 𝑚 . (38)

Why do we write 𝑐2 𝑚 instead of 𝑚𝑐2 ? Because in formulae, constant factors come always
first. The factor 𝑐2 is not central; the essence of the expression is the relation between
energy 𝐸 and mass 𝑚. 𝑐2 is simply the conversion factor between the two quantities.
The mass-energy relation 𝐸 = 𝑐2 𝛾𝑚 is one of the most beautiful and famous discover-
ies of modern physics. In simple words, the existence of a maximum speed implies that
every mass has energy, and that energy has mass. Mass and energy are two terms for the
same basic concept:

⊳ Mass and energy are equivalent.

Since mass and energy are equivalent, energy has all properties of mass. In particular,
energy has inertia and weight. For example, a full battery is more massive and heavier
than an empty one, and a warm glass of water is heavier than a cold one. Radio waves
70 2 relativistic

and light have weight. They can fall.
Conversely, mass has all properties of energy. For example, we can use mass to make
engines run. But this is no news, as the process is realized in every engine we know of!
Muscles, car engines, and nuclear ships work by losing a tiny bit of mass and use the
corresponding energy to overcome friction and move the person, car or ship.
The conversion factor 𝑐2 is large: 1 kg of rock, if converted to electric energy, would be
worth around 8 000 million Euro. In this unit, even the largest financial sums correspond
to modest volumes of rock. Since 𝑐2 is so large, we can also say:

⊳ Mass is concentrated energy.

Increasing the energy of a system increases its mass a little bit, and decreasing the energy
content decreases the mass a little bit. If a bomb explodes inside a closed box, the mass,
weight and momentum of the box are the same before and after the explosion, but the
combined mass of the debris inside the box will be a little bit smaller than before. All

Motion Mountain – The Adventure of Physics
bombs – not only nuclear ones – thus take their power of destruction from a reduction
in mass. In fact, every activity of a system – such as a caress, a smile or a look – takes its
energy from a reduction in mass.
The kinetic energy 𝑇 is thus given by the difference between total energy and rest en-
ergy. This gives

1 1 ⋅ 3 𝑣4 1 ⋅ 3 ⋅ 5 𝑣6
𝑇 = 𝑐2 𝛾𝑚 − 𝑐2 𝑚 = 𝑚𝑣2 + 𝑚 + 𝑚 + ... (39)
2 2 ⋅ 4 𝑐2 2 ⋅ 4 ⋅ 6 𝑐4

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 86 e (using the binomial theorem). The expression reduces to the well-known Galilean value
𝑇Galilean = 12 𝑚𝑣2 only for low, everyday speeds.
The mass–energy equivalence 𝐸 = 𝑐2 𝛾𝑚 implies that extracting any energy from a
material system results in a mass decrease. When a person plays the piano, thinks or
runs, her mass decreases. When a cup of tea cools down or when a star shines, its mass
decreases. When somebody uses somebody else’s electric power, he is taking away some
mass: electric power theft is thus mass theft! The mass–energy equivalence pervades all
of nature.
There is just one known way to transform the full mass of a body into kinetic, in
this case electromagnetic, energy: we annihilate it with the same amount of antimatter.
Fortunately, there is almost no antimatter in the universe, so that the process does not
occur in everyday life. Indeed, the energy content of even a speck of dust is so substantial
that the annihilation with the same amount of antimatter would already be a dangerous
Challenge 87 e event.
The equivalence of mass and energy suggests that it is possible to ‘create’ massive
particles by manipulating light or by extracting kinetic energy in collisions. This is indeed
correct; the transformation of other energy forms into matter particles is occurring, as
we speak, in the centre of galaxies, in particle accelerators, and whenever a cosmic ray
hits the Earth’s atmosphere. The details of these processes will become clear when we
explore quantum physics.
The mass–energy equivalence 𝐸 = 𝑐2 𝛾𝑚 means the death of many science fiction
mechanics 71

fantasies. It implies that there are no undiscovered sources of energy on or near Earth. If
such sources existed, they would be measurable through their mass. Many experiments
have looked for, and are still looking for, such effects. All had a negative result. There is
no freely available energy in nature.
In summary, the mass-energy equivalence is a fact of nature. But many scientists can-
not live long without inventing mysteries. Two different, extremely diluted forms of en-
ergy, called dark matter and (confusingly) dark energy, were found to be distributed
throughout the universe in the 1990s, with a density of about 1 nJ/m3 . Their existence
Page 220 is deduced from quite delicate measurements in the sky that detected their mass. Both
dark energy and dark matter must have mass and particle properties. But so far, their
nature and origin has not yet been resolved.

Weighing light
The mass–energy equivalence 𝐸 = 𝑐2 𝛾𝑚 also implies that one needs about 90 thousand
Challenge 88 e million kJ (or 21 thousand million kcal) to increase one’s weight by one single gram. Of

Motion Mountain – The Adventure of Physics
course, dieticians have slightly different opinions on this matter! As mentioned, humans
do get their everyday energy from the material they eat, drink and breathe by reducing its
combined mass before expelling it again; however, this chemical mass defect cannot yet
be measured by weighing the materials before and after the reaction: the difference is too
small, because of the large conversion factor 𝑐2 . Indeed, for any chemical reaction, bond
energies are about 1 aJ (6 eV) per bond; this gives a weight change of the order of one part
in 1010 , too small to be measured by weighing people or determining mass differences
between food and excrement. Therefore, for everyday chemical reactions mass can be
taken to be constant, in accordance with Galilean physics.
The mass–energy equivalence 𝐸 = 𝑐2 𝛾𝑚 has been confirmed by all experiments per-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
formed so far. The measurement is simplest for the nuclear mass defect. The most precise
Ref. 81 experiment, from 2005, compared the masses difference of nuclei before and after neut-
ron capture on the one hand, and emitted gamma ray energy on the other hand. The
mass–energy relation was confirmed to a precision of more than 6 digits.
Modern methods of mass measurement of single molecules have even made it pos-
sible to measure the chemical mass defect: it is now possible to compare the mass of
a single molecule with that of its constituent atoms. David Pritchard’s research group
has developed so-called Penning traps, which allow masses to be determined from the
measurement of frequencies; the attainable precision of these cyclotron resonance ex-
Ref. 82 periments is sufficient to confirm Δ𝐸0 = 𝑐2 Δ𝑚 for chemical bonds. In the future, bond
energies will be determined in this way with high precision. Since binding energy is often
radiated as light, we can also say that these modern techniques make it possible to weigh
light.
In fact, thinking about light and its mass was the basis for Einstein’s derivation of the
mass–energy relation. When an object of mass 𝑚 emits two equal light beams of total
energy 𝐸 in opposite directions, its own energy decreases by the emitted amount. Let
us look at what happens to its mass. Since the two light beams are equal in energy and
momentum, the body does not move, and we cannot deduce anything about its mass
change. But we can deduce something if we describe the same situation when moving
with the non-relativistic velocity 𝑣 along the beams. We know that due to the Doppler
72 2 relativistic

effect one beam is red-shifted and the other blue-shifted, by the factors 1 + 𝑣/𝑐 and 1 −
Challenge 89 e 𝑣/𝑐. The blue-shifted beam therefore acquires an extra momentum 𝑣𝐸/2𝑐2 and the red-
shifted beam loses momentum by the same amount. In nature, momentum is conserved.
Therefore, after emission, we find that the body has a momentum 𝑝 = 𝑚𝑣 − 𝑣𝐸/𝑐2 =
𝑣(𝑚 − 𝐸/𝑐2). We thus conclude that a body that loses an energy 𝐸 reduces its mass by 𝐸/𝑐2 .
This is the equivalence of mass and energy.
In short, we find that the rest energy 𝐸0 of an object, the maximum energy that can be
extracted from a mass 𝑚, is
𝐸0 = 𝑐2 𝑚 . (40)

We saw above that the Doppler effect is a consequence of the invariance of the speed of
light. We conclude: when the invariance of the speed of light is combined with energy
and momentum conservation we find the equivalence of mass and energy.
How are momentum and energy related? The definitions of momentum (34) and en-
Challenge 90 e ergy (37) lead to two basic relations. First of all, their magnitudes are related by

Motion Mountain – The Adventure of Physics
𝑐4 𝑚2 = 𝐸2 − 𝑐2 𝑝2 (41)

for all relativistic systems, be they objects or, as we will see below, radiation. For the
momentum vector we get the other important relation

𝐸
𝑝= 𝑣, (42)
𝑐2
which is equally valid for any type of moving energy, be it an object or a beam or pulse

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 91 e of radiation.* We will use both relations often in the rest of our adventure, including the
following discussion.

C ollisions, virtual objects and tachyons
We have just seen that in relativistic collisions the conservation of total energy and mo-
mentum are intrinsic consequences of the definition of mass. Let us now have a look at
collisions in more detail. A collision is a process, i.e., a series of events, for which
— the total momentum before the interaction and after the interaction is the same;
— the momentum is exchanged in a small region of space-time;
— for small velocities, the Galilean description is valid.
In everyday life, an impact is the event at which both objects change momentum. But
the two colliding objects are located at different points when this happens. A collision is
Ref. 83 therefore described by a space-time diagram such as the left-hand one in Figure 45; it is
reminiscent of the Orion constellation. It is easy to check that the process described by
Challenge 92 e such a diagram is, according to the above definition, a collision.
The right-hand side of Figure 45 shows the same process seen from another, Greek,
frame of reference. The Greek observer says that the first object has changed its mo-

* Using 4-vector notation, we can write 𝑣/𝑐 = 𝑝/𝑃0 , where 𝑃0 = 𝐸/𝑐.
mechanics 73

time time 𝑡 τ

𝐸󸀠2 , 𝑝2󸀠
𝐸󸀠1 , 𝑝1󸀠
𝐸
𝑝
𝐸2 , 𝑝2
𝐸1 , 𝑝1
F I G U R E 45
object 1 Space-time
object 2 object 1 object 2 diagrams of
the same
ξ collision for
space 𝑥 two different
observers.

Motion Mountain – The Adventure of Physics
mentum before the second one. That would mean that there is a short interval when
momentum and energy are not conserved!
The only way to make sense of the situation is to assume that there is an exchange
of a third object, drawn with a dotted line. Let us find out what the properties of this
object are. We give numerical subscripts to the masses, energies and momenta of the
Challenge 93 e two bodies, and give them a prime after the collision. Then the unknown mass 𝑚 obeys

1 − 𝑣1 𝑣1󸀠
𝑚2 𝑐4 = (𝐸1 − 𝐸󸀠1 )2 − (𝑝1 − 𝑝1󸀠 )2 𝑐2 = 2𝑚21 𝑐4 − 2𝐸1 𝐸󸀠1 (

This is a strange result, because it means that the unknown mass is an imaginary num-
ber!* On top of that, we also see directly from the second graph that the exchanged object
moves faster than light. It is a tachyon, from the Greek ταχύς ‘rapid’. In other words,

⊳ Collisions involve motion that is faster than light.

We will see later that collisions are indeed the only processes where tachyons play a role
in nature. Since the exchanged objects appear only during collisions, never on their own,
they are called virtual objects, to distinguish them from the usual, real objects, which we
Vol. IV, page 64 observe everyday.** We will study the properties of virtual particles later on, when we
come to discuss quantum theory.

* It is usual to change the mass–energy and mass–momentum relation of tachyons to 𝐸 = ±𝑐2 𝑚/√𝑣2 /𝑐2 − 1
and 𝑝 = ±𝑚𝑣/√𝑣2 /𝑐2 − 1 ; this amounts to a redefinition of 𝑚. After the redefinition, tachyons have real
mass. The energy and momentum relations show that tachyons lose energy and momentum when they get
faster. (Provocatively, a single tachyon in a box could provide humanity with all the energy we need.) Both
signs for the energy and momentum relations must be retained, because otherwise the equivalence of all
inertial observers would not be generated. Tachyons thus do not have a minimum energy or a minimum
momentum.
** More precisely, a virtual particle does not obey the relation 𝑚2 𝑐4 = 𝐸2 − 𝑝2 𝑐2 , valid for real particles.
74 2 relativistic

In nature, a tachyon is always a virtual object. Real objects are always bradyons – from
the Greek βραδύς ‘slow’ – or objects moving slower than light. Note that tachyons, des-
pite their high velocity, do not allow the transport of energy faster than light; and that
they do not violate causality if and only if they are emitted or absorbed with equal prob-
Challenge 94 e ability. Can you confirm all this?
When we will study quantum theory, we will also discover that a general contact in-
teraction between objects is described not by the exchange of a single virtual object, but
by a continuous stream of virtual particles. For standard collisions of everyday objects,
the interaction turns out to be electromagnetic. In this case, the exchanged particles are
virtual photons. In other words, when one hand touches another, when it pushes a stone,
or when a mountain supports the trees on it, streams of virtual photons are continuously
Vol. IV, page 64 exchanged.
Ref. 84 As a curiosity, we mention that the notion of relative velocity exists also in relativity.
Challenge 95 e Given two particles 1 and 2, the magnitude of the relative velocity is given by

√(v1 − v2 )2 − (v1 × v2 )2 /𝑐2

Motion Mountain – The Adventure of Physics
𝑣rel = . (44)
1 − v1 ⋅ v2 /𝑐2

The value is never larger than 𝑐, even if both particles depart into opposite directions
with ultrarelativistic speed. The expression is also useful for calculating the relativistic
cross sections for particle collisions. If we determine the relative 4-velocity, we get the
interesting result that in general, v12 ≠ −v21 , i.e., the two relative velocities do not point
in opposite directions – except when the particle velocities are collinear. Nevertheless,
Challenge 96 e the relation 𝑣12 = 𝑣21 = 𝑣rel is satisfied in all cases.
There is an additional secret hidden in collisions. In the right-hand side of Figure 45,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the tachyon is emitted by the first object and absorbed by the second one. However, it is
Challenge 97 s easy to imagine an observer for which the opposite happens. In short, the direction of
travel of a tachyon depends on the observer! In fact, this is a hint about antimatter. In
space-time diagrams, matter and antimatter travel in opposite directions. The connection
Vol. IV, page 192 between relativity and antimatter will become more apparent in quantum theory.

Systems of particles – no centre of mass
Relativity also forces us to eliminate the cherished concept of centre of mass. We can see
this already in the simplest example possible: that of two equal objects colliding.
Figure 46 shows that from the viewpoint in which one of two colliding particles is at
rest, there are at least three different ways to define the centre of mass. In other words, the
Ref. 85 centre of mass is not an observer-invariant concept. We can deduce from the figure that
the concept only makes sense for systems whose components move with small velocities
relative to each other. An atom is an example. For more general systems, centre of mass
is not uniquely definable.
Will the issues with the centre of mass hinder us in our adventure? No. We are more
interested in the motion of single particles than that of composite objects or systems.
mechanics 75

A CM-0 B
𝑣 𝑣

transformed CM
A CM-1 B

𝑣=0 𝑣 2𝑣/(1 + 𝑣2 /𝑐2 )

geometrical CM
A CM-2 B

𝑣=0 𝑣/(1 + 𝑣2 /𝑐2 ) 2𝑣/(1 + 𝑣2 /𝑐2 )

Motion Mountain – The Adventure of Physics
momentum CM
A CM-3 B
F I G U R E 46 There is no consistent
𝑣=0 2𝑣/√1 − 𝑣2 /𝑐2 2𝑣/(1 + 𝑣2 /𝑐2 ) way to deﬁne a relativistic centre
of mass.

Why is most motion so slow?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
For most everyday systems, dilation factors 𝛾 are very near to 1; noticeable departures
from 1, thus speeds of more than a few per cent of the speed of light, are uncommon. Most
such situations are microscopic. We have already mentioned the electrons inside a particle
accelerator or inside a cathode ray tube found in the first colour televisions. The particles
making up cosmic radiation are another example; cosmic rays are important because
their high energy has produced many of the mutations that are the basis of evolution of
animals and plants on this planet. Later we will discover that the particles involved in
radioactivity are also relativistic.
But why don’t we observe any relativistic macroscopic bodies? Because the universe
exists since as long time! Bodies that collide with relativistic velocities undergo processes
not found in everyday life: when they collide, part of their kinetic energy is converted
into new matter via 𝐸 = 𝑐2 𝛾𝑚. In the history of the universe this has happened so many
times that practically all macroscopic bodies move with low speed with respect to their
environment, and practically all of the bodies still in relativistic motion are microscopic
particles.
A second reason for the disappearance of rapid relative motion is radiation damping.
Can you imagine what happens to relativistic charges during collisions, or in a bath of
Challenge 98 s light? Radiation damping also slows down microscopic particles.
In short, almost all matter in the universe moves with small velocity relative to other
matter. The few known counter-examples are either very old, such as the quasar jets men-
tioned above, or stop after a short time. For example, the huge energies necessary for
76 2 relativistic

macroscopic relativistic motion are available in supernova explosions, but the relativ-
istic motion ceases to exist after a few weeks. In summary, the universe is mainly filled
Page 230 with slow motion because it is old. In fact, we will determine its age shortly.

The history of the mass–energy equivalence formula
Albert Einstein took several months after his first paper on special relativity to deduce
the expression
𝐸 = 𝑐2 𝛾𝑚 (45)

which is often called the most famous formula of physics. We write it in this slightly un-
usual, but clear way to stress that 𝑐2 is a unit-dependent and thus unimportant factor.
Such factors are always put first in physical formulae.* Einstein published this formula
Ref. 19 in a separate paper towards the end of 1905. Arguably, the formula could have been dis-
covered thirty years earlier, from the theory of electromagnetism.
In fact, several persons deduced similar results before Einstein. In 1903 and 1904, be-

Motion Mountain – The Adventure of Physics
Ref. 86 fore Einstein’s first relativity paper, Olinto De Pretto, a little-known Italian engineer, cal-
culated, discussed and published the formula 𝐸 = 𝑐2 𝑚. It might well be that Einstein got
the idea for the formula from De Pretto,** possibly through Einstein’s friend Michele
Besso or other Italian-speaking friends he met when he visited his parents, who were
living in Italy at the time. Of course, the value of Einstein’s efforts is not diminished by
this.
Ref. 86 In fact, a similar formula had also been deduced in 1904 by Friedrich Hasenöhrl and
published again in Annalen der Physik in 1905, before Einstein, though with an incor-
rect numerical factor, due to a calculation mistake. The formula 𝐸 = 𝑐2 𝑚 is also part of
several expressions in two publications in 1900 by Henri Poincaré. Also Paul Langevin

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
knew the formula, and Einstein said of him that he would surely have discovered the
theory of special relativity had it not been done before. Also Tolver Preston discussed
the equivalence of mass and energy, already in 1875, in his book Physics of the Ether. The
real hero of the story might be the Swiss chemist Jean Charles Gallisard de Marignac;
already in 1861 he published the now accepted idea about the formation of the elements:
whenever protons form elements, the condensation leads to a lower total mass, and the
energy difference is emitted as energy. The mass–energy equivalence was thus indeed
floating in the air, waiting to be understood and put into the correct context.
Vol. V, page 146 In the 1970s, a similar story occurred: a simple relation between the acceleration and
the temperature of the vacuum was discovered. The result had been waiting to be dis-
covered for over 50 years. Indeed, a number of similar, anterior results were found in the
Challenge 99 s libraries. Could other simple relations be hidden in modern physics waiting to be found?

4-vectors
How can we describe motion consistently for all observers, even for those moving at
speeds near that of light? We have to introduce a simple idea: 4-vectors. We already know

* Examples are 𝐴 = 4π𝑟2 , 𝑎 = 𝐺𝑚/𝑟2 , 𝑈 = 𝑅𝐼, 𝐹 = (1/4π𝜀0 )𝑄2 /𝑟2 , 𝑝𝑉 = 𝑅𝑇 or 𝑆 = 𝑘 ln 𝑊.
** Umberto Bartocci, mathematics professor of the University of Perugia in Italy, published the details of
Ref. 87 this surprising story in several papers and in a book.
mechanics 77

Inside the lightcone, or
future and past:
events with timelike
interval from event E Lightcone:
events with
t
time null interval
from event E
fu
tu

T
re
lig

future
ht

Outside the lightcone,
co

or elsewhere: events with
ne

spacelike interval from E
elsewhere
E space
y
ne
co
ht

space
lig
st

x
pa

past

Motion Mountain – The Adventure of Physics
F I G U R E 47 The space-time
diagram of a moving object T,
with one spatial dimension
missing.

that the motion of a particle can be seen as a sequence of events. Events are points in
space-time. To describe events with precision, we introduce event coordinates, also called
4-coordinates. These are written as

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑋 = (𝑐𝑡, 𝑥) = (𝑐𝑡, 𝑥, 𝑦, 𝑧) = 𝑋𝑖 . (46)

In this way, an event is a point in four-dimensional space-time, and is described by four
coordinates. The four coordinates are called the zeroth, namely time 𝑋0 = 𝑐𝑡, the first,
usually called 𝑋1 = 𝑥, the second, 𝑋2 = 𝑦, and the third, 𝑋3 = 𝑧. In fact, 𝑋 is the simplest
example of a 4-vector. The usual vectors 𝑥 of Galilean physics are also called 3-vectors.
We see that time is treated like the zeroth of four dimensions.
We can now define a space-time distance or space-time interval between two events as
the length of the difference vector 𝑋. In fact, we usually use the square of the length, the
magnitude, to avoid those unwieldy square roots. In special relativity, the magnitude 𝑋2
of any 4-vector 𝑋 is defined as

𝑋2 = 𝑋0 2 − 𝑋1 2 − 𝑋2 2 − 𝑋3 2 = 𝑐2 𝑡2 − 𝑥2 − 𝑦2 − 𝑧2 = 𝑋𝑎 𝑋𝑎 = 𝜂𝑎𝑏 𝑋𝑎 𝑋𝑏 = 𝜂𝑎𝑏 𝑋𝑎 𝑋𝑏 .(47)

The squared space-time interval is thus the squared time interval minus the squared
Page 42 length interval. We have seen above that this minus sign results from the invariance of
the speed of light. In contrast to a squared space interval, a squared space-time interval
can be positive, negative or even zero.
How can we imagine the space-time interval? The magnitude of the space-time inter-
val is the square of 𝑐 times the proper time. The proper time is the time shown by a clock
78 2 relativistic

moving in a straight line and with constant velocity between two events in space-time.
For example, if the start and end events in space-time require motion with the speed of
light, the proper time and the space-time interval vanish. This situation defines the so-
called null vectors or light-like intervals. We call the set of all null vector end points the
Page 47 light cone; it is shown in Figure 47. If the motion between two events is slower than the
speed of light, the squared proper time is positive and the space-time interval is called
time-like. For negative space-time intervals the interval is called space-like. In this last
case, the negative of the magnitude, which then is a positive number, is called the squared
proper distance. The proper distance is the length measured by an odometer as the object
moves along.
We note that the definition of the light cone, its interior and its exterior, are observer-
Challenge 100 e invariant. We therefore use these concepts regularly.
In the definition for the space-time interval we have introduced for the first time two
notations that are useful in relativity. First of all, we automatically sum over repeated
indices. Thus, 𝑋𝑎 𝑋𝑎 means the sum of all products 𝑋𝑎 𝑋𝑎 as 𝑎 ranges over all indices.
Secondly, for every 4-vector 𝑋 we distinguish two ways to write the coordinates, namely

Motion Mountain – The Adventure of Physics
coordinates with superscripts and coordinates with subscripts. (For 3-vectors, we only
use subscripts.) They are related by the following general relation

𝑋𝑏 = (𝑐𝑡, 𝑥, 𝑦, 𝑧)
𝑋𝑎 = (𝑐𝑡, −𝑥, −𝑦, −𝑧) = 𝜂𝑎𝑏 𝑋𝑏 , (48)

where we have introduced the so-called metric 𝜂𝑎𝑏 , an abbreviation of the matrix*

1 0 0 0

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑎𝑏 0 −1 0 0
𝜂 = 𝜂𝑎𝑏 = ( ) . (49)
0 0 −1 0
0 0 0 −1

Don’t panic: this is all, and it won’t get more difficult! (A generalization of this matrix is
used later on, in general relativity.) We now go back to physics; in particular, we are now
ready to describe motion in space-time.

4-velo cit y
We now define velocity of an body in a way that is useful for all observers. We cannot
define the velocity as the derivative of its coordinates with respect to time, since time
and temporal sequences depend on the observer. The solution is to define all observables
with respect to the just-mentioned proper time 𝜏, which is defined as the time shown by
a clock attached to the body. In relativity, motion and change are always measured with
respect to clocks attached to the moving system.
Therefore the relativistic velocity or 4-velocity 𝑈 of an body is defined as the rate of

Ref. 88 * This is the so-called time-like convention, used in about 70 % of all physics texts worldwide. Note that 30 %
of all physics textbooks use the negative of 𝜂 as the metric, the so-called space-like convention, and thus have
opposite signs in this definition.
mechanics 79

change of its 4-coordinates 𝑋 = (𝑐𝑡, 𝑥) with respect to proper time, i.e., as

d𝑋
𝑈= . (50)
d𝜏
The coordinates 𝑋 are measured in the coordinate system defined by the chosen inertial
observer. The value of the 4-velocity 𝑈 depends on the observer or coordinate system
used, as does usual velocity in everyday life. Using d𝑡 = 𝛾 d𝜏 and thus

d𝑥 d𝑥 d𝑡 d𝑥 1
= =𝛾 , where as usual 𝛾= , (51)
d𝜏 d𝑡 d𝜏 d𝑡 √1 − 𝑣2 /𝑐2

we get the relation of 4-velocity with the 3-velocity 𝑣 = d𝑥/d𝑡:

𝑈0 = 𝛾𝑐 , 𝑈𝑖 = 𝛾𝑣𝑖 or 𝑈 = (𝛾𝑐, 𝛾𝑣) . (52)

Motion Mountain – The Adventure of Physics
For small velocities we have 𝛾 ≈ 1, and then the last three components of the 4-velocity
are those of the usual, Galilean 3-velocity. For the magnitude of the 4-velocity 𝑈 we find
𝑈𝑈 = 𝑈𝑎 𝑈𝑎 = 𝜂𝑎𝑏 𝑈𝑎 𝑈𝑏 = 𝑐2 , which is therefore independent of the magnitude of the
3-velocity 𝑣 and makes it a time-like vector, i.e., a vector inside the light cone.
In general, a 4-vector is defined as a quantity (𝐻0 , 𝐻1 , 𝐻2 , 𝐻3 ) that transforms under
boosts as

𝐻𝑉0 = 𝛾𝑉 (𝐻0 − 𝐻1 𝑉/𝑐)
𝐻𝑉1 = 𝛾𝑉 (𝐻1 − 𝐻0 𝑉/𝑐)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝐻𝑉2 = 𝐻2
𝐻𝑉3 = 𝐻3 (53)

when changing from one inertial observer to another moving with a relative velocity
𝑉 in the 𝑥 direction; the corresponding generalizations for the other coordinates are
understood. This relation allows us to deduce the relativistic transformation laws for any
Challenge 101 s 3-vector. Can you deduce the 3-velocity composition formula (10) from this definition?
We know that the magnitude of a 4-vector can be zero even though all its components
are different from zero. Such a vector is called null. Which motions have a null velocity
Challenge 102 s vector?

4-acceleration and proper acceleration
Similarly to 4-velocity, the 4-acceleration 𝐵 of a body is defined as

d𝑈 d2 𝑋
𝐵= = . (54)
d𝜏 d𝜏2
80 2 relativistic

Using d𝛾/d𝜏 = 𝛾d𝛾/d𝑡 = 𝛾4 𝑣𝑎/𝑐2 , we get the following relations between the four com-
Ref. 89 ponents of 𝐵 and the 3-acceleration 𝑎 = d𝑣/d𝑡:

𝑣𝑎 (𝑣𝑎)𝑣𝑖
𝐵0 = 𝛾 4 , 𝐵𝑖 = 𝛾2 𝑎𝑖 + 𝛾4 . (55)
𝑐 𝑐2

Challenge 103 e The magnitude 𝐵 of the 4-acceleration is easily found via 𝐵𝐵 = 𝜂𝑐𝑑 𝐵𝑐 𝐵𝑑 = −𝛾4 (𝑎2 +
𝛾2 (𝑣𝑎)2 /𝑐2 ) = −𝛾6 (𝑎2 − (𝑣 × 𝑎)2 /𝑐2 ). Note that the magnitude does depend on the value
of the 3-acceleration 𝑎. We see that a body that is accelerated for one inertial observer is
also accelerated for all other inertial observers. We also see directly that 3-accelerations
are not Lorentz invariant, unless the velocities are small compared to the speed of light.

⊳ Different inertial observers measure different 3-accelerations.

This is in contrast to our everyday experience and to Galilean physics, where accelera-
tions are independent of the speed of the observer.

Motion Mountain – The Adventure of Physics
We note that 4-acceleration lies outside the light cone, i.e., that it is a space-like vector.
We also note that 𝐵𝑈 = 𝜂𝑐𝑑 𝐵𝑐 𝑈𝑑 = 0, which means that the 4-acceleration is always
perpendicular to the 4-velocity.*
When the 3-acceleration 𝑎 is parallel to the 3-velocity 𝑣, we get 𝐵 = 𝛾3 𝑎; when 𝑎 is
perpendicular to 𝑣, as in circular motion, we get 𝐵 = 𝛾2 𝑎. We will use this result shortly.
How does the 3-acceleration change from one inertial observer to another? To sim-
plify the discussion, we introduce the so-called comoving observer, the observer for which
a particle is at rest. We call the magnitude of the 3-acceleration for the comoving observer
the comoving or proper acceleration; in this case 𝐵 = (0, 𝑎) and 𝐵2 = −𝑎2 . Proper acce-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
leration describes what the comoving observer feels: proper acceleration describes the
experience of being pushed into the back of the accelerating seat. Proper acceleration is
the most important and useful concept when studying accelerated motion in relativity.
Proper acceleration is an important quantity, because no observer, whatever his speed
relative to the moving body, ever measures a 3-acceleration that is higher than the proper
acceleration, as we will see now.
We can calculate how the value of 3-acceleration 𝑎 measured by a general inertial ob-
Ref. 91 server is related to the proper acceleration 𝑎c measured by the comoving observer using
expressions (55) and (53). In this case 𝑣 is both the relative speed of the two observers

* Similarly, the 4-jerk 𝐽 of a body is defined as

𝐽 = d𝐵/d𝜏 = d2 𝑈/d𝜏2 . (56)

Challenge 104 e For the relation with the 3-jerk 𝑗 = d𝑎/d𝑡 we then get

𝛾5 (𝑣𝑎)2 𝛾5 (𝑣𝑎)2 𝑣𝑖
𝐽 = (𝐽0 , 𝐽𝑖 ) = ( (𝑗𝑣 + 𝑎2 + 4𝛾2 2 ) , 𝛾3 𝑗𝑖 + 2 ((𝑗𝑣)𝑣𝑖 + 𝑎2 𝑣𝑖 + 4𝛾2 + 3(𝑣𝑎)𝑎𝑖) ) (57)
𝑐 𝑐 𝑐 𝑐2

Challenge
Page
10594e which we will use later on. Surprisingly, 𝐽 does not vanish when the 3-jerk 𝑗 vanishes. Why not? For this
Ref. 90 reason, slightly amended definitions of 4-jerk have been proposed.
mechanics 81

time
(𝐸/𝑐, 𝑝)

space

F I G U R E 48 Energy–momentum is tangent
to the world line.

Motion Mountain – The Adventure of Physics
and the speed of the accelerated particle. We get

1 (𝑎c 𝑣)2
𝑎2 = (𝑎2
− ) , (58)
𝛾𝑣4 c 𝑐2

Page 80 which we know already in a slightly different form. It shows (again):

⊳ The comoving or proper 3-acceleration is always larger than the 3-

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acceleration measured by any other inertial observer.

that the comoving or proper 3-acceleration is always larger than the 3-acceleration meas-
ured by any other inertial observer. The faster an inertial observer is moving relative to
Challenge 106 e the accelerated system, the smaller the 3-acceleration he observes. The expression also
confirms that whenever the speed is perpendicular to the acceleration, a boost yields a
factor 1/𝛾𝑣2 , whereas a speed parallel to the acceleration yields a factor 1/𝛾𝑣3 .
The maximum property of proper acceleration implies that accelerations, in contrast
to velocities, cannot be called relativistic. In other words, accelerations require relativistic
treatment only when the involved velocities are relativistic. If the velocities involved are
low, even the highest accelerations can be treated with Galilean physics.

4-momentum or energy–momentum or momenergy
To describe motion, we need the concept of momentum. The 4-momentum is defined as

𝑃 = 𝑚𝑈 (59)

and is therefore related to the 3-momentum 𝑝 by

𝑃 = (𝛾𝑚𝑐, 𝛾𝑚𝑣) = (𝐸/𝑐, 𝑝) . (60)
82 2 relativistic

For this reason 4-momentum is also called the energy–momentum 4-vector. In short,

⊳ The 4-momentum or energy–momentum of a body is given by its mass
times the 4-displacement per proper time.

This is the simplest possible definition of momentum and energy. The concept was in-
troduced by Max Planck in 1906.
The energy–momentum 4-vector, sometimes also called momenergy, is, like the 4-
velocity, tangent to the world line of a particle. This connection, shown in Figure 48,
follows directly from the definition, since

(𝐸/𝑐, 𝑝) = (𝛾𝑚𝑐, 𝛾𝑚𝑣) = 𝑚(𝛾𝑐, 𝛾𝑣) = 𝑚(𝑐d𝑡/d𝜏, d𝑥/d𝜏) . (61)

The (square of the) length of momenergy, namely 𝑃𝑃 = 𝜂𝑎𝑏 𝑃𝑎 𝑃𝑏 , is, like any squared
length of a 4-vector, the same for all inertial observers; it is found to be

Motion Mountain – The Adventure of Physics
𝐸2 /𝑐2 − 𝑝2 = 𝑐2 𝑚2 , (62)

thus confirming a result given above. We have already mentioned that energies or situ-
ations are called relativistic if the kinetic energy 𝑇 = 𝐸 − 𝐸0 is not negligible when com-
pared to the rest energy 𝐸0 = 𝑐2 𝑚. A particle whose kinetic energy is much higher than
its rest mass is called ultrarelativistic. Particles in accelerators or in cosmic rays fall into
Challenge 107 s this category. What is their energy–momentum relation?
The conservation of energy, momentum and mass of Galilean mechanics thus merge,
in special relativity, into the conservation of momenergy:

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⊳ In nature, energy–momentum, or momenergy, is conserved.

In particular, mass is not a conserved quantity any more.
In contrast to Galilean mechanics, relativity implies an absolute zero for the energy.
We cannot extract more energy than 𝑐2 𝑚 from a system of mass 𝑚 at rest. In particular,
an absolute zero value for potential energy is fixed in this way. In short, relativity shows
that energy is bounded from below. There is no infinite amount of energy available in
nature.
Not all Galilean energy contributes to mass: potential energy in an outside field does
not. Relativity forces us into precise energy bookkeeping. We keep in mind for later that
‘potential energy’ in relativity is an abbreviation for ‘energy reduction of the outside
field’.
Can you show that for two particles with 4-momenta 𝑃1 and 𝑃2 , one has 𝑃1 𝑃2 =
𝑚1 𝐸2 = 𝑚2 𝐸1 = 𝑐2 𝛾12 𝑚1 𝑚2 , where 𝛾12 is the Lorentz factor due to their relative velocity
Challenge 108 s 𝑣12 ?
Note that by the term ‘mass’ 𝑚 we always mean what is sometimes called the rest
mass. This name derives from the bad habit of many science fiction and secondary-school
books of calling the product 𝛾𝑚 the relativistic mass. Workers in the field usually (but not
Ref. 92 unanimously) reject this concept, as did Einstein himself, and they also reject the often-
mechanics 83

heard expression that ‘(relativistic) mass increases with velocity’. Relativistic mass and
energy would then be two words for the same concept: this way to talk is at the level of
the tabloid press.

4-force – and the nature of mechanics
The 4-force 𝐾 is defined with 4-momentum 𝑃 as

𝐾 = d𝑃/d𝜏 = 𝑚𝐵 , (63)

where 𝐵 is 4-acceleration. Therefore force remains equal to mass times acceleration in
Ref. 89, Ref. 93 relativity. From the definition of 𝐾 we deduce the relation with 3-force 𝐹 = d𝑝/d𝑡 =
𝑚d(𝛾𝑣)/d𝑡, namely*

𝑚𝑣𝑎 2 𝑚𝑣𝑎 𝛾 d𝐸 d𝑝 𝐹𝑣
𝐾 = (𝐾0 , 𝐾𝑖 ) = (𝛾4 , 𝛾 𝑚𝑎𝑖 + 𝛾4 𝑣𝑖 2 ) = ( , 𝛾 ) = (𝛾 , 𝛾𝐹) . (64)
𝑐 𝑐 𝑐 d𝑡 d𝑡 𝑐

Motion Mountain – The Adventure of Physics
Challenge 109 e The 4-force, like the 4-acceleration, is orthogonal to the 4-velocity. The meaning of the
zeroth component of the 4-force can easily be discerned: it is the power required to ac-
celerate the object. Indeed, we have 𝐾𝑈 = 𝑐2 d𝑚/d𝜏 = 𝛾2 (d𝐸/d𝑡 − 𝐹𝑣): this is the proper
rate at which the internal energy of a system increases. The product 𝐾𝑈 vanishes only
for rest-mass-conserving forces. Many particle collisions lead to reactions and thus do
not belong to this class of forces; such collisions and forces do not conserve rest mass. In
everyday life however, the rest mass is preserved, and then we get the Galilean expression
for power given by 𝐹𝑣 = d𝐸/d𝑡.
Challenge 110 s For rest-mass-preserving forces we get 𝐹 = 𝛾𝑚𝑎 + (𝐹𝑣)𝑣/𝑐2 . In other words, in the

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general case, 3-force and 3-acceleration are neither parallel nor proportional to each
other. In contrast, we saw above that 3-momentum is parallel, but not proportional to
3-velocity.
We note that 3-force has the largest possible value, the proper force, in the comoving
frame. A boost keeps the component of the force in the direction of the boost unchanged,
Challenge 111 e and reduces the components in the perpendicular directions. In particular, boost cannot
be used to increase 3-force values beyond all bounds. (Though they appear to allow to
increase the value of 4-force beyond all bounds.) The situation somewhat resembles the
Page 80 situation for 3-acceleration, though the transformation behaviour differs.
The 4-force can thus also be called the power–force 4-vector. In Galilean mechanics,
when we defined force, we also explored potentials. However, we cannot do this easily
in special relativity. In contrast to Galilean mechanics, where interactions and poten-
tials can have almost any desired behaviour, special relativity has strict requirements for
them. There is no way to define potentials and interactions in a way that makes sense
Ref. 94 for all observers – except if the potentials are related to fields that can carry energy and
momentum. In other terms,

⊳ Relativity only allows potentials related to radiation.

* Some authors define 3-force as d𝑝/d𝜏; then 𝐾 looks slightly different.
84 2 relativistic

A 𝑣

𝑣
𝑣󸀠
B 𝑣󸀠
D
F I G U R E 49 On the deﬁnition of
relative velocity (see text).

In fact, only two type of potentials are allowed by relativity in everyday life: those due
to electromagnetism and those due to gravity. (In the microscopic domain, also the two
nuclear interactions are possible.) In particular, this result implies that when two every-

Motion Mountain – The Adventure of Physics
day objects collide, the collision is either due to gravitational or to electric effects. To
put it even more bluntly: relativity forbids ‘purely mechanical’ interactions. Mechanics
is not a fundamental part of nature. Indeed, in the volume on quantum theory we will
confirm that everything that we call mechanical in everyday life is, without exception,
electromagnetic. Every caress and every kiss is an electromagnetic process. To put it in
another way, and using the fact that light is an electromagnetic process, we can say: if we
bang any two objects hard enough onto each other, we will inevitably produce light.
The inclusion of gravity into relativity yields the theory of general relativity. In general
relativity, the just defined power–force vector will play an important role. It will turn out
that in nature, the 3-force 𝐹 and the 3-power 𝐹𝑣 are limited in magnitude. Can you guess

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 112 d how?

Rotation in relativity
If at night we turn around our own axis while looking at the sky, the stars move with a
velocity much higher than that of light. Most stars are masses, not images. Their speed
should be limited by that of light. How does this fit with special relativity?
This example helps to clarify in another way what the limit velocity actually is. Physic-
ally speaking, a rotating sky does not allow superluminal energy transport, and thus does
not contradict the concept of a limit speed. Mathematically speaking, the speed of light
limits relative velocities only between objects that come near to each other, as shown on
the left of Figure 49. To compare velocities of distant objects, like between ourselves and
the stars, is only possible if all velocities involved are constant in time; this is not the
case if we turn. The differential version of the Lorentz transformations make this point
particularly clear. Indeed, the relative velocities of distant objects are frequently higher
Page 60 than the speed of light. We encountered one example earlier, when discussing the car in
Page 100 the tunnel, and we will encounter more examples shortly.
With this clarification, we can now briefly consider rotation in relativity. The first ques-
tion is how lengths and times change in a rotating frame of reference. You may want to
check that an observer in a rotating frame agrees with a non-rotating colleague on the
radius of a rotating body; however, both find that the rotating body, even if it is rigid, has
mechanics 85

O3 O2 O
1
On
On–1

F I G U R E 50 Observers on a rotating object.

Challenge 113 e a circumference different from the one it had before it started rotating. Sloppily speaking,
the value of π changes for rotating observers! For the rotating observer, the ratio between

Motion Mountain – The Adventure of Physics
the circumference 𝑐 and the radius 𝑟 turns out to be 𝑐/𝑟 = 2π𝛾: the ratio increases with
Challenge 114 e rotation speed. This counter-intuitive result is often called Ehrenfest’s paradox. It shows
Ref. 95 that space-time for a rotating observer is not the flat Minkowski space-time of special
relativity. The paradox also shows that rigid bodies do not exist.
Rotating bodies behave strangely in many ways. For example, we get into trouble when
we try to synchronize clocks mounted on a rotating circle, as shown in Figure 50. If we
start synchronizing the clock at position O2 with that at O1 , and so on, continuing up
to last clock On , we find that the last clock is not synchronized with the first. This result
reflects the change in circumference just mentioned. In fact, a careful study shows that

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the measurements of length and time intervals lead all observers Ok to conclude that
they live in a rotating space-time, one that is not flat. Rotating discs can thus be used
as an introduction to general relativity, where spatial curvature and its effects form the
central topic. More about this in the next chapter.
Ref. 26 In relativity, rotation and translation combine in strange ways. Imagine a cylinder
in uniform rotation along its axis, as seen by an observer at rest. As Max von Laue has
discussed, the cylinder will appear twisted to an observer moving along the rotation axis.
Challenge 115 e Can you confirm this?
For train lovers, here is a well-known puzzle. A train travels on a circular train track.
The train is as long as the track, so that it forms a circle. What happens if the same train
runs at relativistic speeds: does the train fall out of the track, remain on the track or fall
Challenge 116 s inside the track?
Is angular velocity limited? Yes: the tangential speed in an inertial frame of reference
cannot reach that of light. The limit on angular velocity thus depends on the size of the
body in question. That leads to a neat puzzle: can we see an object that rotates very rap-
Challenge 117 s idly?
We mention that 4-angular momentum is defined naturally as

𝑙𝑎𝑏 = 𝑥𝑎 𝑝𝑏 − 𝑥𝑏 𝑝𝑎 . (65)

The two indices imply that the 4-angular momentum is a tensor, not a vector. Angular
86 2 relativistic

Challenge 118 e momentum is conserved, also in special relativity. The moment of inertia is naturally
defined as the proportionality factor between angular velocity and angular momentum.
By the way, how would you determine whether a microscopic particle, too small to be
Challenge 119 s seen, is rotating?
For a rotating particle, the rotational energy is part of the rest mass. You may want to
Challenge 120 e calculate the fraction for the Earth and the Sun. It is not large.
Here are some puzzles about relativistic rotation. We know that velocity is relative:
its measured value depends on the observer. Is this the case also for angular velocity?
Challenge 121 s What is the expression for relativistic rotational energy, and for its relation to 4-angular
Challenge 122 s momentum?
Rotation also yields the rotational Doppler effect. To observe it is tricky but nowadays
a regular feat in precision laser laboratories. To see it, one needs a circularly polarized
light beam; such beams are available in many laboratories. When such a light beam is re-
flected from a polarizable rotating surface, the frequency of the reflected beam is shifted
in a certain percentage of the light. This rotational Doppler shift is given by the rotation
frequency of the surface. The effect is important in the theory of the Faraday effect; it

Motion Mountain – The Adventure of Physics
has already been used to measure the rotation of various optical elements and even the
rotation of molecules. One day, the effect might be useful in engineering or astronomy,
to measure the rotation velocity of distant or delicate spinning objects.

Wave motion
Vol. I, page 293 Waves also move. We saw in Galilean physics that a harmonic or sine wave is described,
among others, by an angular frequency 𝜔 = 2π𝜈 and by a wave vector 𝑘, with 𝑘 = 2π/𝜆.
In special relativity, the two quantities are combined in the wave 4-vector 𝐿 that is given
by

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝜔
𝐿𝑎 = ( , 𝑘) . (66)
𝑐

As usual, the phase velocity of a harmonic wave is 𝜔/𝑘 = 𝜆𝜈. The wave 4-vector for light
has magnitude 0, it is a null vector. For slower waves, such as sound waves, the wave
Challenge 123 e 4-vector is time-like.
The phase 𝜑 of a wave can now be defined as

𝜑 = 𝐿 𝑎 𝑥𝑎 = 𝐿𝑎 𝑥𝑎 . (67)

Being a scalar, as expected, the phase of any wave, be it light, sound or any other type, is
Challenge 124 e the same for all observers: the phase is a relativistic invariant.*
Suppose an observer with 4-velocity 𝑈 finds that a wave with wave 4-vector 𝐿 has
frequency 𝜈. Show that
𝜈 = 𝐿𝑈 (68)

Challenge 125 s must be obeyed.

* In component notation, the important relations are (𝜔/𝑐, k)(𝑐𝑡, x) = 𝜑, then (𝜔/𝑐, k)(𝑐, vphase ) = 0 and
finally (d𝜔/𝑐, dk)(𝑐, vgroup ) = 0.
mechanics 87

time

1h30min
B
1h29min 1h29min
1h28min 1h28min

space

0 150 Gm

A
F I G U R E 51 The straight motion between

Motion Mountain – The Adventure of Physics
two points A and B is the motion that
requires the longest proper time.

Interestingly, the wave phase 4-velocity 𝜔/𝑘 transforms in a different way than particle
Ref. 25 velocity, except in the case 𝜔/𝑘 = 𝑐. Also the aberration formula for wave motion differs
Challenge 126 ny from that for particle motion, except in the case 𝜔/𝑘 = 𝑐. Can you find the two relations?

The action of a free particle – how d o things move?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
If we want to describe relativistic motion of a free particle in terms of the least action
Vol. I, page 248 principle, we need a definition of the action. We already know that physical action is a
measure of the change occurring in a system. For an inertially moving or free particle, the
only change is the ticking of its proper clock. As a result, the action of a free particle will
be proportional to the elapsed proper time. In order to get the standard unit of energy
times time, or Js, for the action, the obvious guess for the action of a free particle is
𝜏2
𝑆 = −𝑐2 𝑚 ∫ d𝜏 , (69)
𝜏1

where 𝜏 is the proper time along its path. This is indeed the correct expression.
In short, in nature,

⊳ All particles move in such a way that the elapsed proper time – or wristwatch
time – is maximal.

In other words, we again find that in nature things change as little as possible. Nature is
like a wise old man: its motions are as slow as possible – it does as little as possible. If you
Vol. I, page 253 prefer, every change in nature is maximally effective. As we mentioned before, Bertrand
Russell called this the ‘law’ of cosmic laziness.
88 2 relativistic

Using the invariance of the speed of light, the principle of least action can thus be
rephrased:

⊳ Bodies idle as much as they can.

Figure 51 shows some examples of values of proper times for a body moving from one
point to another in free space. The straight motion, the one that nature chooses, is the
Page 49 motion with the longest proper time. (Recall the result given above: travelling more keeps
you younger.) However, this difference in proper time is noticeable only for relativistic
speeds and large distances – such as those shown in the figure – and therefore we do not
experience any such effect in everyday, non-relativistic life.
We note that maximum proper time is equivalent to minimum action. Both state-
ments have the same content. Both statements express the principle of least action. For a
free body, the change in proper time is maximal, and the action minimal, for straight-line
motion with constant velocity. The principle of least action thus implies conservation of

Motion Mountain – The Adventure of Physics
Challenge 127 e (relativistic) energy and momentum. Can you confirm this?
The expression (69) for the action is due to Max Planck. In 1906, by exploring it in
detail, he found that the quantum of action ℏ, which he had discovered together with the
Boltzmann constant 𝑘, is a relativistic invariant (like the Boltzmann constant). Can you
Challenge 128 s imagine how he did this?
The action can also be written in more complex, seemingly more frightening ways.
These equivalent ways to write it are particularly appropriate to prepare us for general
relativity:

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑡2 𝜏2 𝑠2
1 d𝑥 d𝑥
𝑆 = ∫ 𝐿 d𝑡 = −𝑐2 𝑚 ∫ d𝑡 = −𝑚𝑐 ∫ √𝑢𝑎 𝑢𝑎 d𝜏 = −𝑚𝑐 ∫ √𝜂𝑎𝑏 𝑎 𝑏 d𝑠 , (70)
𝑡1 𝛾 𝜏1 𝑠1 d𝑠 d𝑠

where 𝑠 is some arbitrary, but monotonically increasing, function of 𝜏, such as 𝜏 itself.
As usual, the metric 𝜂𝛼𝛽 of special relativity is

1 0 0 0
𝑎𝑏 0 −1 0 0
𝜂 = 𝜂𝑎𝑏 = ( ) . (71)
0 0 −1 0
0 0 0 −1

You can easily confirm the form of the action (70) by deducing the equation of motion
Challenge 129 e in the usual way.
In short, nature is not in a hurry: every object moves in a such way that its own clock
shows the longest delay possible, compared with any alternative motion nearby. This gen-
eral principle is also valid for particles under the influence of gravity, as we will see in
the section on general relativity, and for particles under the influence of electric or mag-
netic interactions. In fact, the principle of maximum proper time, i.e., the least action
principle, is valid in all cases of motion found in nature, as we will discover step by step.
For the moment, we just note that the longest proper time is realized when the average
Challenge 130 e difference between kinetic and potential energy is minimal. (Can you confirm this?) We
mechanics 89

thus recover the principle of least action in its everyday formulation.
Vol. I, page 248 Earlier on, we saw that the action measures the change going on in a system. Special
relativity shows that nature minimizes change by maximizing proper time. In nature,
proper time is always maximal. In other words, things move along paths defined by the
principle of maximal ageing. Can you explain why ‘maximal ageing’ and ‘cosmic laziness’
Challenge 131 e are equivalent?
When you throw a stone, the stone follows more or less a parabolic path. Had it flown
higher, it would have to move faster, which slows down its aging. Had it flown lower, it
Page 149 would also age more slowly, because at lower height you stay younger, as we will see. The
actual path is thus indeed the path of maximum aging.
We thus again find that nature is the opposite of a Hollywood film: nature changes in
the most economical way possible – all motion realizes the smallest possible amount of
action. Exploring the deeper meaning of this result is left to you: enjoy it!

C onformal transformations

Motion Mountain – The Adventure of Physics
The distinction between space and time in special relativity depends on the inertial ob-
server. On the other hand, all inertial observers agree on the position, shape and ori-
entation of the light cone at a point. Thus, in the theory of relativity, the light cones are
the basic physical ‘objects’. For any expert of relativity, space-time is a large collection
of light cones. Given the importance of light cones, we might ask if inertial observers are
the only ones that observe the same light cones. Interestingly, it turns out that additional
observers do as well.
The first category of additional observers that keep light cones invariant are those
using units of measurement in which all time and length intervals are multiplied by a
scale factor 𝜆. The transformations among these observers or points of view are given by

and are called dilations or scaling transformations.
A second category of additional observers are found by applying the so-called special
conformal transformations. These are compositions of an inversion
𝑥𝑎
𝑥𝑎 󳨃→ (73)
𝑥2
with a translation by a 4-vector 𝑏𝑎 , namely

𝑥𝑎 󳨃→ 𝑥𝑎 + 𝑏𝑎 , (74)

Challenge 132 e and a second inversion. Therefore the special conformal transformations are

𝑥𝑎 + 𝑏𝑎 𝑥2
𝑥𝑎 󳨃→ . (75)
1 + 2𝑏𝑎 𝑥𝑎 + 𝑏2 𝑥2

These transformations are called conformal because they do not change angles of (infin-
90 2 relativistic

Challenge 133 e itesimally) small shapes, as you may want to check. The transformations therefore leave
the form (of infinitesimally small objects) unchanged. For example, they transform infin-
itesimal circles into infinitesimal circles, and infinitesimal (hyper-)spheres into infinites-
imal (hyper-)spheres. The transformations are called special because the full conformal
group includes the dilations and the inhomogeneous Lorentz transformations as well.*
Note that the way in which special conformal transformations leave light cones in-
Challenge 135 e variant is rather subtle. Explore the issue!
Since dilations do not commute with time translations, there is no conserved quantity
associated with this symmetry. (The same is true of Lorentz boosts.) In contrast, rotations
and spatial translations do commute with time translations and thus do lead to conserved
quantities.
In summary, vacuum is conformally invariant – in the special sense just mentioned
– and thus also dilation invariant. This is another way to say that vacuum alone is not
sufficient to define lengths, as it does not fix a scale factor. As we would expect, matter
is necessary to do so. Indeed, (special) conformal transformations are not symmetries
of situations containing matter. Vacuum is conformally invariant; nature as a whole is

Motion Mountain – The Adventure of Physics
not.**
However, conformal invariance, or the invariance of light cones, is sufficient to al-
low velocity measurements. Conformal invariance is also necessary for velocity meas-
Challenge 137 e urements, as you might want to check.
We have seen that conformal invariance implies inversion symmetry: that is, that the
large and small scales of a vacuum are related. This suggests that the invariance of the
speed of light is related to the existence of inversion symmetry. This mysterious connec-
tion gives us a first glimpse of the adventures that we will encounter in the final part of
our adventure.

F I G U R E 52 The animation shows an
observer accelerating down the road in a
desert, until he reaches relativistic speeds.
The inset shows the position along the
road. Note how things seem to recede,
despite the advancing motion. (QuickTime
ﬁlm © Anthony Searle and Australian
National University, from www.anu.edu.au/
Physics/Savage/TEE.)

Motion Mountain – The Adventure of Physics
Accelerating observers
So far, we have only studied what inertial, or free-flying, observers say to each other
when they talk about the same observation. For example, we saw that moving clocks
always run slow. The story gets even more interesting when one or both of the observers
are accelerating.
One sometimes hears that special relativity cannot be used to describe accelerating
observers. That is wrong, just as it is wrong to say that Galilean physics cannot be used
for accelerating observers. Special relativity’s only limitation is that it cannot be used
in non-flat, i.e., curved, space-time. Accelerating bodies do exist in flat space-time, and

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
therefore they can be discussed in special relativity.
As an appetizer, let us see what an accelerating, Greek, observer says about the clock
Ref. 96 of an inertial, Roman, one, and vice versa. We assume that the Greek observer, shown in

Challenge 134 e * The set of all special conformal transformations forms a group with four parameters; adding dilations
and the inhomogeneous Lorentz transformations one gets fifteen parameters for the full conformal group.
Mathematically speaking, the conformal group is locally isomorphic to SU(2,2) and to the simple group
Vol. V, page 358 SO(4,2). These concepts are explained later on. Note that all this is true only for four space-time dimensions.
In two dimensions – the other important case – the conformal group is isomorphic to the group of arbitrary
analytic coordinate transformations, and is thus infinite-dimensional.
** A field that has mass cannot be conformally invariant; therefore conformal invariance is not an exact
symmetry of all of nature. Can you confirm that a mass term 𝑚𝜑2 in a Lagrangian density is not conformally
Challenge 136 e invariant?
We note that the conformal group does not appear only in the kinematics of special relativity and thus
is not only a symmetry of the vacuum: the conformal group is also the symmetry group of physical inter-
actions, such as electromagnetism, as long as the involved radiation bosons have zero mass, as is the case
for the photon. In simple words, both the vacuum and all those radiation fields that are made of massless
particles are conformally invariant. Fields due to massive particles are not.
We can go even further. All elementary particles observed up to now have masses that are many orders of
magnitude smaller than the Planck mass √ℏ𝑐/𝐺 . Thus it can be said that they have almost vanishing mass;
conformal symmetry can then be seen as an approximate symmetry of nature. In this view, all massive
particles can be seen as small corrections, or perturbations, of massless, i.e., conformally invariant, fields.
Therefore, for the construction of a fundamental theory, conformally invariant Lagrangians are often as-
sumed to provide a good starting approximation.
92 2 relativistic

observer (Greek)
𝑣
light
𝑐
observer (Roman)
F I G U R E 53 The simplest situation for
an inertial and an accelerated observer.

Figure 53, moves along the path 𝑥(𝑡), as observed by the inertial Roman one. In general,
the Greek–Roman clock rate ratio is given by Δ𝜏/Δ𝑡 = (𝜏2 − 𝜏1 )/(𝑡2 − 𝑡1 ). Here the Greek
coordinates are constructed with a simple procedure: take the two sets of events defined
by 𝑡 = 𝑡1 and 𝑡 = 𝑡2 , and let 𝜏1 and 𝜏2 be the points where these sets intersect the time

Motion Mountain – The Adventure of Physics
axis of the Greek observer.*
We first briefly assume that the Greek observer is also inertial and moving with velo-
city 𝑣 as observed by the Roman one. The clock ratio of a Greek observer is then given
by
Δ𝜏 d𝜏 √ 1
= = 1 − 𝑣2 /𝑐2 = , (76)
Δ𝑡 d𝑡 𝛾𝑣

Challenge 138 e a formula we are now used to. We find again that inertially moving clocks run slow.
For accelerated motions of the Greek observer, the differential version of the above
Ref. 96 reasoning is necessary. The Greek/Roman clock rate ratio is d𝜏/d𝑡, and 𝜏 and 𝜏 + d𝜏 are

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
calculated in the same way from the times 𝑡 and 𝑡 + d𝑡. To do this, we assume again that
the Greek observer moves along the path 𝑥(𝑡), as measured by the Roman one. We find
directly that
𝜏
= 𝑡 − 𝑥(𝑡)𝑣(𝑡)/𝑐2 (77)
𝛾𝑣

and thus
𝜏 + d𝜏
= (𝑡 + d𝑡) − [𝑥(𝑡) + d𝑡𝑣(𝑡)][𝑣(𝑡) + d𝑡𝑎(𝑡)]/𝑐2 . (78)
𝛾𝑣

Together, and to first order, these equations yield

‘d𝜏/d𝑡’ = 𝛾𝑣 (1 − 𝑣𝑣/𝑐2 − 𝑥𝑎/𝑐2 ) . (79)

This result shows that accelerated clocks can run fast or slow, depending on their position
𝑥 and the sign of their acceleration 𝑎. There are quotes in the above equation because we
can see directly that the Greek observer notes

‘d𝑡/d𝜏’ = 𝛾𝑣 , (80)

* These sets form what mathematicians call hypersurfaces.
mechanics 93

F I G U R E 54 An observer
accelerating down a road
in a city. The ﬁlm shows
the 360° view around the
observer; the borders thus
show the situation behind
his back, where the
houses, located near the
event horizon, remain at
constant size and distance.
(Mpg ﬁlm © Anthony
Searle and Australian
National University.)

which is not the inverse of equation (79). This difference becomes most apparent in the

Motion Mountain – The Adventure of Physics
simple case of two clocks with the same velocity, one of which has a constant acceleration
Ref. 96 𝑔 towards the origin, whereas the other moves inertially. We then have

‘d𝜏/d𝑡’ = 1 + 𝑔𝑥/𝑐2 (81)

and
‘d𝑡/d𝜏’ = 1 . (82)

Page 99 We will discuss this situation in more detail shortly. But first we must clarify the concept
of acceleration.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Accelerating frames of reference
How do we check whether we live in an inertial frame of reference? Let us first define
the term. An inertial frame (of reference) has two defining properties. First, lengths and
distances measured with a ruler are described by Euclidean geometry. In other words,
rulers behave as they do in daily life. In particular, distances found by counting how
many rulers (rods) have to be laid down end to end to reach from one point to another –
the so-called rod distances – behave as in everyday life. For example, rod distances obey
Pythagoras’ theorem in the case of right-angled triangles. Secondly, in inertial frames,
the speed of light is invariant. In other words, any two observers in that frame, independ-
ent of their time and of the position, make the following observation: the ratio 𝑐 between
twice the rod distance between two points and the time taken by light to travel from one
point to the other and back is always the same.
Equivalently, an inertial frame is one for which all clocks always remain synchron-
ized and whose geometry is Euclidean. In particular, in an inertial frame all observers at
fixed coordinates always remain at rest with respect to each other. This last condition is,
however, a more general one. There are other, non-inertial, situations where this is still
the case.
Non-inertial frames, or accelerating frames, are a useful concept in special relativity.
In fact, we all live in such a frame. And we can use special relativity to describe motion
94 2 relativistic

in such an accelerating frame, in the same way that we used Galilean physics to describe
it at the beginning of our journey.
A general frame of reference is a continuous set of observers remaining at rest with
Ref. 97 respect to each other. Here, ‘at rest with respect to each other’ means that the time for a
light signal to go from one observer to another and back again is constant over time, or
equivalently, that the rod distance between the two observers is constant. Any frame of
reference can therefore also be called a rigid collection of observers. We therefore note
that a general frame of reference is not the same as a general set of coordinates; the latter
is usually not rigid. But if all the rigidly connected observers have constant coordinate
values, we speak of a rigid coordinate system. Obviously, these are the most useful when
it comes to describing accelerating frames of reference.*
Ref. 97 Note that if two observers both move with a velocity 𝑣, as measured in some inertial
frame, they observe that they are at rest with respect to each other only if this velocity
Challenge 139 e is constant. Again we find, as above, that two people tied to each other by a rope, and at
Page 54 a distance such that the rope is under tension, will see the rope break (or hang loose) if
they accelerate together to (or decelerate from) relativistic speeds in precisely the same

Motion Mountain – The Adventure of Physics
way. Acceleration in relativity requires careful thinking.
Page 66 Can you state how the acceleration ratio enters into the definition of mass in special
Challenge 140 ny relativity?

C onstant acceleration
Acceleration is a tricky topic. An observer who always feels the same force on his body is
called uniformly accelerating. His proper acceleration is constant. More precisely, a uni-
formly accelerating observer is an observer whose acceleration at every moment, meas-
ured by the inertial frame with respect to which the observer is at rest at that moment,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
always has the same value 𝐵. It is important to note that uniform acceleration is not
uniformly accelerating when always observed from the same inertial frame. This is an
important difference from the Galilean case.
For uniformly accelerated motion in the sense just defined, 4-jerk is zero, and we need

𝐵 ⋅ 𝐵 = −𝑔2 , (83)

Ref. 99 where 𝑔 is a constant independent of 𝑡. The simplest case is uniformly accelerating mo-
tion that is also rectilinear, i.e., for which the acceleration 𝑎 is parallel to 𝑣 at one instant of
time and (therefore) for all other times as well. In this case we can write, using 3-vectors,
Challenge 141 e
d𝛾𝑣
𝛾3 𝑎 = 𝑔 or =𝑔. (84)
d𝑡

Ref. 98 * There are essentially only two other types of rigid coordinate frames, apart from the inertial frames:

— The frame d𝑠2 = d𝑥2 + d𝑦2 + d𝑧2 − 𝑐2 d𝑡2 (1 + 𝑔𝑘 𝑥𝑘 /𝑐2 )2 with arbitrary, but constant, acceleration of the
origin. The acceleration is 𝑎 = −𝑔(1 + 𝑔𝑥/𝑐2 ).
— The uniformly rotating frame d𝑠2 = d𝑥2 + d𝑦2 + d𝑧2 + 2𝜔(−𝑦 d𝑥 + 𝑥 d𝑦)d𝑡 − (1 − 𝑟2 𝜔2 /𝑐2 )d𝑡. Here the
𝑧-axis is the rotation axis, and 𝑟2 = 𝑥2 + 𝑦2 .
mechanics 95

𝑡 𝜏

on
II

riz
𝜉

ho
re
tu
fu
Ω
O 𝑐2 /𝑔
III 𝑥
I

pa
st
ho
IV

riz
on
F I G U R E 55 The hyperbolic motion of an
observer Ω that accelerates rectilinearly
and uniformly with acceleration 𝑔.

Motion Mountain – The Adventure of Physics
Challenge 142 e Taking the direction we are talking about to be the 𝑥-axis, and solving for 𝑣(𝑡), we get

𝑔𝑡
𝑣= , (85)
√1 + 𝑔2 𝑡2
𝑐2

where it was assumed that 𝑣(0) = 0. We note that for small times we get 𝑣 = 𝑔𝑡 and for
large times 𝑣 = 𝑐, both as expected. The momentum of the accelerated observer increases
Challenge 143 e linearly with time, again as expected. Integrating, we find that the accelerated observer

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
moves along the path
𝑐2 √ 𝑔2 𝑡2
𝑥(𝑡) = 1+ 2 , (86)
𝑔 𝑐

where we assumed that 𝑥(0) = 𝑐2 /𝑔, in order to keep the expression simple. Because of
this result, visualized in Figure 55, a rectilinearly and uniformly accelerating observer is
said to undergo hyperbolic motion. For small times, the world-line reduces to the usual
𝑥 = 𝑔𝑡2 /2 + 𝑥0 , whereas for large times it is 𝑥 = 𝑐𝑡, as expected. The motion is thus
uniformly accelerated only for the moving body itself, but not for an outside observer,
again as expected.
The proper time 𝜏 of the accelerated observer is related to the time 𝑡 of the inertial
frame in the usual way by d𝑡 = 𝛾d𝜏. Using the expression for the velocity 𝑣(𝑡) of equation
Ref. 99, Ref. 100 (85) we get*
𝑐 𝑔𝜏 𝑐2 𝑔𝜏
𝑡 = sinh and 𝑥 = cosh (87)
𝑔 𝑐 𝑔 𝑐

Ref. 101 * Use your favourite mathematical formula collection – every person should have one – to deduce this. The
hyperbolic sine and the hyperbolic cosine are defined by sinh 𝑦 = (e𝑦 − e−𝑦 )/2 and cosh 𝑦 = (e𝑦 + e−𝑦 )/2.
They imply that ∫ d𝑦/√𝑦2 + 𝑎2 = arsinh 𝑦/𝑎 = Arsh 𝑦/𝑎 = ln(𝑦 + √𝑦2 + 𝑎2 ).
96 2 relativistic

for the relationship between proper time 𝜏 and the time 𝑡 and position 𝑥 measured by
the external, inertial Roman observer. We will encounter this relation again during our
study of black holes.
Does the last formula sound boring? Just imagine accelerating on your motorbike at
𝑔 = 10 m/s2 for the proper time 𝜏 of 25 years. That would bring you beyond the end of the
known universe! Isn’t that worth a try? Unfortunately, neither motorbikes nor missiles
Challenge 144 s that accelerate like this exist, as their fuel tanks would have to be enormous. Can you
confirm this?
For uniform rectilinear acceleration, the coordinates transform as

𝑐 𝜉 𝑔𝜏
𝑡=( + ) sinh
𝑔 𝑐 𝑐
2
𝑐 𝑔𝜏
𝑥 = ( + 𝜉) cosh
𝑔 𝑐
𝑦=𝜐

Motion Mountain – The Adventure of Physics
𝑧=𝜁, (88)

where 𝜏 now is the time coordinate in the Greek, accelerated frame. We note also that
the space-time interval d𝜎 satisfies

d𝜎2 = (1 + 𝑔𝜉/𝑐2 )2 𝑐2 d𝜏2 − d𝜉2 − d𝜐2 − d𝜁2 = 𝑐2 d𝑡2 − d𝑥2 − d𝑦2 − d𝑧2 , (89)

and since for d𝜏 = 0 distances are given by Pythagoras’ theorem, the Greek, accelerated
Ref. 102 reference frame is indeed rigid.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
After this forest of formulae, let’s tackle a simple question, shown in Figure 55. The
inertial, Roman observer O sees the Greek observer Ω departing under continuous acce-
leration, moving further and further away, following equation (86). What does the Greek
observer say about his Roman colleague? With all the knowledge we have now, that is
easy to answer. At each point of his trajectory Ω sees that O has the coordinate 𝜏 = 0
Challenge 145 e (can you confirm this?), which means that the distance to the Roman observer, as seen
by the Greek one, is the same as the space-time interval OΩ. Using expression (86), we
Ref. 103 see that this is
𝑑OΩ = √𝜉2 = √𝑥2 − 𝑐2 𝑡2 = 𝑐2 /𝑔 , (90)

which, surprisingly enough, is constant in time! In other words, the Greek observer will
observe that he stays at a constant distance from the Roman one, in complete contrast to
what the Roman observer says. Take your time to check this strange result in some other
way. We will need it again later on, to explain why the Earth does not explode. (Can you
Challenge 146 s guess how that is related to this result?)

Event horizons
We now explore one of the most surprising consequences of accelerated motion, one that
is intimately connected with the result just deduced. We explore the trajectory, in the
coordinates 𝜉 and 𝜏 of the rigidly accelerated frame, of an object located at the departure
mechanics 97

𝑡 𝜏

on
quadrant II

riz
𝜉

ho
re
tu
fu
Ω
quadrant III
O 𝑐2 /𝑔 𝑥
quadrant I

pa
st
ho
riz
on
quadrant IV
F I G U R E 56 Hyperbolic motion and event
horizons.

Motion Mountain – The Adventure of Physics
Challenge 147 ny point 𝑥 = 𝑥0 = 𝑐2 /𝑔 at all times 𝑡. We get the two relations*

𝑐2 𝑔𝜏
𝜉=− (1 − sech )
𝑔 𝑐
𝑔𝜏 𝑔𝜏
d𝜉/d𝜏 = −𝑐 sech tanh . (92)
𝑐 𝑐

These equations are strange. For large times 𝜏 the coordinate 𝜉 approaches the limit value
−𝑐2 /𝑔 and d𝜉/d𝜏 approaches zero. The situation is similar to that of riding a car acceler-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ating away from a woman standing on a long road. For the car driver, the woman moves
away; however, after a while, the only thing the driver notices is that she is slowly ap-
proaching the horizon. In everyday life, both the car driver and the woman on the road
see the other person approaching their respective horizon; in special relativity, only the
accelerated observer makes a observation of this type.
A graph of the situation helps to clarify the result. In Figure 56 we can see that light
emitted from any event in regions II and III cannot reach the Greek observer. Those
events are hidden from him and cannot be observed. The boundary between the part
of space-time that can be observed and the part that cannot is called the event horizon.
Strangely enough, however, light from the Greek observer can reach region II. Event
horizons thus act like one-way gates for light and other signals. For completeness, the
graph also shows the past event horizon. We note that an event horizon is a surface. It is
thus a different phenomenon than the everyday horizon, which is a line. Can you confirm
Challenge 148 e that event horizons are black, as illustrated in Figure 57?

* The functions appearing above, the hyperbolic secant and the hyperbolic tangent, are defined using the
expressions from the footnote on page 95:

1 sinh 𝑦
sech 𝑦 = and tanh 𝑦 = . (91)
cosh 𝑦 cosh 𝑦
98 2 relativistic

F I G U R E 57 How an event horizon looks like
according to special (and general) relativity.

Motion Mountain – The Adventure of Physics
So, not all events observed in an inertial frame of reference can be observed in a uni-
formly accelerating frame of reference. Accelerated observers are limited. In particular,
uniformly accelerating frames of reference produce event horizons at a distance −𝑐2 /𝑔.
For example, a person who is standing can never see further than this distance below his
feet.
By the way, is it true that a light beam cannot catch up with a massive observer in
Challenge 149 s hyperbolic motion, if the observer has a sufficient head start?
Here is a more advanced challenge, which prepares us for general relativity. What is
Challenge 150 s the two-dimensional shape of the horizon seen by a uniformly accelerated observer?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 151 s Another challenge: what horizon is seen by an observer on a carousel?

The importance of horizons
In special relativity, horizons might seem to play a secondary role. But this impression
is wrong. Horizons are frequent and important. In principle, if you want to observe a
horizon somewhere, just accelerate in the opposite direction and look back.
In fact, the absence of horizons is rare: it implies the lack of acceleration. And we
know that uniform, inertial motion is limited in nature: it is limited by gravity and other
interactions. Since in everyday life we are not moving inertially, there are horizons every-
where. In other words, space is not really infinite in everyday life.
Whenever you accelerate, there is a horizon behind you. Now, gravity and acceleration
are equivalent, as they locally just differ by change of reference frame. Therefore, gravity
is inextricably linked with horizons.
Horizons are everywhere – because gravity is everywhere. The relativistic description
of gravity is called general relativity. We will find that in general relativity, horizons be-
come even more important and frequent: the night sky is an example of a horizon. Yes,
the sky is dark at night because the universe is not of infinite size. Also the surface of a
black hole is a horizon. And there are literally billions of black holes in the universe. We
will explore these topics below.
But horizons are interesting for a further reason. Two and a half thousand years
mechanics 99

ago, Leucippus of Elea (c. 490 to c. 430 b ce) and Democritus of Abdera (c. 460 to
c. 356 or 370 b ce) founded atomic theory. In particular, they made the statement that
everything found in nature is – in modern words – particles and empty space. For many
centuries, modern physics corroborated this statement. For example, all matter turned
out to be made of particles. Also light and all other types of radiation are made of
particles. But then came relativity and its discovery of horizons.
Horizons show that atomism is wrong. Horizons can be observed and measured. On
the one hand, horizons are extended, not localized systems, and they have two spatial
dimensions. On the other hand, we will discover that horizons are not completely black,
but have a slight colour, and that they can have mass, spin and charge. In short, horizons
are neither particles nor space. Horizons are something new.
Later in our adventure, when we combine general relativity and quantum theory, we
will discover that horizons are effectively intermediate between space and particles. Ho-
rizons can also be seen as a mixture of space and particles. We will need some time to
find out what this means exactly. So far, our exploration of the speed limit in nature only
tells us that horizons are a further phenomenon in nature, an unexpected addition to

Motion Mountain – The Adventure of Physics
particles and vacuum.

Acceleration changes colours
Page 31 We saw above that a moving receiver sees different colours than the sender. So far, we
discussed this colour shift, or Doppler effect, for inertial motion only. For accelerating
frames the situation is even stranger: sender and receiver do not agree on colours even
Ref. 99, Ref. 104 if they are at rest with respect to each other. Indeed, if light is emitted in the direction of
the acceleration, the formula for the space-time interval gives

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑔0 𝑥 2 2 2
d𝜎2 = (1 + ) 𝑐 d𝑡 (93)
𝑐2

in which 𝑔0 is the proper acceleration of an observer located at 𝑥 = 0. We can deduce in
Challenge 152 e a straightforward way that
𝑓r 𝑔ℎ 1
= 1 − r2 = 𝑔ℎ
(94)
𝑓s 𝑐 1 + s2 𝑐

where ℎ is the rod distance between the source and the receiver, and where
𝑔s = 𝑔0 /(1 + 𝑔0 𝑥s /𝑐2 ) and 𝑔r = 𝑔0 /(1 + 𝑔o 𝑥r /𝑐2 ) are the proper accelerations meas-
ured at the source and at the detector. In short, the frequency of light decreases when
light moves in the direction of acceleration. By the way, does this have an effect on the
Challenge 153 s colour of trees along their vertical extension?
The formula usually given, namely

𝑓r 𝑔ℎ
=1− 2 , (95)
𝑓s 𝑐

is only correct to a first approximation. In accelerated frames of reference, we have to
be careful about the meaning of every quantity. For everyday accelerations, however, the
100 2 relativistic

Challenge 154 e differences between the two formulae are negligible. Can you confirm this?

C an light move faster than 𝑐?
What speed of light does an accelerating observer measure? Using expression (95) above,
an accelerated observer deduces that

𝑔ℎ
𝑣light = 𝑐 (1 + ) (96)
𝑐2

which is higher than 𝑐 for light moving in front of or ‘above’ him, and lower than 𝑐 for
light moving behind or ‘below’ him. This strange result follows from a basic property of
any accelerating frame of reference: in such a frame, even though all observers are at rest
with respect to each other, clocks do not remain synchronized. This predicted change of
the speed of light has also been confirmed by experiment: the propagation delays to be
Page 163 discussed in general relativity can be seen as confirmations of this effect.
In short, the speed of light is only invariant when it is defined as 𝑐 = d𝑥/d𝑡, and if d𝑥

Motion Mountain – The Adventure of Physics
is measured with a ruler located at a point inside the interval d𝑥, and if d𝑡 is measured
with a clock read off during the interval d𝑡. In other words, the speed of light is only
invariant if measured locally.
If, however, the speed of light is defined as Δ𝑥/Δ𝑡, or if the ruler measuring distances
or the clock measuring times is located away from the propagating light, the speed of
light is different from 𝑐 for accelerating observers! This is the same effect you can exper-
ience when you turn around your vertical axis at night: the star velocities you observe
are much higher than the speed of light. In short,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
⊳ The value 𝑐 is the speed of light only relative to nearby matter.

In other cases, light can move faster than 𝑐. Note that this result does not imply that
Challenge 155 s signals or energy can be moved faster than 𝑐. You may want to check this for yourself.
In practice, non-local effects on the speed of light are negligible for distances 𝑙 that
are much less than 𝑐2 /𝑎. For an acceleration of 9.5 m/s2 (about that of free fall), distances
would have to be of the order of one light year, or 9.5 ⋅ 1012 km, in order for any sizeable
effects to be observed.
By the way, everyday gravity is equivalent to a constant acceleration. So, why then do
Challenge 156 s distant objects, such as stars, not move faster than light, following expression (96)?

The composition of accelerations
To get a better feeling for acceleration, we explore another topic: the composition the-
orem for accelerations. This situation is more complex than for velocities, and is often
Ref. 105 avoided. However, a good explanation of this was published by Mishra.
If we call 𝑎𝑛𝑚 the acceleration of system 𝑛 by observer 𝑚, we are seeking to express
the object acceleration 𝑎01 as function of the value 𝑎02 measured by the other observer,
the relative acceleration 𝑎12 , and the proper acceleration 𝑎22 of the other observer: see
Figure 58. Here we will only study one-dimensional situations, where all observers and
all objects move along one axis. (For clarity, we also write 𝑣12 = 𝑣 and 𝑣02 = 𝑢.)
mechanics 101

𝑣0𝑛 velocity of object 0 seen by observer n

𝑎0𝑛 acceleration of object 0
Object 0 seen by observer n
𝑦

𝑦
𝑣22 = 0
𝑎22 proper acceleration 𝑣11 = 0
𝑎11 proper acceleration
Observer 2
𝑥
Observer 1
𝑥

Motion Mountain – The Adventure of Physics
F I G U R E 58 The deﬁnitions necessary to deduce the composition behaviour of accelerations.

Challenge 157 e In Galilean physics we have the general connection

𝑎01 = 𝑎02 − 𝑎12 + 𝑎22 (97)

because accelerations behave simply. In special relativity, we get

(1 − 𝑣2 /𝑐2 )3/2 (1 − 𝑢2 /𝑐2 )(1 − 𝑣2 /𝑐2 )−1/2 (1 − 𝑢2 /𝑐2 )(1 − 𝑣2 /𝑐2 )3/2

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑎01 = 𝑎02 − 𝑎12 + 𝑎22
(1 − 𝑢𝑣/𝑐2 )3 (1 − 𝑢𝑣/𝑐2 )2 (1 − 𝑢𝑣/𝑐2 )3
(98)
Challenge 158 e and you might enjoy checking the expression.

Limits on the length of solid b odies
An everyday solid object breaks when some part of it moves with respect to some nearby
part with more than the speed of sound 𝑐 of the material.* For example, when an object
hits the floor and its front end is stopped within a distance 𝑑, the object breaks at the
latest when
𝑣2 2𝑑
⩾ . (99)
𝑐2 𝑙
In this way, we see that we can avoid the breaking of fragile objects by packing them
into foam rubber – which increases the stopping distance. This may explain why boxes
containing presents are usually so much larger than their contents.
The fracture limit can also be written in a different way. To avoid breaking, the acce-

* The (longitudinal) speed of sound is about 5.9 km/s for glass, iron or steel; about 4.5 km/s for gold; and
Vol. I, page 294 about 2 km/s for lead. More sound speed values were given earlier on.
102 2 relativistic

leration 𝑎 of a solid body with length 𝑙 must obey

𝑙𝑎 < 𝑐2 , (100)

where 𝑐 is the speed of sound, which is the speed limit for the material parts of solids.
Let us now repeat the argument in relativity, using the speed of light instead of that of
Ref. 106 sound. Imagine accelerating the front of a solid body with some proper acceleration 𝑎.
The back end cannot move with an acceleration 𝛼 equal or larger than infinity, or more
Challenge 159 s precisely, it cannot move with more than the speed of light. A quick check shows that
therefore the length 𝑙 of a solid body must obey

𝑙𝑎 < 𝑐2 , (101)

where 𝑐 is now the speed of light.

Motion Mountain – The Adventure of Physics
⊳ The speed of light thus limits the size of accelerated solid bodies.

For example, for 9.8 m/s2 , the acceleration of good motorbike, this expression gives a
length limit of 9.2 Pm, about a light year. Not a big restriction: most motorbikes are
shorter. However, there are other, more interesting situations. Today, high accelerations
are produced in particle accelerators. Atomic nuclei have a size of a few femtometres.
Challenge 160 ny Can you deduce at which energies they break when smashed together in an acceler-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ator? In fact, inside a nucleus, the nucleons move with accelerations of the order of
𝑣2 /𝑟 ≈ ℏ2 /𝑚2 𝑟3 ≈ 1031 m/s2 ; this is one of the highest values found in nature. Is the
Challenge 161 s length limit also obeyed by nuclei?
We find that Galilean physics and relativity produce similar conclusions: a limiting
speed, be it that of sound or that of light, makes it impossible for solid bodies to be rigid.
When we push one end of a body, the other end always can move only a little bit later.
A puzzle: does the speed limit imply a relativistic ‘indeterminacy relation’

Δ𝑙 Δ𝑎 ⩽ 𝑐2 (102)

Challenge 162 s for the length and acceleration indeterminacies?
What does all this mean for the size of elementary particles? Take two electrons a
distance 𝑑 apart, and call their size 𝑙. The acceleration due to electrostatic repulsion then
Challenge 163 ny leads to an upper limit for their size given by

4π𝜀0 𝑐2 𝑑2 𝑚
𝑙< . (103)
𝑒2
The nearer electrons can get, the smaller they must be. The present experimental limit
gives a size smaller than 10−19 m. Can electrons be exactly point-like? We will come back
to this question several times in the rest of our adventure.
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103
mechanics
Chapter 3

SPE C IA L R E L AT I V I T Y I N F OU R
SE N T E NC E S

The results that we encountered so far can be summarized in four statements:
— All nearby observers observe that there is a unique, maximal and invariant energy
speed in nature, the ‘perfect’ speed 𝑣max = 𝑐 = 299 792 458 m/s ≈ 0.3 Gm/s. The
maximum speed is realized by massless radiation such as light or radio signals, but
cannot be achieved by material systems. This observation defines special relativity.

Motion Mountain – The Adventure of Physics
— Therefore, even though space-time is the same for every observer, measured times
and length values – thus also angles and colours – vary from one observer to an-
Page 44 other, as described by the Lorentz transformations (15) and (16), and as confirmed
by experiment.
— Collisions show that the maximum energy speed implies that mass is equivalent to
energy, that the total energy of a moving massive body is given by 𝐸 = 𝑐2 𝛾𝑚, and that
mass is not conserved.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
— Applied to accelerated objects, these results lead to numerous counter-intuitive con-
sequences, such as the twin paradox, the appearance of event horizons and the ap-
pearance of short-lived, i.e., virtual, tachyons in collisions.
Not only is all motion of radiation and matter is limited in speed, but all speeds are
defined and measured using the propagation of light. The other properties of everyday
motion remain. In particular, the six basic properties of everyday motion that follow from
Vol. I, page 29 its predictability are still valid: also relativistic motion is continuous, conserves energy–
momentum and angular momentum, is relative, is reversible, is mirror-invariant (ex-
cept for the weak interaction, where a generalized way to predict mirror-inverse motion
Vol. V, page 245 holds). Above all, also relativistic motion is lazy: it minimizes action.

C ould the speed of light vary?
The speed of massless light and radiation is the limit speed of energy in nature. Could the
limit speed change from place to place, or change as time goes by? This tricky question
still makes a fool out of many physicists. The first answer is often a loud: ‘Yes, of course!
Ref. 107 Just look at what happens when the value of 𝑐 is changed in formulae.’ Several such
‘variable speed of light’ conjectures have even been explored by researchers. However,
this often-heard answer is wrong.
Since the speed of light enters into our definition of time and space, it thus enters,
even if we do not notice it, into the construction of all rulers, all measurement standards
in four sentences 105

and all measuring instruments. Therefore there is no way to detect whether the value
actually varies.

⊳ A change in the speed of light cannot be measured.

No imaginable experiment could detect a variation of the limit speed, as the limit speed
Challenge 164 s is the basis for all measurements. ‘That is intellectual cruelty!’, you might say. ‘All exper-
iments show that the speed of light is invariant; we had to swallow one counter-intuitive
result after another to accept the invariance of the speed of light, and now we are even
supposed to admit that there is no other choice?’ Yes, we are. That is the irony of pro-
gress in physics. There is no way to detect variations – in time or across space – of a
Challenge 165 e measurement standard. Just try!
The observer-invariance of the speed of light is counter-intuitive and astonishing
when compared to the observer-dependence of everyday, Galilean speeds. But had we
taken into account that every speed measurement is – whether we like it or not – a com-

Motion Mountain – The Adventure of Physics
parison with the speed of light, we would not have been astonished by the invariance of
the speed of light at all; rather, we would have been astonished by the speed limit – and
by the strange properties of small speeds.
In summary, there is, in principle, no way to falsify the invariance of a measurement
standard. To put it another way, the truly surprising aspect of relativity is not the in-
variance of 𝑐; it is the disappearance of the limit speed 𝑐 from the formulae of everyday
motion.

Where d oes special relativit y break d own?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The maximum local energy speed is confirmed by all experiments. The speed limit is
thus correct: the local energy speed limit is a fundamental truth about nature. Indeed, it
remains valid throughout the rest of our adventure.
As we approach the speed of light, the Lorentz factor and the quantities in the Lorentz
transformation exceed all bounds. However, in nature, no observable actually reaches ar-
bitrary large values. For example, no elementary particle with an energy or a momentum
above – or even close to – the (corrected) Planck limits

ℏ𝑐5
𝐸Planck = √ = 9.8 ⋅ 108 J = 0.60 ⋅ 1019 GeV
4𝐺
ℏ𝑐3
𝑝Planck = √ = 3.2 kg m/s = 0.60 ⋅ 1019 GeV/c (104)
4𝐺

has ever been observed. In fact, the record values observed so far are one million times
smaller than the Planck limits. The reason is simple: when the speed of light is ap-
proached as closely as possible, special relativity breaks down as a description of nature.
How can the maximum speed limit remain valid, and special relativity break down
nevertheless? At highest energies, special relativity is not sufficient to describe nature.
There are two reasons.
In the case of extreme Lorentz contractions, we must take into account the curvature
106 3 special relativity

of space-time that the moving energy itself generates: gravitation needs to be included.
Equivalently, we recall that so far, we assumed that point masses are possible in nature.
However, point masses would have infinite mass density, which is impossible: gravity,
characterized by the gravitational constant 𝐺, prevents infinite mass densities through
the curvature of space, as we will find out.
In addition, in the case of extreme Lorentz contractions, we must take into account
the fluctuations in speed and position of the moving particles: quantum theory needs
to be included. We recall that so far, we assumed that measurements can have infinite
precision in nature. However, this is not the case: quantum theory, characterized by the
smallest action value ℏ, prevents infinite measurement precision, as we will find out.
In summary, the two fundamental constants 𝐺, the gravitational constant, and ℏ, the
quantum of action, limit the validity of special relativity. Both constants appear in the
Planck limits. The gravitational constant 𝐺 modifies the description of motion for power-
ful and large movements. The quantum of action ℏ modifies the description of motion
for tiny movements. The exploration of these two kinds of motions define the next two
stages of our adventure. We start with gravitation.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Chapter 4

SI M PL E G E N E R A L R E L AT I V I T Y:
G R AV I TAT ION , M A X I M UM SPE E D
A N D M A X I M UM F ORC E

G
eneral relativity is easy. Nowadays, it can be made as intuitive as universal
ravity and its inverse square law, so that the important ideas of
eneral relativity, like those of special relativity, are accessible to secondary-
school students. In particular, black holes, gravitational waves, space-time curvature and
the limits of the universe can then be understood as easily as the twins paradox.

Motion Mountain – The Adventure of Physics
In the following pages we will discover that, just as special relativity is based on and
derives from a maximum speed 𝑐,

⊳ General relativity is based on and derives from a maximum momentum
change or maximum force 𝑐4 /4𝐺 – equivalently, from a maximum power
𝑐5 /4𝐺.

We first show that all known experimental data are consistent with these limits. Then
we find that the maximum force and the maximum power are achieved only on insur-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
mountable limit surfaces.

⊳ The surfaces that realize maximum force – or maximum momentum flow –
and maximum power – or maximum energy flow – are called horizons.

Horizons are simple generalizations of those horizons that we encountered in special
Page 96 relativity. We will find out shortly why the maximum values are related to them. Horizons
play the role in general relativity that is played by light beams in special relativity: they are
the systems that realize the limit. A horizon is the reason that the sky is dark at night and
that the universe is of finite size. Horizons tell us that in general, space-time is curved.
And horizons will allow us to deduce the field equations of general relativity.
We also discuss the main counter-arguments and paradoxes arising from the force
and power limits. The resolutions of the paradoxes clarify why the limits have remained
dormant for so long, both in experiments and in teaching.
After this introduction, we will study the effects of relativistic gravity in detail. We
will explore the consequences of space-time curvature for the motions of bodies and of
light in our everyday environment. For example, the inverse square law will be modified.
Challenge 166 s (Can you explain why this is necessary in view of what we have learned so far?) Most
fascinating of all, we will discover how to move and bend the vacuum. Then we will
study the universe at large. Finally, we will explore the most extreme form of gravity:
black holes.
108 4 simple general relativity

F I G U R E 59 Effects of gravity: a dripping stalactite (© Richard Cindric) and the rings of Saturn,
photographed when the Sun is hidden behind the planet (courtesy CICLOPS, JPL, ESA, NASA).

Maximum force – general relativity in one statement

Motion Mountain – The Adventure of Physics
“
One of the principal objects of theoretical
research in any department of knowledge is to
find the point of view from which the subject

”
Ref. 108 appears in its greatest simplicity.
Willard Gibbs

We just saw that the theory of special relativity appears when we recognize the speed
limit 𝑐 in nature and take this limit as a basic principle. At the turn of the twenty-first
Ref. 109 century it was shown that general relativity can be approached by using a similar basic
principle:

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
⊳ There is in nature a maximum force, or maximum momentum change per time:

𝑐4
𝐹⩽ = 3.0258(4) ⋅ 1043 N . (105)
4𝐺
In nature, no force in any muscle, machine or system can exceed this value. For the curi-
ous, the value of the force limit is the energy of a (Schwarzschild) black hole divided by
twice its radius. The force limit can be understood intuitively by noting that (Schwarz-
schild) black holes are the densest bodies possible for a given mass. Since there is a limit
to how much a body can be compressed, forces – whether gravitational, electric, centri-
petal or of any other type – cannot be arbitrary large.
Alternatively, it is possible to use another, equivalent statement as a basic principle:
⊳ There is a maximum power, or energy change per time, in nature:

𝑐5
𝑃⩽ = 9.071(1) ⋅ 1051 W . (106)
4𝐺
No power of any lamp, engine or explosion can exceed this value. It is equivalent to
1.2 ⋅ 1049 horsepower. Another way to visualize the value is the following: the maximum
power corresponds to converting 50 solar masses into massless radiation within a milli-
second. The maximum power is realized when a (Schwarzschild) black hole is radiated
gravitation, maximum speed and maximum force 109

TA B L E 3 How to convince yourself and others that there is a maximum
force 𝑐4 /4𝐺 or a maximum power 𝑐5 /4𝐺 in nature. Compare this table with
the table about maximum energy speed, on page 26 above, and with the
table about a smallest action, on page 19 in volume IV.

S tat e m e n t Te s t

The maximum force value 𝑐4 /4𝐺 is Check all observations.
observer-invariant.
Force values > 𝑐4 /4𝐺 are not Check all observations.
observed.
Force values > 𝑐4 /4𝐺 cannot be Check all attempts.
produced.
Force values > 𝑐4 /4𝐺 cannot even Solve all paradoxes.
be imagined.
The maximum force value 𝑐4 /4𝐺 is Deduce the theory of
a principle of nature. general relativity from it.

Motion Mountain – The Adventure of Physics
Show that all consequences,
however weird, are
confirmed by observation.

away in the time that light takes to travel along a length corresponding to its diameter.
We will see below precisely what black holes are and why they are connected to these
limits.
Yet another, equivalent limit appears when the maximum power is divided by 𝑐2 .

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
⊳ There is a maximum rate of mass change in nature:

d𝑚 𝑐3
⩽ = 1.000 93(1) ⋅ 1035 kg/s . (107)
d𝑡 4𝐺
This bound on mass flow imposes a limit on pumps, jet engines and fast eaters. Indeed,
the rate of flow of water or any other material through tubes is limited. The mass flow
limit is obviously equivalent to either the force or the power limit.
The existence of a maximum force, power or mass flow implies the full theory of gen-
eral relativity. In order to prove the correctness and usefulness of this approach, a se-
quence of arguments is required. This sequence of arguments, listed in Table 3, is the
Page 26 same as the sequence that we used for the establishment of the limit speed in special re-
lativity. The basis is to recognize that the maximum force value is invariant. This follows
from the invariance of 𝑐 and 𝐺. For the first argument, we need to gather all observational
evidence for the claimed limit and show that it holds in all cases. Secondly, we have to
show that the limit applies in all possible and imaginable situations; any apparent para-
doxes will need to be resolved. Finally, in order to establish the limit as a principle of
nature, we have to show that general relativity follows from it.
These three steps structure this introduction to general relativity. We start the story
by explaining the origin of the idea of a limiting value.
110 4 simple general relativity

The meaning of the force and power limits
In the nineteenth and twentieth centuries many physicists took pains to avoid the
concept of force. Heinrich Hertz made this a guiding principle of his work, and wrote
an influential textbook on classical mechanics without ever using the concept. The fath-
ers of quantum theory, who all knew this text, then dropped the term ‘force’ completely
from the vocabulary of microscopic physics. Meanwhile, the concept of ‘gravitational
force’ was eliminated from general relativity by reducing it to a ‘pseudo-force’. Force fell
out of fashion.
Nevertheless, the maximum force principle does make sense, provided that we visu-
alize it by means of the definition of force:

⊳ Force is the flow of momentum per unit time.

In nature, momentum cannot be created or destroyed. We use the term ‘flow’ to remind
Ref. 110 us that momentum, being a conserved quantity, can only change by inflow or outflow. In

Motion Mountain – The Adventure of Physics
other words,

⊳ Change of momentum, and thus force, always takes place through some
boundary surface.

This connection is of central importance. Whenever we think about force at a point, we
really mean the momentum ‘flowing’ through a surface at that point. And that amount
is limited.

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⊳ Force is a relative concept.

Any force measurement is relative to a surface. Any momentum flow measurement is
relative. In special relativity, speed is relative; nevertheless, speed is limited. In general
relativity, force is relative; nevertheless, force is limited. That is the fascination of the
force limit.
General relativity usually explains the concept of force as follows: a force keeps bodies
from following geodesics. (A geodesic is a path followed by a freely falling particle.) The
mechanism underlying a measured force is not important; in order to have a concrete
example to guide the discussion it can be helpful to imagine force as electromagnetic in
origin. However, any type of force or momentum flow is limited, relative to any surface.
It is not important whether the surface, i.e., the observer, or the body does not follow
geodesics.
The maximum force principle boils down to the following statement: if we ima-
gine any physical surface (and cover it with observers), the integral of momentum flow
through the surface (measured by all those observers) never exceeds the limit value
𝑐4 /4𝐺. It does not matter how the surface is chosen, as long as it is physical:

⊳ A surface is physical as long as we can fix observers onto it.

We stress that observers in general relativity, like in special relativity, are massive physical
gravitation, maximum speed and maximum force 111

systems that are small enough so that their influence on the system under observation is
negligible.
The principle of maximum force imposes a limit on muscles, the effect of hammers,
the flow of material, the acceleration of massive bodies, and much more. No system can
create, measure or experience a force above the limit. No particle, no galaxy and no bull-
dozer can exceed it.
The existence of a force limit has an appealing consequence. In nature, forces can be
measured. Every measurement is a comparison with a standard.

⊳ The force limit provides a natural unit of force: the Planck force.

The force unit fits into the system of natural units that Max Planck derived from the
speed of light 𝑐, the gravitational constant 𝐺 and the quantum of action ℎ (nowadays
ℏ = ℎ/2π is preferred).* The maximum force thus provides a measurement standard for
force that is valid in every place and at every instant of time.
The maximum force value 𝑐4 /4𝐺 differs from Planck’s originally proposed unit in

Motion Mountain – The Adventure of Physics
two ways. First, the numerical factor is different (Planck had in mind the value 𝑐4 /𝐺).
Secondly, the force unit is a limiting value. In this respect, the maximum force plays the
Ref. 111 same role as the maximum speed. As we will see later on, this limit property is valid for
Vol. VI, page 27 all other Planck units as well, once the numerical factors have been properly corrected.
The factor 1/4 has no deeper meaning: it is just the value that leads to the correct form
of the field equations of general relativity. The factor 1/4 in the limit is also required to
Page 129 recover, in everyday situations, the inverse square law of universal gravitation. When the
factor is properly taken into account, the maximum force (or power) is simply given by
the (corrected) Planck energy divided by the (corrected) Planck length or Planck time.

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The expression 𝑐4 /4𝐺 for the maximum force involves the speed of light 𝑐 and the
gravitational constant 𝐺; it thus qualifies as a statement on relativistic gravitation. The
fundamental principle of special relativity states that speed 𝑣 obeys 𝑣 ⩽ 𝑐 for all observ-
ers. Analogously, the basic principle of general relativity states that in all cases force 𝐹
and power 𝑃 obey 𝐹 ⩽ 𝑐4 /4𝐺 and 𝑃 ⩽ 𝑐5 /4𝐺. It does not matter whether the observer
measures the force or power while moving with high velocity relative to the system un-
der observation, during free fall, or while being strongly accelerated. However, it does
matter that the observer records values measured at his own location and that the ob-
server is realistic, i.e., made of matter and not separated from the system by a horizon.
These conditions are the same that must be obeyed by observers measuring velocity in
special relativity.
The force limit concerns 3-force, or what we call ‘force’ in everyday life, and that the
power limit concerns what we call ‘power’ in everyday life. In other words, in nature,
both 3-velocity and 3-force are limited.
Since physical power is force times speed, and since nature provides a speed limit, the
force bound and the power bound are equivalent. We have already seen that force and
Page 83 power appear together in the definition of 4-force. The statement of a maximum 3-force

* When Planck discovered the quantum of action, he noticed at once the possibility to define natural units
Vol. IV, page 20 for all observable quantities. Indeed, on a walk with his seven-year-old son Erwin in the forest around
Berlin, he told him that he had made a discovery as important as the discovery of universal gravity.
112 4 simple general relativity

is valid for every component of the 3-force, as well as for its magnitude. (As we will see
Page 122 below, a boost to an observer with high 𝛾 value cannot be used to overcome the force or
power limits.) The power bound limits the output of car and motorcycle engines, lamps,
lasers, stars, gravitational radiation sources and galaxies. The maximum power principle
states that there is no way to move or get rid of energy more quickly than that.
The power limit can be understood intuitively by noting that every engine produces
exhausts, i.e., some matter or energy that is left behind. For a lamp, a star or an evapor-
ating black hole, the exhausts are the emitted radiation; for a car or jet engine they are
hot gases; for a water turbine the exhaust is the slowly moving water leaving the turbine;
for a rocket it is the matter ejected at its back end; for a photon rocket or an electric mo-
tor it is electromagnetic energy. Whenever the power of an engine gets close to the limit
value, the exhausts increase dramatically in mass–energy. For extremely high exhaust
masses, the gravitational attraction from these exhausts – even if they are only radiation
– prevents further acceleration of the engine with respect to them.

Motion Mountain – The Adventure of Physics
⊳ The maximum power principle thus expresses there is a built-in braking
mechanism in nature; this braking mechanism is gravity.

The claim of a maximum force, a maximum power or a maximum mass flow in nature
seems almost too fantastic to be true. Our first task is therefore to check it empirically as
thoroughly as we can.

The experimental evidence
Like the maximum speed principle, the maximum force principle must first of all be

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checked experimentally. We recall that Michelson spent a large part of his research life
looking for possible changes in the value of the speed of light. No one has yet dedicated
so much effort to testing the maximum force or power. However, it is straightforward
to confirm that no experiment, whether microscopic, macroscopic or astronomical, has
ever measured force values larger than the stated limit. In the past, many people have
claimed to have produced energy speeds higher than that of light. So far, nobody has
Challenge 167 s ever claimed to have produced or observed a force higher than the limit value.
The large accelerations that particles undergo in collisions inside the Sun, in the most
powerful accelerators or in reactions due to cosmic rays correspond to force values much
smaller than the force limit. The same is true for neutrons in neutron stars, for quarks
inside protons, and for all matter that has been observed to fall towards black holes.
Furthermore, the search for space-time singularities, which would allow forces to achieve
or exceed the force limit, has been fruitless.
In the astronomical domain, all forces between stars or galaxies are below the limit
value, as are the forces in their interior. Not even the interactions between any two halves
of the universe exceed the limit, whatever physically sensible division between the two
Page 127 halves is taken. (The meaning of ‘physically sensible division’ will be defined below; for
divisions that are not sensible, exceptions to the maximum force claim can be construc-
Challenge 168 s ted. You might enjoy searching for such an exception.)
Astronomers have also failed to find any region of space-time whose curvature (a
concept to be introduced below) is large enough to allow forces to exceed the force limit.
gravitation, maximum speed and maximum force 113

Indeed, none of the numerous recent observations of black holes has brought to light
forces larger than the limit value or objects smaller than the corresponding black hole
radii.
Also the power limit can be checked experimentally. It turns out that the power –
or luminosity – of stars, quasars, binary pulsars, gamma-ray bursters, galaxies or galaxy
clusters can indeed be a sizeable fraction of the power limit. However, no violation of
Ref. 113 the limit has ever been observed. In fact, the sum of all light output from all stars in the
universe does not exceed the limit. Similarly, even the brightest sources of gravitational
waves, merging black holes, do not exceed the power limit. For example, the black hole
Ref. 112 merger published in 2016, possibly the most powerful event observed so far, transformed
about 3 solar masses into radiation in 0.2 s. Its power was therefore about three thousand
times lower than the power limit 𝑐5 /4𝐺; the peak power possibly was around three hun-
dred times lower than the limit. It might well be that only the brightness of evaporating
black holes in their final phase can equal the power limit. However, no such event has
ever been observed yet. (Given that several nearby localised sources can each approach
Page 127 the power limit, the so-called power paradox arises, which will be discussed below.)

Motion Mountain – The Adventure of Physics
Similarly, all observed mass flow rates are orders of magnitude below the correspond-
ing limit. Even physical systems that are mathematical analogues of black holes – for
Ref. 114 example, silent acoustical black holes or optical black holes – do not invalidate the force
and power limits that hold in the corresponding systems.
In summary, the experimental situation is somewhat disappointing. Experiments do
not contradict the limit values. But neither do the data do much to confirm the limits.
The reason is the lack of horizons in everyday life and in experimentally accessible sys-
tems. The maximum speed at the basis of special relativity is found almost everywhere;
maximum force and maximum power are found almost nowhere. Below we will propose

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Page 133 some dedicated tests of the limits that could be performed in the near future.

Deducing general relativit y*
In order to establish the maximum force and power limits as fundamental physical prin-
ciples, it is not sufficient to show that they are consistent with what we observe in nature.
It is necessary to show that they imply the complete theory of general relativity. (This sec-
tion is only for readers who already know the field equations of general relativity. Other
Page 117 readers may skip to the next section.)
In order to derive the theory of relativity we need to study those systems that realize
the limit under scrutiny. In the case of the special theory of relativity, the main system
that realizes the limit speed is light. For this reason, light is central to the exploration
of special relativity. In the case of general relativity, the systems that realize the limit are
less obvious. We note first that a maximum force (or power) cannot be realized through-
out a volume of space. If this were possible, a simple boost** could transform the force
(or power) to a higher value. Therefore, nature can realize maximum force and power
only on surfaces, not volumes. In addition, these surfaces must be unattainable. These
Ref. 109, Ref. 111 unattainable surfaces are basic to general relativity; they are called horizons.

* This section can be skipped at first reading. The proof mentioned in it dates from December 2003.
** A boost was defined in special relativity as a change of viewpoint to a second observer moving in relation
to the first.
114 4 simple general relativity

Maximum force c4/4G, First law of horizon Field
are is equations
mechanics
Maximum power c5/4G, equivalent equivalent of general
to to relativity
(horizon equation)
Maximum mass rate c3/4G

F I G U R E 60 Showing the equivalence of the maximum force or power with the ﬁeld equations of
general relativity.

⊳ Maximum force and power only appear on horizons.

Page 97 We have encountered horizons in special relativity, where they were defined as surfaces
that impose limits to observation. (Note the contrast with everyday life, where a horizon
is only a line, not a surface.) The present definition of a horizon as a surface of maximum

Motion Mountain – The Adventure of Physics
force (or power) is equivalent to the definition as a surface beyond which no signal may
be received. In both cases, a horizon is a surface beyond which any interaction is im-
possible.
The connection between horizons and the maximum force is a central point of re-
lativistic gravity. It is as important as the connection between light and the maximum
speed in special relativity. In special relativity, we used the limit property of the speed
of light to deduce the Lorentz transformations. In general relativity, we will now prove
that the maximum force in nature, which we can also call the horizon force, implies the
field equations of general relativity. To achieve this aim, we start by recognizing that all
horizons have an energy flow across them. The flow depends on the horizon curvature,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
as we will see. This connection implies that horizons cannot be planes, as an infinitely
extended plane would imply an infinite energy flow.
The deduction of the equations of general relativity has only two steps, as shown in
Figure 60. In the first step, we show that the maximum force or power principle implies
the first ‘law’ of horizon mechanics. In the second step, we show that the first ‘law’ im-
plies the field equations of general relativity.
The simplest finite horizon is a static sphere, corresponding to a Schwarzschild black
hole. A spherical horizon is characterized by its radius of curvature 𝑅, or equivalently, by
its surface gravity 𝑎; the two quantities are related by 2𝑎𝑅 = 𝑐2 . Now, the energy flowing
through any horizon is always finite in extension, when measured along the propaga-
tion direction. We can thus speak more specifically of an energy pulse. Any energy pulse
through a horizon is thus characterized by an energy 𝐸 and a proper length 𝐿. When the
energy pulse flows perpendicularly through a horizon, the rate of momentum change, or
force, for an observer at the horizon is

𝐸
𝐹= . (108)
𝐿
Our goal is to show that the existence of a maximum force implies general relativity. Now,
maximum force is realized on horizons. We thus need to insert the maximum possible
values on both sides of equation (108) and to show that general relativity follows.
gravitation, maximum speed and maximum force 115

Using the maximum force value and the area 4π𝑅2 for a spherical horizon we get

𝑐4 𝐸
= 4π𝑅2 . (109)
4𝐺 𝐿𝐴
The fraction 𝐸/𝐴 is the energy per area flowing through any area 𝐴 that is part of a
horizon. The insertion of the maximum values is complete when we note that the length
𝐿 of the energy pulse is limited by the radius 𝑅. The limit 𝐿 ⩽ 𝑅 follows from geometrical
considerations: seen from the concave side of the horizon, the pulse must be shorter than
the radius of curvature. An independent argument is the following. The length 𝐿 of an
Ref. 115 object accelerated by 𝑎 is limited, by special relativity, by 𝐿 ⩽ 𝑐2 /2𝑎. Already special
relativity shows that this limit is related to the appearance of a horizon. Together with
relation (109), the statement that horizons are surfaces of maximum force leads to the
following important relation for static, spherical horizons:

𝑐2

Motion Mountain – The Adventure of Physics
𝐸= 𝑎𝐴 . (110)
8π𝐺
This horizon equation relates the energy flow 𝐸 through an area 𝐴 of a spherical horizon
with surface gravity 𝑎. It states that the energy flowing through a horizon is limited, that
this energy is proportional to the area of the horizon, and that the energy flow is propor-
tional to the surface gravity. The horizon equation is also called the first law of black hole
Ref. 116 mechanics or the first law of horizon mechanics.
The above derivation also yields the intermediate result

𝑐4 𝐴

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𝐸⩽ . (111)
16π𝐺 𝐿
This form of the horizon equation states more clearly that no surface other than a horizon
can achieve the maximum energy flow, when the area and pulse length (or surface grav-
ity) are given. Gravity limits energy flow. No other domain of physics makes comparable
statements: they are intrinsic to the theory of gravitation.
An alternative derivation of the horizon equation starts with the emphasis on power
instead of on force, using 𝑃 = 𝐸/𝑇 as the initial equation.
It is important to stress that the horizon equation in its forms (110) and (111) follows
from only two assumptions: first, there is a maximum speed in nature, and secondly,
there is a maximum force (or power) in nature. No specific theory of gravitation is as-
sumed. The horizon equation might even be testable experimentally, as argued below.
Next, we have to generalize the horizon equation from static and spherical horizons
to general horizons. Since the maximum force is assumed to be valid for all observers,
whether inertial or accelerating, the generalization is straightforward. For a horizon that
is irregularly curved or time-varying the horizon equation becomes

𝑐2
𝛿𝐸 = 𝑎 𝛿𝐴 . (112)
8π𝐺
116 4 simple general relativity

This differential relation – it might be called the general horizon equation – is valid for any
kind of horizon. It can be applied separately for every piece 𝛿𝐴 of a dynamic or spatially
changing horizon.
The general horizon equation (112) has been known to be equivalent to general relativ-
Ref. 117 ity at least since 1995, when this equivalence was (implicitly) shown by Jacobson. We will
show that the differential horizon equation has the same role for general relativity as the
equation d𝑥 = 𝑐 d𝑡 has for special relativity. From now on, when we speak of the horizon
equation, we mean the general, differential form (112) of the relation.
It is instructive to restate the behaviour of energy pulses of length 𝐿 in a way that holds
for any surface, even one that is not a horizon. Repeating the above derivation, we get
the energy limit
𝛿𝐸 𝑐4 1
⩽ . (113)
𝛿𝐴 16π𝐺 𝐿
Equality is only realized when the surface 𝐴 is a horizon. In other words, whenever the
value 𝛿𝐸/𝛿𝐴 in a physical system approaches the right-hand side, a horizon starts to

Motion Mountain – The Adventure of Physics
form. This connection will be essential in our discussion of apparent counter-examples
to the limit principles.
If we keep in mind that on a horizon the pulse length 𝐿 obeys 𝐿 ⩽ 𝑐2 /2𝑎, it becomes
clear that the general horizon equation is a consequence of the maximum force 𝑐4 /4𝐺
or the maximum power 𝑐5 /4𝐺. In addition, the horizon equation takes also into account
maximum speed, which is at the origin of the relation 𝐿 ⩽ 𝑐2 /2𝑎. The horizon equation
thus follows purely from these two limits of nature. We also note that the horizon equa-
tion – or, equivalently, the force or power limit – implies a maximum mass change rate
in nature given by d𝑚/d𝑡 ⩽ 𝑐3 /4𝐺.

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The remaining, second step of the argument is the derivation of general relativity
Ref. 117 from the general horizon equation. This derivation was provided by Jacobson, and the
essential points are given in the following paragraphs. To see the connection between
the general horizon equation (112) and the field equations, we only need to generalize
the general horizon equation to general coordinate systems and to general directions of
energy–momentum flow. This is achieved by introducing tensor notation that is adapted
to curved space-time.
To generalize the general horizon equation, we introduce the general surface element
dΣ and the local boost Killing vector field 𝑘 that generates the horizon (with suitable
norm). Jacobson uses these two quantities to rewrite the left-hand side of the general
horizon equation (112) as
𝛿𝐸 = ∫ 𝑇𝑎𝑏 𝑘𝑎 dΣ𝑏 , (114)

where 𝑇𝑎𝑏 is the energy–momentum tensor. This expression obviously gives the energy
at the horizon for arbitrary coordinate systems and arbitrary energy flow directions.
Jacobson’s main result is that the factor 𝑎 𝛿𝐴 in the right hand side of the general hori-
zon equation (112) can be rewritten, making use of the (purely geometric) Raychaudhuri
equation, as
𝑎 𝛿𝐴 = 𝑐2 ∫ 𝑅𝑎𝑏 𝑘𝑎 dΣ𝑏 , (115)
gravitation, maximum speed and maximum force 117

where 𝑅𝑎𝑏 is the Ricci tensor describing space-time curvature. This relation describes
how the local properties of the horizon depend on the local curvature.
Combining these two steps, the general horizon equation (112) becomes

𝑐4
∫ 𝑇𝑎𝑏 𝑘𝑎 dΣ𝑏 = ∫ 𝑅𝑎𝑏 𝑘𝑎 dΣ𝑏 . (116)
8π𝐺

Jacobson then shows that this equation, together with local conservation of energy (i.e.,
vanishing divergence of the energy–momentum tensor) can only be satisfied if

𝑐4 𝑅
𝑇𝑎𝑏 = (𝑅𝑎𝑏 − ( + Λ)𝑔𝑎𝑏 ) , (117)
8π𝐺 2
where 𝑅 is the Ricci scalar and Λ is a constant of integration the value of which is not
determined by the problem. The above equations are the full field equations of general
relativity, including the cosmological constant Λ. The field equations thus follow from

Motion Mountain – The Adventure of Physics
the horizon equation. They are therefore shown to be valid at horizons.
Since it is possible, by choosing a suitable coordinate transformation, to position a
Page 98 horizon at any desired space-time point (just accelerate away, as explained above), the
field equations must be valid over the whole of space-time. This observation completes
Jacobson’s argument. Since the field equations follow, via the horizon equation, from the
maximum force principle, we have also shown that at every space-time point in nature
the same maximum force holds: the value of the maximum force is an invariant and a
constant of nature.
In other words, the field equations of general relativity are a direct consequence of

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the limit on energy flow at horizons, which in turn is due to the existence of a maximum
force (or power). In fact, Jacobson’s derivation shows that the argument works in both
directions. In summary, maximum force (or power), the horizon equation, and general
relativity are equivalent.
We note that the deduction of general relativity’s field equations from the maximum
power of force is correct only under the assumption that gravity is purely geometric.
And indeed, this is the essential statement of general relativity. If the mechanism of grav-
ity would be based on other fields, such as hitherto unknown particles, the equivalence
between gravity and a maximum force would not be given.
Since the derivation of general relativity from the maximum force principle or from
the maximum power principle is now established, we can rightly call these limits hori-
zon force and horizon power. Every experimental or theoretical confirmation of the field
equations indirectly confirms the existence of the horizon limits.

Gravit y, space-time curvature, horizons and maximum force
Let us repeat the results of the previous section in simple terms. Imagine two observers
who start moving freely and parallel to each other. Both continue straight ahead. If after
a while they discover that they are not moving parallel to each other any more, then they
Challenge 169 s can deduce that they have moved on a curved surface (try it!) or in a curved space. Such
deviations from parallel free motion are observed near masses and other localized en-
118 4 simple general relativity

ergy. We conclude that space-time is curved near masses. Or, simply put: gravity curves
space.
Gravitation leads to acceleration. And acceleration leads to a horizon at distance 𝑐2 /𝑎.
No horizon occurs in everyday life, because the resulting distances are not noticeable;
but horizons do occur around bodies whose mass is concentrated in a sphere of radius
𝑟 = 2𝐺𝑚/𝑐2 . Such bodies are called (Schwarzschild) black holes. The spatial curvature
around a black hole of mass 𝑚 is the maximum curvature possible around a body of that
mass.
Black holes can be seen as matter in permanent free fall. We will study black holes in
Page 262 detail below. In case of a black hole, like for any horizon, it is impossible to detect what
is ‘behind’ the boundary.*
Black holes are characterized by a surface gravity 𝑎 and an energy flow 𝐸.

⊳ The maximum force principle is a simple way to state that, on horizons, en-
ergy flow is proportional to area and surface gravity.

Motion Mountain – The Adventure of Physics
This connection makes it possible to deduce the full theory of general relativity. In par-
ticular, a maximum force value is sufficient to tell space-time how to curve. We will ex-
plore the details of this relation shortly.
If no force limit existed in nature, it would be possible to ‘pump’ any desired amount
of energy through a given surface, including any horizon. In this case, the energy flow
would not be proportional to area, horizons would not have the properties they have, and
general relativity would not hold. We thus get an idea how the maximum flow of energy,
the maximum flow of momentum and the maximum flow of mass are all connected to
horizons. The connection is most obvious for black holes, where the energy, momentum

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Page 265 or mass are those falling into the black hole.
The analogy between special and general relativity can be carried further. In spe-
cial relativity, maximum speed implies d𝑥 = 𝑐 d𝑡, and that time depends on the ob-
server. 2In general relativity, maximum force (or power) implies the horizon equation
𝑐
𝛿𝐸 = 8π𝐺 𝑎 𝛿𝐴 and the observation that space-time is curved. The horizon equation im-
plies the field equations of general relativity. In short:

⊳ The existence of a maximum force implies that space-time is curved near
masses, and it implies how it is curved.

The maximum force (or power) thus has the same double role in general relativity as
the maximum speed has in special relativity. In special relativity, the speed of light is the
maximum speed; it is also the proportionality constant that connects space and time, as
the equation d𝑥 = 𝑐 d𝑡 makes apparent. In general relativity, the horizon force is the
maximum force; it also appears (with a factor 2π) in the field equations as the propor-
tionality constant connecting energy and curvature. The maximum force thus describes
both the elasticity of space-time and – if we use the simple image of space-time as a me-
Ref. 109 dium – the maximum tension to which space-time can be subjected. This double role of

* Analogously, in special relativity it is impossible to detect what moves faster than the light barrier.
gravitation, maximum speed and maximum force 119

a material constant as proportionality factor and as limit value is well known in materials
science.
Why is the maximum force also the proportionality factor between curvature and
energy? Imagine space as an elastic material.* The elasticity of a material is described
by a numerical material constant. The simplest definition of this material constant is the
ratio of stress (force per area) to strain (the proportional change of length). An exact
definition has to take into account the geometry of the situation. For example, the shear
modulus 𝐺 (or 𝜇) describes how difficult it is to move two parallel surfaces of a material
against each other. If the force 𝐹 is needed to move two parallel surfaces of area 𝐴 and
length 𝑙 against each other by a distance Δ𝑙, we define the shear modulus 𝐺 by

𝐹 Δ𝑙
=𝐺 . (118)
𝐴 𝑙
The value of the shear modulus 𝐺 for metals and alloys ranges between 25 and 80 GPa.
The continuum theory of solids shows that for any crystalline solid without any defect

Motion Mountain – The Adventure of Physics
(a ‘perfect’ solid) there is a so-called theoretical shear stress: when stresses higher than
this value are applied, the material breaks. The theoretical shear stress, in other words, the
maximum stress in a material, is given by

𝐺
𝐺tss = . (119)
2π
The maximum stress is thus essentially given by the shear modulus. This connection is
similar to the one we found for the vacuum. Indeed, imagining the vacuum as a material
Ref. 118 that can be bent is a helpful way to understand general relativity. We will use it regularly

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
in the following.
What happens when the vacuum is stressed with the maximum force? Is it also torn
apart like a solid? Almost: in fact, when vacuum is torn apart, particles appear. We will
find out more about this connection later on: since particles are quantum entities, we
need to study quantum theory first, before we can describe the tearing effect in the last
Vol. VI, page 303 part of our adventure.

C onditions of validit y for the force and power limits
The maximum force value is valid only under three conditions. To clarify this point, we
can compare the situation to the maximum speed. There are three conditions for the
validity of maximum speed.
First of all, the speed of light (in vacuum) is an upper limit for motion of systems with
momentum or energy only. It can, however, be exceeded for motions of non-material
points. Indeed, the cutting point of a pair of scissors, a laser light spot on the Moon,
shadows, or the group velocity or phase velocity of wave groups can exceed the speed of
Page 58 light. The limit speed is valid for motion of energy only.

* Does this analogy make you think about aether? Do not worry: physics has no need for the concept of
Vol. III, page 136 aether, because it is indistinguishable from vacuum. General relativity does describe the vacuum as a sort
of material that can be deformed and move – but it does not need nor introduce the aether.
120 4 simple general relativity

Secondly, the speed of light is a limit only if measured near the moving mass or energy:
the Moon moves faster than light if one turns around one’s axis in a second; distant
points in a Friedmann universe move apart from each other with speeds larger than the
speed of light. The limit speed is only a local limit.
Thirdly, the observer measuring speeds must be physical: also the observer must be
made of matter and energy, thus must move more slowly than light, and must be able to
Ref. 119 observe the system. No system moving at or above the speed of light can be an observer.
The limit speed is only for physical observers.
The same three conditions apply for the validity of maximum force and power. The
third point is especially important. In particular, relativistic gravity forbids point-like ob-
servers and point-like test masses: they are not physical. Surfaces moving faster than light
are also not physical. In such cases, counter-examples to the maximum force claim can
Challenge 170 s be found. Try and find one – many are possible, and all are fascinating. We now explore
some of the most important cases.

Gedanken experiments and parad oxes ab ou t the force limit

Motion Mountain – The Adventure of Physics
“
Wenn eine Idee am Horizonte eben aufgeht, ist
gewöhnlich die Temperatur der Seele dabei sehr
kalt. Erst allmählich entwickelt die Idee ihre
Wärme, und am heissesten ist diese (das heisst
sie tut ihre grössten Wirkungen), wenn der

”
Glaube an die Idee schon wieder im Sinken ist.
Friedrich Nietzsche*

The last, but central, step in our discussion of the force limit is the same as in the dis-
cussion of the speed limit. We saw that no real experiment has ever led to a force value

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
larger than the force limit. But we also need to show that no imaginable experiment can
overcome the force limit. Following a tradition dating back to the early twentieth cen-
tury, such an imagined experiment is called a Gedanken experiment, from the German
Gedankenexperiment, meaning ‘thought experiment’.
A limit to speed is surprising at first, because speed is relative, and therefore it should
be possible to let speed take any imaginable value. The situation for force is similar: force
is relative, and therefore it should be possible to let force take any imaginable value.
In order to dismiss all imaginable attempts to exceed the maximum speed, it was suf-
ficient to study the properties of velocity addition and the divergence of kinetic energy
near the speed of light. In the case of maximum force, the task is more involved. In-
deed, stating a maximum force, a maximum power and a maximum mass change easily
provokes numerous attempts to contradict them.
∗∗
The brute force approach. The simplest attempt to exceed the force limit is to try to ac-
celerate an object with a force larger than the maximum value. Now, acceleration implies
* ‘When an idea is just rising on the horizon, the soul’s temperature with respect to it is usually very cold.
Only gradually does the idea develop its warmth, and it is hottest (which is to say, exerting its greatest influ-
ence) when belief in the idea is already once again in decline.’ Friedrich Nietzsche (1844–1900), philosopher
and scholar. This is aphorism 207 – Sonnenbahn der Idee – from his text Menschliches Allzumenschliches –
Der Wanderer und sein Schatten.
gravitation, maximum speed and maximum force 121

the transfer of energy. This transfer is limited by the horizon equation (112) or the energy
limit (113). For any attempt to exceed the force limit, the flowing energy results in the
appearance of a horizon. The horizon then prevents the force from exceeding the limit,
because it imposes a limit on interaction.
Page 101 Let us explore the interaction limit. In special relativity we found that the acceleration
of an object is limited by its length. Indeed, at a distance given by 𝑐2 /2𝑎 in the direction
opposite to the acceleration 𝑎, a horizon appears. In other words, an accelerated body
breaks, at the latest, at that point. The force 𝐹 on a body of mass 𝑀 and radius 𝑅 is thus
limited by
𝑀 2
𝐹⩽ 𝑐 . (120)
2𝑅
It is straightforward to add the (usually small) effects of gravity. To be observable, an ac-
celerated body must remain larger than a black hole; inserting the corresponding radius
𝑅 = 2𝐺𝑀/𝑐2 we get the force limit (105). Dynamic attempts to exceed the force limit
thus fail.

Motion Mountain – The Adventure of Physics
∗∗
The rope attempt. We can also try to generate a higher force in a static situation, for ex-
ample by pulling two ends of a rope in opposite directions. We assume for simplicity
that an unbreakable rope exists. Any rope works because the potential energy between
its atoms can produce high forces between them. To produce a rope force exceeding the
limit value, we need to store large (elastic) energy in the rope. This energy must enter
from the ends. When we increase the tension in the rope to higher and higher values,
more and more (elastic) energy must be stored in smaller and smaller distances. To ex-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ceed the force limit, we would need to add more energy per distance and area than is
allowed by the horizon equation. A horizon thus inevitably appears. But there is no way
to stretch a rope across a horizon, even if it is unbreakable! A horizon leads either to the
breaking of the rope or to its detachment from the pulling system.
⊳ Horizons thus make it impossible to generate forces larger than the force limit.
In fact, the assumption of infinite wire strength is unnecessary: the force limit cannot be
exceeded even if the strength of the wire is only finite.
We note that it is not important whether an applied force pulls – as for ropes or wires
– or pushes. Also in the case of pushing two objects against each other, an attempt to
increase the force value without end will equally lead to the formation of a horizon, due
to the limit provided by the horizon equation. By definition, this happens precisely at
the force limit. As there is no way to use a horizon to push (or pull) on something, the
attempt to achieve a higher force ends once a horizon is formed. In short, static forces
cannot exceed the maximum force.
∗∗
The braking attempt. A force limit provides a maximum momentum change per time. We
can thus search for a way to stop a moving physical system so abruptly that the maximum
force might be exceeded. The non-existence of rigid bodies in nature, already known
Page 101 from special relativity, makes a completely sudden stop impossible; but special relativity
122 4 simple general relativity

on its own provides no lower limit to the stopping time. However, the inclusion of gravity
does. Stopping a moving system implies a transfer of energy. The energy flow per area
cannot exceed the value given by the horizon equation. Thus we cannot exceed the force
limit by stopping an object.
Similarly, if a rapid system is reflected instead of stopped, a certain amount of energy
needs to be transferred and stored for a short time. For example, when a tennis ball is
reflected from a large wall its momentum changes and a force is applied. If many such
balls are reflected at the same time, surely a force larger than the limit can be realized? It
turns out that this is impossible. If we attempted it, the momentum flow at the wall would
reach the limit given by the horizon equation and thus create a horizon. In that case, no
reflection is possible any more. So the limit cannot be exceeded through reflection.
∗∗
The classical radiation attempt. Instead of systems that pull, push, stop or reflect mat-
ter, we can explore systems where radiation is involved. However, the arguments hold
in exactly the same way, whether photons, gravitons or other particles are involved. In

Motion Mountain – The Adventure of Physics
particular, mirrors, like walls, are limited in their capabilities: it is impossible to use light
and mirrors to create a momentum change larger than 𝑐4 /4𝐺.
It is even impossible to create a force larger than the maximum force by concentrating
a large amount of light onto a surface. The same situation as for tennis balls arises: when
the limit value 𝐸/𝐴 given by the horizon equation (113) is reached, a horizon appears
that prevents the limit from being broken.
∗∗
The brick attempt. The force and power limits can also be tested with more concrete

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Gedanken experiments. We can try to exceed the force limit by stacking weight. But even
building an infinitely high brick tower does not generate a sufficiently strong force on its
foundations: integrating the weight, taking into account its decrease with height, yields a
finite value that cannot reach the force limit. If we continually increase the mass density
of the bricks, we need to take into account that the tower and the Earth will change into
a black hole. And black holes do not allow the force limit to be exceeded.
∗∗
The boost attempt. A boost can apparently be chosen in such a way that a 3-force value
Ref. 120 𝐹 in one frame is transformed into any desired value 𝐹󸀠 in another frame. This turns out
to be wrong. In relativity, 3-force cannot be increased beyond all bounds using boosts.
Page 83 In all reference frames, the measured 3-force can never exceed the proper force, i.e., the
3-force value measured in the comoving frame. (The situation can be compared to 3-
velocity, where a boost cannot be used to exceed the value 𝑐, whatever boost we may
choose; however, there is no strict equivalence, as the transformation behaviour of 3-
force and of 3-velocity differ markedly.)
∗∗
The divergence attempt. The force on a test mass 𝑚 at a radial distance 𝑑 from a Schwarz-
gravitation, maximum speed and maximum force 123

Ref. 113 schild black hole (for Λ = 0) is given by

𝐺𝑀𝑚
𝐹= . (121)
2𝐺𝑀
𝑑 2 √1 − 𝑑𝑐2

Similarly, the inverse square expression of universal gravitation states that the force
between two masses 𝑚 and 𝑀 is
𝐺𝑀𝑚
𝐹= . (122)
𝑑2
Both expressions can take any value; this suggest that no maximum force limit exists.
However, gravitational force can diverge only for non-physical, point-like masses.
However, there is a minimum approach distance to a mass 𝑚 given by

2𝐺𝑚
𝑑min = . (123)
𝑐2

Motion Mountain – The Adventure of Physics
The minimum approach distance is the corresponding black hole radius. Black hole
formation makes it impossible to achieve zero distance between two masses. Black hole
formation also makes it impossible to realize point-like masses. Point-like masses are
unphysical. As a result, in nature there is a (real) minimum approach distance, propor-
tional to the mass. If this minimum approach distance is introduced in equations (121)
and (122), we get
𝑐4 𝑀𝑚 1 𝑐4
𝐹= ⩽ . (124)
4𝐺 (𝑀 + 𝑚)2 √1 − 𝑀 4𝐺

The approximation of universal gravitation yields

𝑐4 𝑀𝑚 𝑐4
𝐹= ⩽ . (125)
4𝐺 (𝑀 + 𝑚)2 4𝐺

In both cases, the maximum force value is never exceeded, as long as we take into account
the physical size of masses or of observers.
∗∗
The consistency problem. If observers cannot be point-like, we might question whether
it is still correct to apply the original definition of momentum change or energy change
as the integral of values measured by observers attached to a given surface. In general
relativity, observers cannot be point-like, but they can be as small as desired. The original
definition thus remains applicable when taken as a limit procedure for ever-decreasing
observer size. Obviously, if quantum theory is taken into account, this limit procedure
comes to an end at the Planck length. This is not an issue in general relativity, as long as
the typical dimensions in the situation are much larger than the Planck value.
∗∗
124 4 simple general relativity

The quantum problem. If quantum effects are neglected, it is possible to construct sur-
Challenge 171 e faces with sharp angles or even fractal shapes that overcome the force limit. However,
such surfaces are not physical, as they assume that lengths smaller than the Planck length
can be realized or measured. The condition that a surface be physical implies that it must
Ref. 109, Ref. 111 have an intrinsic indeterminacy given by the Planck length. A detailed study shows that
quantum effects do not allow the horizon force to be exceeded.
∗∗
The relativistically extreme observer attempt. Any extreme observer, whether in rapid
inertial or in accelerated motion, has no chance to beat the force limit. In classical physics
we are used to thinking that the interaction necessary for a measurement can be made
as small as desired. This statement, however, is not valid for all observers; in particular,
extreme observers cannot fulfil it. For them, the measurement interaction is large. As a
result, a horizon forms that prevents the limit from being exceeded.
∗∗

Motion Mountain – The Adventure of Physics
The microscopic attempt. We can attempt to exceed the force limit by accelerating a small
particle as strongly as possible or by colliding it with other particles. High forces do in-
deed appear when two high energy particles are smashed against each other. However,
if the combined energy of the two particles became high enough to challenge the force
limit, a horizon would appear before they could get sufficiently close.
In fact, quantum theory gives exactly the same result. Quantum theory by itself
Ref. 121 already provides a limit to acceleration. For a particle of mass 𝑚 it is given by

2𝑚𝑐3
𝑎⩽ . (126)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ℏ
Here, ℏ = 1.1 ⋅ 10−34 Js is the quantum of action, a fundamental constant of nature. In
particular, this acceleration limit is satisfied in particle accelerators, in particle collisions
and in pair creation. For example, the spontaneous generation of electron–positron pairs
in intense electromagnetic fields or near black hole horizons does respect the limit (126).
Inserting the maximum possible mass for an elementary particle, namely the (corrected)
Vol. VI, page 40 Planck mass, we find that equation (126) then states that the horizon force is the upper
bound for elementary particles.
∗∗
The compaction attempt. Are black holes really the most dense form of matter or energy?
The study of black hole thermodynamics shows that mass concentrations with higher
Ref. 113 density than black holes would contradict the principles of thermodynamics. In black
hole thermodynamics, surface and entropy are related: reversible processes that reduce
entropy could be realized if physical systems could be compressed to smaller values than
the black hole radius. As a result, the size of a black hole is the limit size for a mass in
nature. Equivalently, the force limit cannot be exceeded in nature.
∗∗
The force addition attempt. In special relativity, composing velocities by a simple vector
gravitation, maximum speed and maximum force 125

addition is not possible. Similarly, in the case of forces such a naive sum is incorrect; any
attempt to add forces in this way would generate a horizon. If textbooks on relativity had
explored the behaviour of force vectors under addition with the same care with which
they explored that of velocity vectors, the force bound would have appeared much earl-
ier in the literature. (Obviously, general relativity is required for a proper treatment.) In
nature, large forces do not add up.
∗∗
In special relativity, a body moving more slowly than light in one frame does so in all
frames. Can you show that a force smaller than the invariant limit 𝑐4 /4𝐺 in one frame of
Challenge 172 s reference is also smaller in any other frame?
∗∗
We could also try to use the cosmological constant to produce forces that exceed the
maximum force. But also this method does not succeed, as discussed by John Barrow
Ref. 122 and Gary Gibbons.

Motion Mountain – The Adventure of Physics
∗∗
Challenge 173 r Can you propose and then resolve an additional attempt to exceed the force limit?

Gedanken experiments with the power and the mass flow limits
Like the force bound, the power bound must be valid for all imaginable systems. Here
are some attempts to refute it.
∗∗

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The cable-car attempt. Imagine an engine that accelerates a mass with an unbreakable
and massless wire (assuming that such a wire could exist). As soon as the engine reached
the power bound, either the engine or the exhausts would reach the horizon equation.
When a horizon appears, the engine cannot continue to pull the wire, as a wire, even
an infinitely strong one, cannot pass a horizon. The power limit thus holds whether the
engine is mounted inside the accelerating body or outside, at the end of the wire pulling
it.
∗∗
The mountain attempt. It is possible to define a surface that is so strangely bent that
it passes just below every nucleus of every atom of a mountain, like the surface A in
Figure 61. All atoms of the mountain above sea level are then just above the surface,
barely touching it. In addition, imagine that this surface is moving upwards with almost
the speed of light. It is not difficult to show that the mass flow through this surface is
higher than the mass flow limit. Indeed, the mass flow limit 𝑐3 /4𝐺 has a value of about
1035 kg/s. In a time of 10−22 s, the diameter of a nucleus divided by the speed of light,
only 1013 kg need to flow through the surface: that is the mass of a mountain.
The surface bent around atoms seems to provide a counter-example to the limit. How-
ever, a closer look shows that this is not the case. The problem is the expression ‘just
below’. Nuclei are quantum particles and have an indeterminacy in their position; this
126 4 simple general relativity

6000 m

mountain

nuclei

surface A

Motion Mountain – The Adventure of Physics
0m F I G U R E 61 The mountain
surface B attempt to exceed the
maximum mass ﬂow value.

indeterminacy is essentially the nucleus–nucleus distance. As a result, in order to be sure
that the surface of interest has all atoms above it, the shape cannot be that of surface A in
Figure 61. It must be a flat plane that remains below the whole mountain, like surface B

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
in the figure. However, a flat surface beneath a mountain does not allow the mass change
limit to be exceeded.
∗∗
The multiple atom attempt. We can imagine a number of atoms equal to the number
of the atoms of a mountain that all lie with large spacing (roughly) in a single plane.
Again, the plane is moving upwards with the speed of light. Again, also in this case the
indeterminacy in the atomic positions makes it impossible to observe or state that the
mass flow limit has been exceeded.
∗∗
The multiple black hole attempt. Black holes are typically large and the indeterminacy in
their position is thus negligible. The mass limit 𝑐3 /4𝐺, or power limit 𝑐5 /4𝐺, corresponds
to the flow of a single black hole moving through a plane at the speed of light. Several
black holes crossing a plane together at just under the speed of light thus seem to beat the
limit. However, the surface has to be physical: an observer must be possible on each of
its points. But no observer can cross a black hole. A black hole thus effectively punctures
the plane surface. No black hole can ever be said to cross a plane surface; even less so a
multiplicity of black holes. The limit remains valid.
∗∗
gravitation, maximum speed and maximum force 127

The multiple neutron star attempt. The mass limit seems to be in reach when several
neutron stars (which are slightly less dense than a black hole of the same mass) cross a
plane surface at the same time, at high speed. However, when the speed approaches the
speed of light, the crossing time for points far from the neutron stars and for those that
actually cross the stars differ by large amounts. Neutron stars that are almost black holes
cannot be crossed in a short time in units of a coordinate clock that is located far from
the stars. Again, the limit is not exceeded.
∗∗
The luminosity attempt. The existence of a maximum luminosity bound has been dis-
Ref. 113 cussed by astrophysicists. In its full generality, the maximum bound on power, i.e., on
energy per time, is valid for any energy flow through any physical surface whatsoever.
The physical surface may even run across the whole universe. However, not even bring-
ing together all lamps, all stars and all galaxies of the universe yields a surface which has
a larger power output than the proposed limit.
The surface must be physical.* A surface is physical if an observer can be placed on

Motion Mountain – The Adventure of Physics
each of its points. In particular, a physical surface may not cross a horizon, or have
local detail finer than a certain minimum length. This minimum length will be intro-
Vol. VI, page 67 duced later on; it is given by the corrected Planck length. If a surface is not physical, it
Challenge 174 s may provide a counter-example to the power or force limits. However, these unphysical
counter-examples make no statements about nature. (Ex falso quodlibet.**)
∗∗
The many lamps attempt, or power paradox. An absolute power limit imposes a limit on
the rate of energy transport through any imaginable, physical surface. At first sight, it may

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
seem that the combined power emitted by two radiation sources that each emit 3/4 of
the maximum value should emit a total of 3/2 times the maximum value, and thus allow
us to overcome the power limit. However, two such lamps would be so massive that they
Challenge 175 e would form a horizon around them – a black hole would form. Again, since the horizon
limit (113) is achieved, the arising horizon swallows parts of the radiation and prevents
the force or power limit from being exceeded. Exploring a numerical simulation of this
Challenge 176 r situation would be instructive. Can you provide one? In short, we can say that large power
values do not add up in nature.
∗∗
The light concentration attempt. Another approach is to shine a powerful, short and
spherical flash of light onto a spherical mass. At first sight it seems that the force and
power limits can be exceeded, because light energy can be concentrated into small
volumes. However, a high concentration of light energy forms a black hole or induces
the mass to form one. There is no way to pump energy into a mass at a faster rate than
that dictated by the power limit. In fact, it is impossible to group light sources in such
a way that their total output is larger than the power limit. Every time the force limit is
approached, a horizon appears that prevents the limit from being exceeded.

* It can also be called physically sensible.
** ‘Anything can be deduced from a falsehood.’
128 4 simple general relativity

∗∗
The black hole attempt. One possible system in nature that actually achieves the power
limit is the final stage of black hole evaporation. However, even in this case the power
limit is not exceeded, but only equalled.
∗∗
The saturation attempt. If the universe already saturates the power limit, any new power
source would break it, or at least imply that another elsewhere must close down. Can you
Challenge 177 s find the fallacy in this argument?
∗∗
The water flow attempt. We could try to pump water as rapidly as possible through a
large tube of cross-section 𝐴. However, when a tube of length 𝐿 filled with water flowing
at speed 𝑣 gets near to the mass flow limit, the gravity of the water waiting to be pumped
through the area 𝐴 will slow down the water that is being pumped through the area. The
limit is again reached when the cross-section 𝐴 turns into a horizon.

Motion Mountain – The Adventure of Physics
∗∗
Checking that no system – from microscopic to astrophysical – ever exceeds the max-
imum power or maximum mass flow is a further test of general relativity. It may seem
easy to find a counter-example, as the surface may run across the whole universe or en-
velop any number of elementary particle reactions. However, no such attempt succeeds.
∗∗
In summary, in all situations where the force, power or mass-flow limits are challenged,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
whenever the energy flow reaches the black hole mass–energy density in space or the cor-
responding momentum flow in time, an event horizon appears; this horizon then makes
it impossible to exceed the limits. All three limits are confirmed both in observation and
in theory. Values exceeding the limits can neither be generated nor measured. Gedanken
experiments also show that the three bounds are the tightest ones possible. Obviously,
all three limits are open to future tests and to further Gedanken experiments. (If you can
Challenge 178 r think of a good one, let the author know.)

Why maximum force has remained undiscovered for so long
The first reason why the maximum force principle remained undiscovered for so long is
the absence of horizons in everyday life. Due to this lack, experiments in everyday life do
not highlight the force or power limits. It took many decades before physicists realized
that the dark night sky is not something unique, but only one example of an observation
that is common in nature: nature is full of horizons. But in everyday life, horizons do not
play an important role – fortunately – because the nearest one is probably located at the
centre of the Milky Way.
The second reason why the principle of maximum force remained hidden is the erro-
neous belief that point particles exist. This is a theoretical prejudice due to a common
idealization used in Galilean physics. For a complete understanding of general relativity
it is essential to remember regularly that point particles, point masses and point-like ob-
gravitation, maximum speed and maximum force 129

servers do not exist. They are approximations that are only applicable in Galilean physics,
in special relativity or in quantum theory. In general relativity, horizons prevent the ex-
istence of point-like systems. The incorrect habit of believing that the size of a system can
be made as small as desired while keeping its mass constant prevents the force or power
limit from being noticed.
The third reason why the principle of maximum force remained hidden are prejudices
against the concept of force. In general relativity, gravitational force is hard to define.
Even in Galilean physics it is rarely stressed that force is the flow of momentum through
a surface. The teaching of the concept of force is incomplete since centuries – with rare
Ref. 123 notable exceptions – and thus the concept is often avoided.
In summary, the principle of maximum force – or of maximum power – has remained
undiscovered for so long because a ‘conspiracy’ of nature and of thinking habits hid it
from most experimental and theoretical physicists.

An intuitive understanding of general relativit y

Motion Mountain – The Adventure of Physics
“
Wir leben zwar alle unter dem gleichen
Himmel, aber wir haben nicht alle den gleichen

”
Horizont.*
Konrad Adenauer

The concepts of horizon force and horizon power can be used as the basis for a direct,
intuitive approach to general relativity.
∗∗
What is gravity? Of the many possible answers we will encounter, we now have the first:
gravity is the ‘shadow’ of the maximum force. Whenever we experience gravity as weak,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
we can remember that a different observer at the same point and time would experi-
ence the maximum force. Searching for the precise properties of that observer is a good
exercise. Another way to put it: if there were no maximum force, gravity would not exist.
∗∗
The maximum force implies universal gravity. To see this, we study a simple planetary
system, i.e., one with small velocities and small forces. A simple planetary system of size
𝐿 consists of a (small) satellite circling a central mass 𝑀 at a radial distance 𝑅 = 𝐿/2.
Let 𝑎 be the acceleration of the object. Small velocity implies the condition 𝑎𝐿 ≪ 𝑐2 , de-
duced from special relativity; small force implies √4𝐺𝑀𝑎 ≪ 𝑐2 , deduced from the force
limit. These conditions are valid for the system as a whole and for all its components.
Both expressions have the dimensions of speed squared. Since the system has only one
characteristic speed, the two expressions 𝑎𝐿 = 2𝑎𝑅 and √4𝐺𝑀𝑎 must be proportional,
yielding
𝐺𝑀
𝑎=𝑓 2 , (127)
𝑅
where the numerical factor 𝑓 must still be determined. To determine it, we study the

* ‘We all live under the same sky, but we do not have the same horizon.’ Konrad Adenauer (1876–1967),
West German Chancellor.
130 4 simple general relativity

escape velocity necessary to leave the central body. The escape velocity must be smaller
than the speed of light for any body larger than a black hole. The escape velocity, derived
2
from expression (127), from a body of mass 𝑀 and radius 𝑅 is given by 𝑣esc = 2𝑓𝐺𝑀/𝑅.
The minimum radius 𝑅 of objects, given by 𝑅 = 2𝐺𝑀/𝑐2 , then implies that 𝑓 = 1.
Therefore, for low speeds and low forces, the inverse square law describes the orbit of a
satellite around a central mass.
∗∗
If empty space-time is elastic, like a piece of metal, it must also be able to oscillate. Any
physical system can show oscillations when a deformation brings about a restoring force.
We saw above that there is such a force in the vacuum: it is called gravitation. In other
words, vacuum must be able to oscillate, and since it is extended, it must also be able to
sustain waves. Indeed, gravitational waves are predicted by general relativity, as we will
Page 174 see below.
∗∗

Motion Mountain – The Adventure of Physics
If curvature and energy are linked, the maximum speed must also hold for gravitational
energy. Indeed, we will find that gravity has a finite speed of propagation. The inverse
square law of everyday life cannot be correct, as it is inconsistent with any speed limit.
More about the corrections induced by the maximum speed will become clear shortly.
In addition, since gravitational waves are waves of massless energy, we would expect the
Page 174 maximum speed to be their propagation speed. This is indeed the case, as we will see.
∗∗
A body cannot be denser than a (non-rotating) black hole of the same mass. The max-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
imum force and power limits that apply to horizons make it impossible to squeeze mass
into smaller horizons. The maximum force limit can therefore be rewritten as a limit for
the size 𝐿 of physical systems of mass 𝑚:

4𝐺𝑚
𝐿⩾ . (128)
𝑐2
If we call twice the radius of a black hole its ‘size’, we can state that no physical system
of mass 𝑚 is smaller than this value.* The size limit plays an important role in general
relativity. The opposite inequality, 𝑚 ⩾ √𝐴/16π 𝑐2 /𝐺, which describes the maximum
‘size’ of black holes, is called the Penrose inequality and has been proven for many phys-
Ref. 124, Ref. 125 ically realistic situations. The Penrose inequality can be seen to imply the maximum force
limit, and vice versa. The maximum force principle, or the equivalent minimum size of
matter–energy systems, thus prevents the formation of naked singularities. (Physicists
call the lack of naked singularities the so-called cosmic censorship. conjecture.)
∗∗
There is a power limit for all energy sources. In particular, the value 𝑐5 /4𝐺 limits the lu-

* The maximum value for the mass to size limit is obviously equivalent to the maximum mass change given
above.
gravitation, maximum speed and maximum force 131

minosity of all gravitational sources. Indeed, all formulae for gravitational wave emission
Ref. 113 imply this value as an upper limit. Furthermore, numerical relativity simulations never
exceed it: for example, the power emitted during the simulated merger of two black holes
is below the limit.
∗∗
Perfectly plane waves do not exist in nature. Plane waves are of infinite extension. But
neither electrodynamic nor gravitational waves can be infinite, since such waves would
carry more momentum per time through a plane surface than is allowed by the force
limit. The non-existence of plane gravitational waves also precludes the production of
singularities when two such waves collide.
∗∗
In nature, there are no infinite forces. There are thus no (naked nor dressed) singularities
in nature. Horizons prevent the appearance of singularities. In particular, the big bang
was not a singularity. The mathematical theorems by Penrose and Hawking that seem to

Motion Mountain – The Adventure of Physics
imply the existence of singularities tacitly assume the existence of point masses – often in
the form of ‘dust’ – in contrast to what general relativity implies. Careful re-evaluation
of each such proof is necessary.
∗∗
The force limit means that space-time has a limited stability. The limit suggests that
space-time can be torn into pieces. In a sense, this is indeed the case, even though hori-
zons usually prevent it. However, the way that this tearing happens is not described by
general relativity. We will study it in the last part of this text.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
The maximum force is the standard of force. This implies that the gravitational constant
𝐺 is constant in space and time – or at least, that its variations across space and time
Ref. 126 cannot be detected. Present data support this claim to a high degree of precision.
∗∗
The maximum force principle implies that gravitational energy – as long as it can be
defined – falls in gravitational fields in the same way as other type of energy. As a result,
Ref. 113 the maximum force principle predicts that the Nordtvedt effect vanishes. The Nordtvedt
effect is a hypothetical periodical change in the orbit of the Moon that would appear if the
gravitational energy of the Earth–Moon system did not fall, like other mass–energy, in
the gravitational field of the Sun. Lunar range measurements have confirmed the absence
of this effect.
∗∗
If horizons are surfaces, we can ask what their colour is. We will explore this question
Page 262 later on.
∗∗
Vol. VI, page 37 Later on we will find that quantum effects cannot be used to exceed the force or power
132 4 simple general relativity

Challenge 179 e limit. (Can you guess why?) Quantum theory also provides a limit to motion, namely
a lower limit to action; however, this limit is independent of the force or power limit.
(A dimensional analysis already shows this: there is no way to define an action by com-
binations of 𝑐 and 𝐺.) Therefore, even the combination of quantum theory and general
relativity does not help in overcoming the force or power limits.
∗∗
Given that the speed 𝑐 and the force value 𝑐4 /4𝐺 are limit values, what can be said about
𝐺 itself? The gravitational constant 𝐺 describes the strength of the gravitational inter-
action. In fact, 𝐺 characterizes the strength of the weakest possible interaction. In other
words, given a central body of mass 𝑀, and given the acceleration 𝑎 of a test body at a
distance 𝑟 due to any interaction whatsoever with the central body, then the ratio 𝑎𝑟2 /𝑀
is at least equal to 𝐺. (Can you show that geostationary satellites or atoms in crystals are
Challenge 180 e not counterexamples?) In summary, also the gravitational constant 𝐺 is a limit value in
nature.

Motion Mountain – The Adventure of Physics
An intuitive understanding of cosmolo gy
Page 240 A maximum power is the simplest possible explanation of Olbers’ paradox. Power and
luminosity are two names for the same observable. The sum of all luminosity values in
the universe is finite; the light and all other energy emitted by all stars, taken together,
is finite. If we assume that the universe is homogeneous and isotropic, the power limit
𝑃 ⩽ 𝑐5 /4𝐺 must be valid across any plane that divides the universe into two halves. The
part of the universe’s luminosity that arrives on Earth is then so small that the sky is dark
at night. In fact, the actually measured luminosity is still smaller than this calculation,
as a large part of the power is not visible to the human eye – and most of the emitted

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
power is matter anyway. In other words, the night is dark because of nature’s power limit.
This explanation is not in contrast to the usual one, which uses the finite lifetime of stars,
their finite density, their finite size, and the finite age and the expansion of the universe.
In fact, the combination of all these usual arguments simply implies and repeats in more
complex words that the power limit cannot be exceeded. However, the much simpler
explanation with the power limit seems to be absent in the literature.
The existence of a maximum force in nature, together with homogeneity and isotropy,
implies that the visible universe is of finite size. The opposite case would be an infinitely
large, homogeneous and isotropic universe of density 𝜌0 . But in this case, any two halves
of the universe would attract each other with a force above the limit (provided the uni-
verse were sufficiently old). This result can be made quantitative by imagining a sphere
of radius 𝑅0 whose centre lies at the Earth, which encompasses all the universe, and
whose radius changes with time (almost) as rapidly as the speed of light. The mass flow
d𝑚/d𝑡 = 𝜌0 𝐴 0 𝑐 through that sphere is predicted to reach the mass flow limit 𝑐3 /4𝐺; thus
we have
𝑐3
𝜌0 4π𝑅20 𝑐 ⩽ . (129)
4𝐺
We can compare this with the Friedmann models, who predict, in a suitable limit, that
Ref. 127 one third of the left hand side saturates the mass flow limit. The precision measurements
gravitation, maximum speed and maximum force 133

of the cosmic background radiation by the WMAP satellite confirm that the present-day
total energy density 𝜌0 – including dark matter and dark energy – and the horizon radius
𝑅0 just reach the Friedmann value. The above argument using the maximum force or
mass flow thus still needs a slight correction.
In summary, the maximum force limit predicts, within a factor of 6, the observed
relation between the size and density of the universe. In particular, the maximum force
principle predicts that the universe is of finite size. By the way, a finite limit for power
also suggests that a finite age for the universe can be deduced. Can you find an argument?
Challenge 181 s

Experimental challenges for the third millennium
The lack of direct tests of the horizon force, power or mass flow is obviously due to the
lack of horizons in the vicinity of researchers. Nevertheless, the limit values are observ-
able and falsifiable.
The force limit might be tested with high-precision measurements in binary pulsars

Motion Mountain – The Adventure of Physics
or binary black holes. Such systems allow precise determination of the positions of the
two stars. The maximum force principle implies a relation between the position error Δ𝑥
Ref. 109, Ref. 111 and the energy error Δ𝐸. For all systems we have

Δ𝐸 𝑐4
⩽ . (130)
Δ𝑥 4𝐺
For example, a position error of 1 mm gives a mass error of below 3 ⋅ 1023 kg. In everyday
life, all measurements comply with this relation. Indeed, the left side is so much smaller
than the right side that the relation is rarely mentioned. For a direct check, only systems

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
which might achieve direct equality are interesting: dual black holes or dual pulsars are
such systems. Pulsar experiments and gravitational wave detectors therefore can test the
power limit in the coming years.
The power limit implies that the highest luminosities are only achieved when systems
emit energy at the speed of light. Indeed, the maximum emitted power is only achieved
when all matter is radiated away as rapidly as possible: the emitted power 𝑃 = 𝑐2 𝑀/(𝑅/𝑣)
cannot reach the maximum value if the body radius 𝑅 is larger than that of a black hole
(the densest body of a given mass) or the emission speed 𝑣 is lower than that of light. The
sources with highest luminosity must therefore be of maximum density and emit entities
without rest mass, such as gravitational waves, electromagnetic waves or (maybe) gluons.
Candidates to detect the limit are black holes in formation, in evaporation or undergoing
mergers. Gravitational wave detectors therefore can test the power limit in the coming
years.
A candidate surface that reaches the power limit is the night sky. The night sky is
a horizon. Provided that light, neutrino, particle and gravitational wave flows are added
together, the limit 𝑐5 /4𝐺 is predicted to be reached. If the measured power is smaller than
the limit (as it seems to be at present), this might even give a hint about new particles
yet to be discovered. If the limit were exceeded or not reached, general relativity would
be shown to be incorrect. This might be an interesting future experimental test.
The power limit implies that a wave whose integrated intensity approaches the force
134 4 simple general relativity

limit cannot be plane. The power limit thus implies a limit on the product of intensity
𝐼 (given as energy per unit time and unit area) and the size (curvature radius) 𝑅 of the
front of a wave moving with the speed of light 𝑐:

𝑐5
4π𝑅2 𝐼 ⩽ . (131)
4𝐺
Obviously, this statement is difficult to check experimentally, whatever the frequency
and type of wave might be, because the value appearing on the right-hand side is ex-
tremely large. Possibly, future experiments with gravitational wave detectors, X-ray de-
tectors, gamma ray detectors, radio receivers or particle detectors might allow us to test
relation (131) with precision.
It might well be that the amount of matter falling into some black hole, such as the one
at the centre of the Milky Way, might be measurable one day. The limit d𝑚/d𝑡 ⩽ 𝑐3 /4𝐺
could then be tested directly.
In short, direct tests of the limits are possible, but not easy. In fact, you might want

Motion Mountain – The Adventure of Physics
Challenge 182 e to predict which of these experiments will confirm the limit first. The scarcity of direct
experimental tests of the force, power and mass flow limits implies that indirect tests be-
come particularly important. All such tests study the motion of matter or energy and
compare it with a famous consequence of the limits: the field equations of general re-
lativity. This will be our next topic.

A summary of general relativit y – and minimum force

“
Non statim pusillum est si quid maximo minus

”
est.*

There is a simple axiomatic formulation of general relativity: the horizon force 𝑐4 /4𝐺 and
the horizon power 𝑐5 /4𝐺 are the highest possible force and power values. No contradict-
ing observation is known. No counter-example has been imagined. General relativity
follows from these limits. Moreover, the limits imply the darkness of the night and the
finiteness of the universe.
The principle of maximum force has obvious applications for the teaching of gen-
eral relativity. The principle brings general relativity to the level of first-year university,
and possibly to well-prepared secondary school, students: only the concepts of max-
imum force and horizon are necessary. Space-time curvature is a consequence of horizon
curvature.
The concept of a maximum force leads us to an additional aspect of gravitation.
The cosmological constant Λ is not fixed by the maximum force principle. (However,
Challenge 183 e the principle does fix its sign to be positive.) Present measurements give the result
Page 243 Λ ≈ 10−52 /m2 . A positive cosmological constant implies the existence of a negative en-
ergy volume density −Λ𝑐4 /𝐺. This value corresponds to a negative pressure, as pressure
and energy density have the same dimensions. Multiplication by the (numerically cor-

* ‘Nothing is negligible only because it is smaller than the maximum.’ Lucius Annaeus Seneca (c. 4 bce
–65), Epistolae 16, 100.
gravitation, maximum speed and maximum force 135

Vol. VI, page 37 rected) Planck area 4𝐺ℏ/𝑐3 , the smallest area in nature, gives a force value

𝐹 = 4Λℏ𝑐 = 1.20 ⋅ 10−77 N . (132)

This is also the gravitational force between two (numerically corrected) Planck masses
√ℏ𝑐/4𝐺 located at the cosmological distance 1/4√Λ .
We conjecture that expression (132) is the minimum force in nature. Proving this con-
jecture is more involved than for the case of maximum force. So far, only some hints
are possible. Like the maximum force, also the minimum force must be compatible
with gravitation, must not be contradicted by any experiment, and must withstand any
thought experiment. A quick check shows that the minimum force allows us to deduce
the cosmological constant of gravitation; minimum force is an invariant and is not con-
tradicted by any experiment. There are also hints that there may be no way to generate or
measure any smaller value. For example, the gravitational force between any two neutral
particles at cosmological distance, such as between two atoms or two neutrinos, is much
smaller than the minimum force; however, it seems impossible to detect experimentally

Motion Mountain – The Adventure of Physics
whether two such particles interact at all: the acceleration is too small to be measured.
Challenge 184 e As another example, the minimum force corresponds to the energy per length contained
by a photon with a wavelength of the size of the universe. It is hard – but maybe not im-
Challenge 185 d possible – to imagine the measurement of a still smaller force. (Can you do so?)
If we leap to the – not completely proven – conclusion that expression (132) is the
smallest possible force in nature (the numerical factors are not yet verified), we get the
fascinating conjecture that the full theory of general relativity, including the cosmological
constant, may be defined by the combination of a maximum and a minimum force in
nature.

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We have seen that both the maximum force principle and general relativity fail to fix
the value of the cosmological constant. Only a unified theory can do so. We thus get two
requirements for such a theory. First, any unified theory must predict the same upper
limit to force as general relativity. Secondly, a unified theory must fix the cosmological
constant. The appearance of ℏ in the conjectured expression for the minimum force sug-
gests that the minimum force is determined by a combination of general relativity and
quantum theory. The proof of this suggestion and the confirmation of the minimum force
are two important challenges for our ascent beyond general relativity. We come back to
the issue in the last part of our adventure.
We are now ready to explore the consequences of general relativity and its field equa-
tions in more detail. We start by focusing on the concept of space-time curvature in
everyday life, and in particular, on its consequences for the observation of motion.
Chapter 5

HOW M A X I M UM SPE E D C HA NG E S
SPAC E , T I M E A N D G R AV I T Y

“ ”
Sapere aude.**
Horace Epistulae, 1, 2, 40.

O
bservation shows that gravitational influences do transport energy.***
ur description of gravity must therefore include the speed limit.

Motion Mountain – The Adventure of Physics
nly a description that takes into account the limit speed for energy transport
can be a precise description of gravity. Henri Poincaré stated this requirement for a
precise description of gravitation as early as 1905. But universal gravity, with its relation
𝑎 = 𝐺𝑀/𝑟2 , allows speeds higher than that of light. For example, in universal gravity,
the speed of a mass in orbit is not limited. In universal gravity it is also unclear how the
values of 𝑎 and 𝑟 depend on the observer. In short, universal gravity cannot be correct.
In order to reach the correct description, called general relativity by Albert Einstein, we
Ref. 128, Ref. 129 have to throw quite a few preconceptions overboard.
The results of combining maximum speed with gravity are fascinating: we will find
that empty space can bend and move, that the universe has a finite age and that objects

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
can be in permanent free fall. We will discover that even though empty space can be bent,
it is much stiffer than steel. Despite the strangeness of these and other consequences, they
have all been confirmed by all experiments performed so far.

R est and free fall
The opposite of motion in daily life is a body at rest, such as a child sleeping or a rock
defying the waves. A body is at rest whenever it is not disturbed by other bodies. In
the everyday description of the world, rest is the absence of velocity. With Galilean and
special relativity, rest became inertial motion, since no inertial observer can distinguish
its own motion from rest: nothing disturbs him. Both the rock in the waves and the rapid
protons crossing the galaxy as cosmic rays are at rest. With the inclusion of gravity, we
are led to an even more general definition of rest.

⊳ Every observer and every body in free fall can rightly claim to be at rest.

Challenge 186 e If any body moving inertially is to be considered at rest, then any body in free fall must
also be. Nobody knows this better than Joseph Kittinger, the man who in August 1960

** ‘Venture to be wise.’ Horace is Quintus Horatius Flaccus, (65–8 bce), the great Roman poet.
*** The details of this statement are far from simple. They are discussed on page 174 and page 204.
how maximum speed changes space, time and gravity 137

Ref. 130 stepped out of a balloon capsule at the record height of 31.3 km. At that altitude, the air
is so thin that during the first minute of his free fall he felt completely at rest, as if he
were floating. Although an experienced parachutist, he was so surprised that he had to
turn upwards in order to convince himself that he was indeed moving away from his
balloon! Despite his lack of any sensation of movement, he was falling at up to 274 m/s
or 988 km/h with respect to the Earth’s surface. He only started feeling something when
he encountered the first substantial layers of air. That was when his free fall started to be
disturbed. Later, after four and a half minutes of fall, his special parachute opened; and
nine minutes later he landed in New Mexico.
Kittinger and all other observers in free fall, such as the cosmonauts circling the Earth
or the passengers in parabolic aeroplane flights,* make the same observation: it is im-
possible to distinguish anything happening in free fall from what would happen at rest.
This impossibility is called the principle of equivalence; it is one of the starting points of
general relativity. It leads to the most precise – and final – definition of rest that we will
encounter in our adventure:

Motion Mountain – The Adventure of Physics
⊳ Rest is free fall.

Rest, like free fall, is the lack of disturbance.
The set of all possible free-falling observers at a point in space-time generalizes the
special-relativistic notion of the set of the inertial observers at a point. This means that
we must describe motion in such a way that not only all inertial but also all freely falling
observers can talk to each other. In addition, a full description of motion must be able to
describe gravitation and the motion it produces, and it must be able to describe motion
for any observer imaginable. General relativity realizes this aim.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
As a first step, we put the result on rest in other words:

⊳ True motion is the opposite of free fall.

This statement immediately rises a number of questions: Most trees or mountains are
Challenge 187 s not in free fall, thus they are not at rest. What motion are they undergoing? And if free
fall is rest, what is weight? And what then is gravity anyway? Let us start with the last
question.

What clo cks tell us ab ou t gravit y
Page 129 Above, we described gravity as the shadow of the maximum force. But there is a second
way to describe it, more closely related to everyday life. As William Unruh likes to
Ref. 131 explain, the constancy of the speed of light for all observers implies a simple conclusion:

⊳ Gravity is the uneven running of clocks at different places.**

* Nowadays it is possible to book such flights at specialized travel agents.
** Gravity is also the uneven length of metre bars at different places, as we will see below. Both effects are
needed to describe it completely; but for daily life on Earth, the clock effect is sufficient, since it is much
Challenge 188 s larger than the length effect, which can usually be neglected. Can you see why?
138 5 how maximum speed changes space, time and gravity

𝑣(𝑡) = 𝑔 𝑡

B light F

F I G U R E 62 Inside an accelerating train or
bus.

Challenge 189 e Of course, this seemingly absurd definition needs to be checked. The definition does not
talk about a single situation seen by different observers, as we often did in special relativ-
ity. The definition depends on the observation that neighbouring, identical clocks, fixed
against each other, run differently in the presence of a gravitational field when watched

Motion Mountain – The Adventure of Physics
by the same observer; moreover, this difference is directly related to what we usually call
gravity. There are two ways to check this connection: by experiment and by reasoning.
Let us start with the latter method, as it is cheaper, faster and more fun.
An observer feels no difference between gravity and constant acceleration. We can
thus study constant acceleration and use a way of reasoning we have encountered already
in the chapter on special relativity. We assume light is emitted at the back end of a train or
bus of length Δℎ that is accelerating forward with acceleration 𝑔, as shown in Figure 62.
The light arrives at the front of the train or bus after a time 𝑡 = Δℎ/𝑐. However, during this
time the accelerating train or bus has picked up some additional velocity, namely Δ𝑣 =

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑔𝑡 = 𝑔Δℎ/𝑐. As a result, because of the Doppler effect we encountered in our discussion
Page 55 of special relativity, the frequency 𝑓 of the light arriving at the front has changed. Using
Challenge 190 e the expression of the Doppler effect, we get*

Δ𝑓 𝑔Δℎ
= 2 . (133)
𝑓 𝑐

The sign of the frequency change depends on whether the light motion and the train
acceleration are in the same or in opposite directions. For actual trains or buses, the
Challenge 192 s frequency change is quite small; nevertheless, it is measurable.

⊳ Acceleration induces frequency changes in light.

Let us compare this first effect of acceleration with the effects of gravity.
To measure time and space, we use light. What happens to light when gravity is
Ref. 132 involved? The simplest experiment is to let light fall or rise. In order to deduce what
must happen, we add a few details. Imagine a conveyor belt carrying masses around two
wheels, a low and a high one, as shown in Figure 63. The descending, grey masses are

* The expression 𝑣 = 𝑔𝑡 is valid only for non-relativistic speeds; nevertheless, the conclusion of this section
Challenge 191 e is not affected by this approximation.
how maximum speed changes space, time and gravity 139

𝑚

𝑚 + 𝐸/𝑐2

ℎ

Motion Mountain – The Adventure of Physics
light

F I G U R E 63 The necessity of blue- and red-shift of
light: why trees are greener at the bottom.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
slightly larger. Whenever such a larger mass is near the bottom, some mechanism – not
shown in the figure – converts the mass surplus to light, in accordance with the equation
𝐸 = 𝑐2 𝑚, and sends the light up towards the top.* At the top, one of the lighter, white
masses passing by absorbs the light and, because of its added weight, turns the conveyor
belt until it reaches the bottom. Then the process repeats.**
As the grey masses on the descending side are always heavier, the belt would turn for
ever and this system could continuously generate energy. However, since energy conser-
Vol. I, page 280 vation is at the basis of our definition of time, as we saw in the beginning of our walk, the
whole process must be impossible. We have to conclude that the light changes its energy
when climbing. The only possibility is that the light arrives at the top with a frequency
different from the one at which it is emitted from the bottom.***
In short, it turns out that

⊳ Rising light is gravitationally red-shifted.

* As in special relativity, here and in the rest of our adventure, the term ‘mass’ always refers to rest mass.
Challenge 193 s ** Can this process be performed with 100 % efficiency?
*** The precise relation between energy and frequency of light is described and explained in the discussion
Vol. IV, page 44 on quantum theory. But we know already from classical electrodynamics that the energy of light depends
on its intensity and on its frequency.
140 5 how maximum speed changes space, time and gravity

Similarly, the light descending from the top of a tree down to an observer is blue-shifted;
this gives a darker colour to the top in comparison with the bottom of the tree. The com-
bination of light speed invariance and gravitation thus imply that trees have different
shades of green along their height.* How big is the effect? The result deduced from the
Challenge 195 e drawing is again the one of formula (133). That is what we would expect, as light mov-
ing in an accelerating train and light moving in gravity are equivalent situations, as you
Challenge 196 s might want to check yourself. The formula gives a relative change of frequency of only
1.1 ⋅ 10−16 /m near the surface of the Earth. For trees, this so-called gravitational red-shift
or gravitational Doppler effect is far too small to be observable, at least using normal light.
Ref. 133 In 1911, Einstein proposed an experiment to check the change of frequency with height
by measuring the red-shift of light emitted by the Sun, using the famous Fraunhofer lines
Vol. IV, page
Page 180
312 as colour markers. The results of the first experiments, by Schwarzschild and others, were
unclear or even negative, due to a number of other effects that induce colour changes at
high temperatures. But in 1920 and 1921, Leonhard Grebe and Albert Bachem, and inde-
Ref. 134 pendently Alfred Perot, confirmed the gravitational red-shift with careful experiments.
In later years, technological advances made the measurements much easier, until it was

Motion Mountain – The Adventure of Physics
even possible to measure the effect on Earth. In 1960, in a classic experiment using the
Mössbauer effect, Pound and Rebka confirmed the gravitational red-shift in their uni-
Ref. 135 versity tower using 𝛾 radiation.
But our two thought experiments tell us much more. Let us use the same argument as
in the case of special relativity: a colour change implies that clocks run differently at dif-
ferent heights, just as they run differently in the front and in the back of a train. The time
difference Δ𝜏 is predicted to depend on the height difference Δℎ and the acceleration of
gravity 𝑔 according to
Δ𝜏 Δ𝑓 𝑔Δℎ
= = 2 . (134)

In simple words,

⊳ In gravity, time is height-dependent.

Challenge 197 e In other words, height makes old. Can you confirm this conclusion?
In 1972, by flying four precise clocks in an aeroplane while keeping an identical one
Ref. 55 on the ground, Hafele and Keating found that clocks indeed run differently at different
Ref. 136 altitudes according to expression (134). Subsequently, in 1976, the team of Vessot shot
a precision clock based on a maser – a precise microwave generator and oscillator –
upwards on a missile. The team compared the maser inside the missile with an identical
maser on the ground and again confirmed the above expression. In 1977, Briatore and
Ref. 137 Leschiutta showed that a clock in Torino indeed ticks more slowly than one on the top of
the Monte Rosa. They confirmed the prediction that on Earth, for every 100 m of height
Challenge 198 e gained, people age more rapidly by about 1 ns per day. This effect has been confirmed for
all systems for which experiments have been performed, such as several planets, the Sun
and numerous other stars.

Challenge 194 ny * How does this argument change if you include the illumination by the Sun?
how maximum speed changes space, time and gravity 141

Do these experiments show that time changes or are they simply due to clocks that
Challenge 199 e function badly? Take some time and try to settle this question. We will give one argument
only: gravity does change the colour of light, and thus really does change time. Clock
precision is not an issue here.
In summary, gravity is indeed the uneven running of clocks at different heights. Note
that an observer at the lower position and another observer at the higher position agree
on the result: both find that the upper clock goes faster. In other words, when gravity is
present, space-time is not described by the Minkowski geometry of special relativity, but
by some more general geometry. To put it mathematically, whenever gravity is present,
the 4-distance d𝑠2 between events is different from the expression without gravity:

d𝑠2 ≠ 𝑐2 d𝑡2 − d𝑥2 − d𝑦2 − d𝑧2 . (135)

We will give the correct expression shortly.
Is this view of gravity as height-dependent time really reasonable? No. It turns out
that it is not yet strange enough! Since the speed of light is the same for all observers,

Motion Mountain – The Adventure of Physics
we can say more. If time changes with height, length must also do so! More precisely, if
clocks run differently at different heights, the length of metre bars must also change with
Challenge 200 s height. Can you confirm this for the case of horizontal bars at different heights?
If length changes with height, the circumference of a circle around the Earth cannot be
given by 2π𝑟. An analogous discrepancy is also found by an ant measuring the radius and
circumference of a circle traced on the surface of a basketball. Indeed, gravity implies that
humans are in a situation analogous to that of ants on a basketball, the only difference
being that the circumstances are translated from two to three dimensions. We conclude
that wherever gravity plays a role, space is curved.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
What tides tell us ab ou t gravit y
During his free fall, Kittinger was able to specify an inertial frame for himself. Indeed,
he felt completely at rest. Does this mean that it is impossible to distinguish acceleration
from gravitation? No: distinction is possible. We only have to compare two (or more)
falling observers, or two parts of one observer.
Kittinger could not have found a frame which is also inertial for a colleague falling
Challenge 201 e on the opposite side of the Earth. Such a common frame does not exist. In general, it is
impossible to find a single inertial reference frame describing different observers freely
falling near a mass. In fact, it is impossible to find a common inertial frame even for
nearby observers in a gravitational field. Two nearby observers observe that during their
Challenge 202 s fall, their relative distance changes. (Why?) The same happens to orbiting observers.
In a closed room in orbit around the Earth, a person or a mass at the centre of the
room would not feel any force, and in particular no gravity. But if several particles are
located in the room, they will behave differently depending on their exact positions in
the room. Only if two particles were on exactly the same orbit would they keep the same
relative position. If one particle is in a lower or higher orbit than the other, they will de-
part from each other over time. Even more interestingly, if a particle in orbit is displaced
Challenge 203 e sideways, it will oscillate around the central position. (Can you confirm this?)
Gravitation leads to changes of relative distance. These changes evince another effect,
142 5 how maximum speed changes space, time and gravity

before

after

F I G U R E 64 Tidal effects: the only effect bodies feel when falling.

shown in Figure 64: an extended body in free fall is slightly squeezed. This effect also

Motion Mountain – The Adventure of Physics
tells us that it is an essential feature of gravity that free fall is different from point to
Vol. I, page 197 point. That rings a bell. The squeezing of a body is the same effect as that which causes
the tides. Indeed, the bulging oceans can be seen as the squeezed Earth in its fall towards
Ref. 138 the Moon. Using this result of universal gravity we can now affirm: the essence of gravity
is the observation of tidal effects.
In other words, gravity is simple only locally. Only locally does it look like acceleration.
Only locally does a falling observer like Kittinger feel at rest. In fact, only a point-like
observer does so! As soon as we take spatial extension into account, we find tidal effects.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
⊳ Gravity is the presence of tidal effects.

The absence of tidal effects implies the absence of gravity. Tidal effects are the everyday
consequence of height-dependent time. Isn’t this a beautiful conclusion from the invari-
ance of the speed of light?
In principle, Kittinger could have felt gravitation during his free fall, even with his eyes
closed, had he paid attention to himself. Had he measured the distance change between
his two hands, he would have found a tiny decrease which could have told him that he
was falling. This tiny decrease would have forced Kittinger to a strange conclusion. Two
inertially moving hands should move along two parallel lines, always keeping the same
distance. Since the distance changes, he must conclude that in the space around him lines
starting out in parallel do not remain so. Kittinger would have concluded that the space
around him was similar to the surface of the Earth, where two lines starting out north,
parallel to each other, also change distance, until they meet at the North Pole. In other
words, Kittinger would have concluded that he was in a curved space.
By studying the change in distance between his hands, Kittinger could even have con-
cluded that the curvature of space changes with height. Physical space differs from a
sphere, which has constant curvature. Physical space is more involved. The effect is ex-
tremely small, and cannot be felt by human senses. Kittinger had no chance to detect any-
thing. However, the conclusion remains valid. Space-time is not described by Minkowski
geometry when gravity is present. Tidal effects imply space-time curvature.
how maximum speed changes space, time and gravity 143

⊳ Gravity is the curvature of space-time.

This is the main and final lesson that follows from the invariance of the speed of light.

Bent space and mat tresses

“
Wenn ein Käfer über die Oberfläche einer Kugel
krabbelt, merkt er wahrscheinlich nicht, dass
der Weg, den er zurücklegt, gekrümmt ist. Ich

”
dagegen hatte das Glück, es zu merken.*
Albert Einstein’s answer to his son Eduard’s
question about the reason for his fame

On the 7th of November 1919, Albert Einstein became world-famous. On that day, an
article in the Times newspaper in London announced the results of a double expedition
to South America under the heading ‘Revolution in science / new theory of the universe /
Newtonian ideas overthrown’. The expedition had shown unequivocally – though not
for the first time – that the theory of universal gravity, essentially given by 𝑎 = 𝐺𝑀/𝑟2 ,

Motion Mountain – The Adventure of Physics
was wrong, and that instead space was curved. A worldwide mania started. Einstein was
presented as the greatest of all geniuses. ‘Space warped’ was the most common headline.
Einstein’s papers on general relativity were reprinted in full in popular magazines. People
could read the field equations of general relativity, in tensor form and with Greek indices,
in Time magazine. Nothing like this has happened to any other physicist before or since.
What was the reason for this excitement?
The expedition to the southern hemisphere had performed an experiment proposed
Ref. 139 by Einstein himself. Apart from seeking to verify the change of time with height, Einstein
had also thought about a number of experiments to detect the curvature of space. In the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
one that eventually made him famous, Einstein proposed to take a picture of the stars
near the Sun, as is possible during a solar eclipse, and compare it with a picture of the
same stars at night, when the Sun is far away. From the equations of general relativity,
Einstein predicted a change in position of 1.75 󸀠󸀠 (1.75 seconds of arc) for star images at
Vol. I, page 201 the border of the Sun, a value twice as large as that predicted by universal gravity. The
Ref. 140 prediction was confirmed for the first time in 1919, and thus universal gravity was ruled
out.
Does this result imply that space is curved? Not by itself. In fact, other explanations
could be given for the result of the eclipse experiment, such as a potential differing from
the inverse square form. However, the eclipse results are not the only data. We already
know about the change of time with height. Experiments show that two observers at
different heights measure the same value for the speed of light 𝑐 near themselves. But
these experiments also show that if an observer measures the speed of light at the position
of the other observer, he gets a value differing from 𝑐, since his clock runs differently.
There is only one possible solution to this dilemma: metre bars, like clocks, also change
with height, and in such a way as to yield the same speed of light everywhere.
If the speed of light is constant but clocks and metre bars change with height, the
Challenge 204 e conclusion must be that space is curved near masses. Many physicists in the twentieth

* ‘When an insect walks over the surface of a sphere it probably does not notice that the path it walks is
curved. I, on the other hand, had the luck to notice it.’
144 5 how maximum speed changes space, time and gravity

image image
of star
position
star of star

Sun
Sun

Mercury Earth
Earth

F I G U R E 65 The mattress model of space: the path of a light beam and of a satellite near a spherical
mass.

Motion Mountain – The Adventure of Physics
century checked whether metre bars really behave differently in places where gravity
is present. And indeed, curvature has been detected around several planets, around all
the hundreds of stars where it could be measured, and around dozens of galaxies. Many
indirect effects of curvature around masses, to be described in detail below, have also
been observed. All results confirm the curvature of space and space-time around masses,
and in addition confirm the curvature values predicted by general relativity. In other
words, metre bars near masses do indeed change their size from place to place, and even
from orientation to orientation. Figure 65 gives an impression of the situation.
But beware: the right-hand figure, although found in many textbooks, can be

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 141 misleading. It can easily be mistaken for a reproduction of a potential around a body. In-
deed, it is impossible to draw a graph showing curvature and potential separately. (Why?)
Challenge 205 s We will see that for small curvatures, it is even possible to explain the change in metre
bar length using a potential only. Thus the figure does not really cheat, at least in the
case of weak gravity. But for large and changing values of gravity, a potential cannot be
defined, and thus there is indeed no way to avoid using curved space to describe grav-
ity. In summary, if we imagine space as a sort of generalized mattress in which masses
produce deformations, we have a reasonable model of space-time. As masses move, the
deformation follows them.
The acceleration of a test particle only depends on the curvature of the mattress. It
does not depend on the mass of the test particle. So the mattress model explains why
all bodies fall in the same way. (In the old days, this was also called the equality of the
inertial and gravitational mass.)
Space thus behaves like a frictionless mattress that pervades everything. We live in-
side the mattress, but we do not feel it in everyday life. Massive objects pull the foam of
the mattress towards them, thus deforming the shape of the mattress. More force, more
energy or more mass imply a larger deformation. (Does the mattress remind you of the
Vol. III, page 136 aether? Do not worry: physics eliminated the concept of aether because it is indistin-
guishable from vacuum.)
If gravity means curved space, then any accelerated observer, such as a man in a de-
parting car, must also observe that space is curved. However, in everyday life we do not
how maximum speed changes space, time and gravity 145

Motion Mountain – The Adventure of Physics
F I G U R E 66 A three-dimensional illustration of the curvature of space around a mass (© Farooq Ahmad
Bhat.

notice any such effect, because for accelerations and sizes of everyday life the curvature
values are too small to be noticed. Could you devise a sensitive experiment to check the
Challenge 206 s prediction?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
You might be led to ask: if the flat space containing a macroscopic body is bent by a
Challenge 207 ny gravitational field, what happens to the body contained in it? (For simplicity, we can ima-
gine that the body is suspended and kept in place by massless ropes.) The gravitational
field will also affected the body, but its bending is not related in a simple way to the bend-
ing of the underlying space. For example, bodies have higher inertia than empty space.
And in static situations, the bending of the body depends on its own elastic properties,
which differ markedly from those of empty space, which is much stiffer.

Curved space-time
Figure 65 and Figure 66 shows the curvature of space only, but in fact the whole of space-
time is curved. We will shortly find out how to describe both the shape of space and the
shape of space-time, and how to measure their curvature.
Let us have a first attempt to describe nature with the idea of curved space-time. In
the case of Figure 65, the best description of events is with the use of the time 𝑡 shown
by a clock located at spatial infinity; that avoids problems with the uneven running of
clocks at different distances from the central mass. For the radial coordinate 𝑟, the most
practical choice to avoid problems with the curvature of space is to use the circumference
of a circle around the central body, divided by 2π. The curved shape of space-time is
best described by the behaviour of the space-time distance d𝑠, or by the wristwatch time
Page 45 d𝜏 = d𝑠/𝑐, between two neighbouring points with coordinates (𝑡, 𝑟) and (𝑡 + d𝑡, 𝑟 + d𝑟).
146 5 how maximum speed changes space, time and gravity

Page 141 As we saw above, gravity means that in spherical coordinates we have

d𝑠2
d𝜏2 = 2
≠ d𝑡2 − d𝑟2 /𝑐2 − 𝑟2 d𝜑2 /𝑐2 . (136)
𝑐
The inequality expresses the fact that space-time is curved. Indeed, the experiments on
time change with height confirm that the space-time interval around a spherical mass is
given by
d𝑠2 2𝐺𝑀 d𝑟2 𝑟2 2
d𝜏2 = 2 = (1 − ) d𝑡2
− − d𝜑 . (137)
𝑐 𝑟𝑐2 𝑐2 − 2𝐺𝑀 𝑐2 𝑟

This expression is called the Schwarzschild metric after one of its discoverers.* The metric
(137) describes the curved shape of space-time around a spherical non-rotating mass. It
is well approximated by the Earth or the Sun. (Why can their rotation be neglected?)
Challenge 208 s Expression (137) also shows that gravity’s strength around a body of mass 𝑀 and radius
𝑅 is measured by a dimensionless number ℎ defined as

Motion Mountain – The Adventure of Physics
2𝐺 𝑀
ℎ= . (138)
𝑐2 𝑅
This ratio expresses the gravitational strain with which lengths and the vacuum are de-
formed from the flat situation of special relativity, and thus also determines how much
clocks slow down when gravity is present. (The ratio also reveals how far one is from any
possible horizon.) On the surface of the Earth the ratio ℎ has the small value of 1.4 ⋅ 10−9 ;
on the surface of the Sun is has the somewhat larger value of 4.2 ⋅ 10−6 . The precision of

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
modern clocks allows detecting such small effects quite easily. The various consequences
and uses of the deformation of space-time will be discussed shortly.
We note that if a mass is highly concentrated, in particular when its radius becomes
equal to its so-called Schwarzschild radius

2𝐺𝑀
𝑅S = , (139)
𝑐2
the Schwarzschild metric behaves strangely: at that location, time disappears (note that
𝑡 is time at infinity). At the Schwarzschild radius, the wristwatch time (as shown by a
clock at infinity) stops – and a horizon appears. What happens precisely will be explored
Page 266 below. This situation is not common: the Schwarzschild radius for a mass like the Earth
is 8.8 mm, and for the Sun is 3.0 km; you might want to check that the object size for
Challenge 209 e every system in everyday life is larger than its Schwarzschild radius. Physical systems
Ref. 143 which realize the Schwarzschild radius are called black holes; we will study them in de-
Page 262 tail shortly. In fact, general relativity states that no system in nature is smaller than its

* Karl Schwarzschild (1873–1916), influential astronomer; he was one of the first people to understand gen-
eral relativity. He published his formula in December 1915, only a few months after Einstein had published
his field equations. He died prematurely, at the age of 42, much to Einstein’s distress. We will deduce the
form of the metric later on, directly from the field equations of general relativity. The other discoverer of
Ref. 142 the metric, unknown to Einstein, was Johannes Droste, a student of Lorentz.
how maximum speed changes space, time and gravity 147

Schwarzschild size, in other words that the ratio ℎ defined by expression (138) is never
above unity.
In summary, the results mentioned so far make it clear that mass generates curvature.
The mass–energy equivalence we know from special relativity then tells us that as a con-
sequence, space should also be curved by the presence of any type of energy–momentum.
Every type of energy curves space-time. For example, light should also curve space-time.
However, even the highest-energy beams we can create correspond to extremely small
masses, and thus to unmeasurably small curvatures. Even heat curves space-time; but in
most systems, heat is only about a fraction of 10−12 of total mass; its curvature effect is
thus unmeasurable and negligible. Nevertheless it is still possible to show experimentally
that energy curves space. In almost all atoms a sizeable fraction of the mass is due to the
electrostatic energy among the positively charged protons. In 1968 Kreuzer confirmed
Ref. 144 that energy curves space with a clever experiment using a floating mass.
It is straightforward to deduce that the temporal equivalent of spatial curvature is the
Challenge 210 e uneven running of clocks. Taking the two curvatures together, we conclude that when
gravity is present, space-time is curved.

Motion Mountain – The Adventure of Physics
Let us sum up our chain of thoughts. Energy is equivalent to mass; mass produces
gravity; gravity is equivalent to acceleration; acceleration is position-dependent time.
Since light speed is constant, we deduce that energy–momentum tells space-time to curve.
This statement is the first half of general relativity.
We will soon find out how to measure curvature, how to calculate it from energy–
momentum and what is found when measurement and calculation are compared. We
will also find out that different observers measure different curvature values. The set of
transformations relating one viewpoint to another in general relativity, the diffeomorph-
ism symmetry, will tell us how to relate the measurements of different observers.

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Since matter moves, we can say even more. Not only is space-time curved near masses,
it also bends back when a mass has passed by. In other words, general relativity states
that space, as well as space-time, is elastic. However, it is rather stiff: quite a lot stiffer
Ref. 145 than steel. To curve a piece of space by 1 % requires an energy density enormously larger
Challenge 211 ny than to curve a simple train rail by 1 %. This and other interesting consequences of the
elasticity of space-time will occupy us for a while.

The speed of light and the gravitational constant

“ ”
Si morior, moror.*
Antiquity

We continue on the way towards precision in our understanding of gravitation. All our
theoretical and empirical knowledge about gravity can be summed up in just two general
statements. The first principle states:

⊳ The speed 𝑣 of a physical system is bounded above:

𝑣⩽𝑐 (140)

* ‘If I rest, I die.’ This is the motto of the bird of paradise.
148 5 how maximum speed changes space, time and gravity

for all observers, where 𝑐 is the speed of light.

The theory following from this first principle, special relativity, is extended to general re-
lativity by adding a second principle, characterizing gravitation. There are several equi-
valent ways to state this principle. Here is one.

⊳ For all observers, the force 𝐹 on a system is limited by

𝑐4
𝐹⩽ , (141)
4𝐺
where 𝐺 is the universal constant of gravitation.

In short, there is a maximum force in nature. Gravitation leads to attraction of masses.
Challenge 212 e However, this force of attraction is limited. An equivalent statement is:

Motion Mountain – The Adventure of Physics
⊳ For all observers, the size 𝐿 of a system of mass 𝑀 is limited by

𝐿 4𝐺
⩾ 2 . (142)
𝑀 𝑐

In other words, a massive system cannot be more concentrated than a non-rotating black
hole of the same mass. Another way to express the principle of gravitation is the follow-
ing:

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
⊳ For all systems, the emitted power 𝑃 is limited by

𝑐5
𝑃⩽ . (143)
4𝐺

In short, there is a maximum power in nature.
The three limits given above are all equivalent to each other; and no exception is
known or indeed possible. The limits include universal gravity in the non-relativistic
case. They tell us what gravity is, namely curvature, and how exactly it behaves. The lim-
its allow us to determine the curvature in all situations, at all space-time events. As we
Page 113 have seen above, the speed limit together with any one of the last three principles imply
all of general relativity.*
For example, can you show that the formula describing gravitational red-shift com-
Challenge 213 ny plies with the general limit (142) on length-to-mass ratios?
We note that any formula that contains the speed of light 𝑐 is based on special re-
lativity, and if it contains the constant of gravitation 𝐺, it relates to universal gravity. If a
formula contains both 𝑐 and 𝐺, it is a statement of general relativity. The present chapter
frequently underlines this connection.

* This didactic approach is unconventional. It is possible that is has been pioneered by the present author,
Ref. 109 though several researchers developed similar ideas before, among them Venzo de Sabbata and C. Sivaram.
how maximum speed changes space, time and gravity 149

Our mountain ascent so far has taught us that a precise description of motion requires
the specification of all allowed viewpoints, their characteristics, their differences, and
the transformations between them. From now on, all viewpoints are allowed, without
exception: anybody must be able to talk to anybody else. It makes no difference whether
an observer feels gravity, is in free fall, is accelerated or is in inertial motion. Furthermore,
people who exchange left and right, people who exchange up and down or people who
say that the Sun turns around the Earth must be able to talk to each other and to us. This
gives a much larger set of viewpoint transformations than in the case of special relativity;
it makes general relativity both difficult and fascinating. And since all viewpoints are
allowed, the resulting description of motion is complete.*

Why d oes a stone thrown into the air fall back to E arth? –
Geodesics

“
A genius is somebody who makes all possible

”
mistakes in the shortest possible time.
Anonymous

Motion Mountain – The Adventure of Physics
In our discussion of special relativity, we saw that inertial or free-floating motion is the
Page 87 motion which connecting two events that requires the longest proper time. In the absence
of gravity, the motion fulfilling this requirement is straight (rectilinear) motion. On the
Vol. I, page 59 other hand, we are also used to thinking of light rays as being straight. Indeed, we are all
accustomed to check the straightness of an edge by looking along it. Whenever we draw
the axes of a physical coordinate system, we imagine either drawing paths of light rays
or drawing the motion of freely moving bodies.
In the absence of gravity, object paths and light paths coincide. However, in the pres-
ence of gravity, objects do not move along light paths, as every thrown stone shows. Light

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
does not define spatial straightness any more. In the presence of gravity, both light and
matter paths are bent, though by different amounts. But the original statement remains
valid: even when gravity is present, bodies follow paths of longest possible proper time.
For matter, such paths are called time-like geodesics. For light, such paths are called light-
like or null geodesics.
We note that in space-time, geodesics are the curves with maximal length. This is in
contrast with the case of pure space, such as the surface of a sphere, where geodesics are
the curves of minimal length.
In simple words, stones fall because they follow geodesics. Let us perform a few checks
of this statement. Since stones move by maximizing proper time for inertial observers,
they also must do so for freely falling observers, like Kittinger. In fact, they must do so
for all observers. The equivalence of falling paths and geodesics is at least consistent.
If falling is seen as a consequence of the Earth’s surface approaching – as we will
Page 158 argue below – we can deduce directly that falling implies a proper time that is as long as
Challenge 214 e possible. Free fall indeed is motion along geodesics.
We saw above that gravitation follows from the existence of a maximum force. The
result can be visualized in another way. If the gravitational attraction between a central
body and a satellite were stronger than it is, black holes would be smaller than they are;

* Or it would be, were it not for a small deviation called quantum theory.
150 5 how maximum speed changes space, time and gravity

height
slow, steep throw c · time
h
d

Motion Mountain – The Adventure of Physics
F I G U R E 67 All paths of ﬂying stones,
rapid, flat throw independently of their speed and angle,
have the same curvature in space-time
throw distance (photograph © Marco Fulle).

in that case the maximum force limit and the maximum speed could be exceeded by
getting close to such a black hole. If, on the other hand, gravitation were weaker than it

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
is, there would be observers for which the two bodies would not interact, thus for which
they would not form a physical system. In summary, a maximum force of 𝑐4 /4𝐺 implies
universal gravity. There is no difference between stating that all bodies attract through
gravitation and stating that there is a maximum force with the value 𝑐4 /4𝐺. But at the
same time, the maximum force principle implies that objects move on geodesics. Can
Challenge 215 ny you show this?
Let us turn to an experimental check. If falling is a consequence of curvature, then
the paths of all stones thrown or falling near the Earth must have the same curvature in
space-time. Take a stone thrown horizontally, a stone thrown vertically, a stone thrown
rapidly, or a stone thrown slowly: it takes only two lines of argument to show that in
Challenge 216 ny space-time all their paths are approximated to high precision by circle segments, as shown
in Figure 67. All paths have the same curvature radius 𝑟, given by

𝑐2
𝑟= ≈ 9.2 ⋅ 1015 m . (144)
𝑔

The large value of the radius, corresponding to a low curvature, explains why we do not
notice it in everyday life. The parabolic shape typical of the path of a stone in everyday
life is just the projection of the more fundamental path in 4-dimensional space-time
into 3-dimensional space. The important point is that the value of the curvature does not
depend on the details of the throw. In fact, this simple result could have suggested the
how maximum speed changes space, time and gravity 151

ideas of general relativity to people a full century before Einstein; what was missing was
the recognition of the importance of the speed of light as limit speed. In any case, this
simple calculation confirms that falling and curvature are connected. As expected, and
as mentioned already above, the curvature diminishes at larger heights, until it vanishes
at infinite distance from the Earth. Now, given that the curvature of all paths for free fall
is the same, and given that all such paths are paths of least action, it is straightforward
that they are also geodesics.
If we describe fall as a consequence of the curvature of space-time, we must show that
the description with geodesics reproduces all its features. In particular, we must be able
to explain that stones thrown with small speed fall back, and stones thrown with high
Challenge 217 ny speed escape. Can you deduce this from space curvature?
In summary, the motion of any particle falling freely ‘in a gravitational field’ is de-
scribed by the same variational principle as the motion of a free particle in special re-
lativity: the path maximizes the proper time ∫ d𝜏. We rephrase this by saying that any
particle in free fall from space-time point 𝐴 to space-time point 𝐵 minimizes the action
𝑆 given by

Motion Mountain – The Adventure of Physics
𝐵
𝑆 = −𝑐2 𝑚 ∫ d𝜏 . (145)
𝐴

That is all we need to know about the free fall of objects. As a consequence, any deviation
from free fall keeps you young. The larger the deviation, the younger you stay.
Page 289 As we will see below, the minimum action description of free fall has been tested
Ref. 146 extremely precisely, and no difference from experiment has ever been observed. We will
also find out that for free fall, the predictions of general relativity and of universal gravity
differ substantially both for particles near the speed of light and for central bodies of

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
high density. So far, all experiments have shown that whenever the two predictions differ,
general relativity is right, and universal gravity and other alternative descriptions are
wrong.
All bodies fall along geodesics. This tells us something important. The fall of bod-
ies does not depend on their mass. The geodesics are like ‘rails’ in space-time that tell
bodies how to fall. In other words, space-time can indeed be imagined as a single, giant,
deformed entity. Space-time is not ‘nothing’; it is an entity of our thinking. The shape of
this entity tells objects how to move. Space-time is thus indeed like an intangible mat-
tress; this deformed mattress guides falling objects along its networks of geodesics.
Moreover, bound energy falls in the same way as mass, as is proven by comparing
the fall of objects made of different materials. They have different percentages of bound
Challenge 218 s energy. (Why?) For example, on the Moon, where there is no air, David Scott from Apollo
15 dropped a hammer and a feather and found that they fell together, alongside each
other. The independence on material composition has been checked and confirmed over
Ref. 147 and over again.

C an light fall?
How does radiation fall? Light, like any radiation, is energy without rest mass. It moves
like a stream of extremely fast and light objects. Therefore deviations from universal
gravity become most apparent for light. How does light fall? Light cannot change speed.
152 5 how maximum speed changes space, time and gravity

Page 137 When light falls vertically, it only changes colour, as we have seen above. But light can also
change direction. Long before the ideas of relativity became current, in 1801, the Prus-
Ref. 148 sian astronomer Johann Soldner understood that universal gravity implies that light is
deflected when passing near a mass. He also calculated how the deflection angle depends
Vol. I, page 201 on the mass of the body and the distance of passage. However, nobody in the nineteenth
century was able to check the result experimentally.
Obviously, light has energy, and energy has weight; the deflection of light by itself
is thus not a proof of the curvature of space. General relativity also predicts a deflec-
tion angle for light passing masses, but of twice the classical Soldner value, because the
curvature of space around large masses adds to the effect of universal gravity. The deflec-
tion of light thus only confirms the curvature of space if the value agrees with the one
predicted by general relativity. This is the case: observations do coincide with predictions.
Page 161 More details will be given shortly.
Simply said, mass is not necessary to feel gravity; energy is sufficient. This result of the
mass–energy equivalence must become second nature when studying general relativity.
In particular, light is not light-weight, but heavy. Can you argue that the curvature of

Motion Mountain – The Adventure of Physics
Challenge 219 ny light near the Earth must be the same as that of stones, given by expression (144)?
In summary, all experiments show that not only mass, but also energy falls along
geodesics, whatever its type (bound or free), and whatever the interaction (be it elec-
tromagnetic or nuclear). Moreover, the motion of radiation confirms that space-time is
curved.
Since experiments show that all particles fall in the same way, independently of their
mass, charge or any other property, we can conclude that the system of all possible tra-
jectories forms an independent structure. This structure is what we call space-time.
We thus find that space-time tells matter, energy and radiation how to fall. This state-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ment is the second half of general relativity. It complements the first half, which states
that energy tells space-time how to curve. To complete the description of macroscopic
motion, we only need to add numbers to these statements, so that they become testable.
As usual, we can proceed in two ways: we can deduce the equations of motion directly,
or we can first deduce the Lagrangian and then deduce the equations of motion from it.
But before we do that, let’s have some fun.

Curiosities and fun challenges ab ou t gravitation

“
Wenn Sie die Antwort nicht gar zu ernst
nehmen und sie nur als eine Art Spaß ansehen,
so kann ich Ihnen das so erklären: Früher hat
man geglaubt, wenn alle Dinge aus der Welt
verschwinden, so bleiben noch Raum und Zeit
übrig. Nach der Relativitätstheorie
verschwinden aber auch Zeit und Raum mit

”
den Dingen.*
Albert Einstein in 1921 in New York
how maximum speed changes space, time and gravity 153

rubber band

cup
ball

hand

wooden
stick,
about
1.5 m
long

Motion Mountain – The Adventure of Physics
F I G U R E 68 A puzzle: what is the simplest way to get the ball attached
to the rubber band into the cup?

Take a plastic bottle and make some holes in it near the bottom. Fill the bottle with water,
closing the holes with your fingers. If you let the bottle go, no water will leave the bottle
Challenge 220 s during the fall. Can you explain how this experiment confirms the equivalence of rest

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
and free fall?
∗∗
On his seventy-sixth birthday, Einstein received a birthday present specially made for
him, shown in Figure 68. A rather deep cup is mounted on the top of a broom stick.
The cup contains a weak piece of elastic rubber attached to its bottom, to which a ball
is attached at the other end. In the starting position, the ball hangs outside the cup. The
rubber is too weak to pull the ball into the cup against gravity. What is the most elegant
Challenge 221 s way to get the ball into the cup?
∗∗
Gravity has the same properties in the whole universe – except in the US patent office.
In 2005, it awarded a patent, Nr. 6 960 975, for an antigravity device that works by dis-
torting space-time in such a way that gravity is ‘compensated’ (see patft.uspto.gov). Do
Challenge 222 s you know a simpler device?
∗∗

* ‘If you do not take the answer too seriously and regard it only for amusement, I can explain it to you in the
following way: in the past it was thought that if all things were to disappear from the world, space and time
would remain. But following relativity theory, space and time would disappear together with the things.’
154 5 how maximum speed changes space, time and gravity

The radius of curvature of space-time at the Earth’s surface is 9.2 ⋅ 1015 m. Can you con-
Challenge 223 e firm this value?
∗∗
Challenge 224 s A piece of wood floats on water. Does it stick out more or less in a lift accelerating up-
wards?
∗∗
Page 55 We saw in special relativity that if two twins are identically accelerated in the same dir-
ection, with one twin some distance ahead of the other, then the twin ahead ages more
than the twin behind. Does this happen in a gravitational field as well? And what happens
Challenge 225 s when the field varies with height, as on Earth?
∗∗
A maximum force and a maximum power also imply a maximum flow of mass. Can you
Challenge 226 s show that no mass flow can exceed 1.1 ⋅ 1035 kg/s?

Motion Mountain – The Adventure of Physics
∗∗
The experiments of Figure 62 and 63 differ in one point: one happens in flat space, the
other in curved space. One seems to be related energy conservation, the other not. Do
Challenge 227 s these differences invalidate the equivalence of the observations?
∗∗
Challenge 228 s How can cosmonauts weigh themselves to check whether they are eating enough?
∗∗

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Is a cosmonaut in orbit really floating freely? No. It turns out that space stations and
satellites are accelerated by several small effects. The important ones are the pressure of
the light from the Sun, the friction of the thin air, and the effects of solar wind. (Micro-
meteorites can usually be neglected.) These three effects all lead to accelerations of the
order of 10−6 m/s2 to 10−8 m/s2 , depending on the height of the orbit. Can you estimate
how long it would take an apple floating in space to hit the wall of a space station, start-
Challenge 229 s ing from the middle? By the way, what is the magnitude of the tidal accelerations in this
situation?
∗∗
Vol. I, page 106 There is no negative mass in nature, as discussed in the beginning of our walk (even
antimatter has positive mass). This means that gravitation cannot be shielded, in contrast
to electromagnetic interactions. Since gravitation cannot be shielded, there is no way to
make a perfectly isolated system. But such systems form the basis of thermodynamics!
Vol. V, page 140 We will study the fascinating implications of this later on: for example, we will discover
an upper limit for the entropy of physical systems.
∗∗
Can curved space be used to travel faster than light? Imagine a space-time in which
two points could be connected either by a path leading through a flat portion, or by a
how maximum speed changes space, time and gravity 155

second path leading through a partially curved portion. Could that curved portion be
used to travel between the points faster than through the flat one? Mathematically, this
is possible; however, such a curved space would need to have a negative energy density.
Such a situation is incompatible with the definition of energy and with the non-existence
Ref. 149 of negative mass. The statement that this does not happen in nature is also called the weak
Challenge 230 ny energy condition. Is it implied by the limit on length-to-mass ratios?
∗∗
The statement of a length-to-mass limit 𝐿/𝑀 ⩾ 4𝐺/𝑐2 invites experiments to try to over-
come it. Can you explain what happens when an observer moves so rapidly past a mass
Challenge 231 ny that the body’s length contraction reaches the limit?
∗∗
There is an important mathematical property of three-dimensional space ℝ3 that singles
it from all other dimensions. A closed (one-dimensional) curve can form knots only in
ℝ3 : in any higher dimension it can always be unknotted. (The existence of knots also

Motion Mountain – The Adventure of Physics
explains why three is the smallest dimension that allows chaotic particle motion.) How-
ever, general relativity does not say why space-time has three plus one dimensions. It is
simply based on the fact. This deep and difficult question will be explored in the last part
of our adventure.
∗∗
Henri Poincaré, who died in 1912, shortly before the general theory of relativity was
finished, thought for a while that curved space was not a necessity, but only a possibility.
He imagined that one could continue using Euclidean space provided light was permitted

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 232 s to follow curved paths. Can you explain why such a theory is impossible?
∗∗
Can two hydrogen atoms circle each other, in their mutual gravitational field? What
Challenge 233 s would the size of this ‘molecule’ be?
∗∗
Challenge 234 s Can two light pulses circle each other, in their mutual gravitational field?
∗∗
The various motions of the Earth mentioned in the section on Galilean physics, such as
Vol. I, page 156 its rotation around its axis or around the Sun, lead to various types of time in physics
and astronomy. The time defined by the best atomic clocks is called terrestrial dynamical
time. By inserting leap seconds every now and then to compensate for the bad definition
Vol. I, page 456 of the second (an Earth rotation does not take 86 400, but 86 400.002 seconds) and, in
minor ways, for the slowing of Earth’s rotation, one gets the universal time coordinate or
UTC. Then there is the time derived from this one by taking into account all leap seconds.
One then has the – different – time which would be shown by a non-rotating clock in
the centre of the Earth. Finally, there is barycentric dynamical time, which is the time
Ref. 150 that would be shown by a clock in the centre of mass of the solar system. Only using
this latter time can satellites be reliably steered through the solar system. In summary,
156 5 how maximum speed changes space, time and gravity

relativity says goodbye to Greenwich Mean Time, as does British law, in one of the rare
cases where the law follows science. (Only the BBC continues to use it.)
∗∗
Space agencies thus have to use general relativity if they want to get artificial satellites to
Mars, Venus, or comets. Without its use, orbits would not be calculated correctly, and
satellites would miss their targets and usually even the whole planet. In fact, space agen-
cies play on the safe side: they use a generalization of general relativity, namely the so-
called parametrized post-Newtonian formalism, which includes a continuous check on
whether general relativity is correct. Within measurement errors, no deviation has been
found so far.*
∗∗
General relativity is also used by space agencies around the world to calculate the exact
Ref. 151 positions of satellites and to tune radios to the frequency of radio emitters on them.
In addition, general relativity is essential for the so-called global positioning system, or

Motion Mountain – The Adventure of Physics
GPS. This modern navigation tool** consists of 24 satellites equipped with clocks that
fly around the world. Why does the system need general relativity to operate? Since all
the satellites, as well as any person on the surface of the Earth, travel in circles, we have
d𝑟 = 0, and we can rewrite the Schwarzschild metric (137) as

d𝜏 2 2𝐺𝑀 𝑟2 d𝜑 2 2𝐺𝑀 𝑣2
( ) =1− − ( ) = 1 − − 2 . (147)
d𝑡 𝑟𝑐2 𝑐2 d𝑡 𝑟𝑐2 𝑐

Challenge 235 e For the relation between satellite time and Earth time we then get

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
2
2𝐺𝑀 𝑣sat
d𝑡sat 2 1− 𝑟sat 𝑐2
− 𝑐2
( ) = 2
𝑣Earth
. (148)
d𝑡Earth 1− 2𝐺𝑀
−
𝑟Earth 𝑐2 𝑐2

Can you deduce how many microseconds a satellite clock gains every day, given that the
Challenge 236 s GPS satellites orbit the Earth once every twelve hours? Since only three microseconds

* To give an idea of what this means, the unparametrized post-Newtonian formalism, based on general
relativity, writes the equation of motion of a body of mass 𝑚 near a large mass 𝑀 as a deviation from the
inverse square expression for the acceleration 𝑎:
𝐺𝑀 𝐺𝑀 𝑣2 𝐺𝑀 𝑣4 𝐺𝑚 𝑣5
𝑎= + 𝑓2 + 𝑓4 + 𝑓5 + ⋅⋅⋅ (146)
𝑟2 𝑟2 𝑐2 𝑟2 𝑐4 𝑟2 𝑐5
Here the numerical factors 𝑓𝑛 are calculated from general relativity and are of order one. The first two odd
terms are missing because of the (approximate) reversibility of general relativistic motion: gravity wave
emission, which is irreversible, accounts for the small term 𝑓5 ; note that it contains the small mass 𝑚 instead
of the large mass 𝑀. All factors 𝑓n up to 𝑓7 have now been calculated. However, in the solar system, only the
term 𝑓2 has ever been detected. This situation might change with future high-precision satellite experiments.
Page 180 Higher-order effects, up to 𝑓5 , have been measured in the binary pulsars, as discussed below.
In a parametrized post-Newtonian formalism, all factors 𝑓𝑛 , including the uneven ones, are fitted
through the data coming in; so far all these fits agree with the values predicted by general relativity.
** For more information, see the www.gpsworld.com website.
how maximum speed changes space, time and gravity 157

would give a position error of one kilometre after a single day, the clocks in the satellites
Ref. 152 must be adjusted to run slow by the calculated amount. The necessary adjustments are
monitored, and so far have confirmed general relativity every single day, within experi-
mental errors, since the system began operation.
∗∗
General relativity is the base of the sport of geocaching, the world-wide treasure hunt
with the help of GPS receivers. See the www.terracaching.com and www.geocaching.com
websites for more details.
∗∗
Ref. 153 The gravitational constant 𝐺 does not seem to change with time. The latest experiments
limit its rate of change to less than 1 part in 1012 per year. Can you imagine how this can
Challenge 237 d be checked?
∗∗

Motion Mountain – The Adventure of Physics
Could our experience that we live in only three spatial dimensions be due to a limitation
Challenge 238 s of our senses? How?
∗∗
Challenge 239 s Can you estimate the effect of the tides on the colour of the light emitted by an atom?
∗∗
The strongest possible gravitational field is that of a small black hole. The strongest grav-
Ref. 154 itational field ever observed is somewhat less though. In 1998, Zhang and Lamb used

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the X-ray data from a double star system to determine that space-time near the 10 km
sized neutron star is curved by up to 30 % of the maximum possible value. What is the
corresponding gravitational acceleration, assuming that the neutron star has the same
Challenge 240 ny mass as the Sun?
∗∗
Ref. 155 Light deflection changes the angular size 𝛿 of a mass 𝑀 with radius 𝑟 when observed at
Challenge 241 e distance 𝑑. The effect leads to the pretty expression

𝑟√1 − 𝑅S /𝑑 2𝐺𝑀
𝛿 = arcsin ( ) where 𝑅S = . (149)
𝑑√1 − 𝑅S /𝑟 𝑐2

Challenge 242 e What percentage of the surface of the Sun can an observer at infinity see? We will exam-
Page 276 ine this issue in more detail shortly.

What is weight?
There is no way for a single (and point-like) observer to distinguish the effects of grav-
ity from those of acceleration. This property of nature allows making a strange state-
ment: things fall because the surface of the Earth accelerates towards them. Therefore,
158 5 how maximum speed changes space, time and gravity

the weight of an object results from the surface of the Earth accelerating upwards and
pushing against the object. That is the principle of equivalence applied to everyday life.
For the same reason, objects in free fall have no weight.
Let us check the numbers. Obviously, an accelerating surface of the Earth produces
a weight for each body resting on it. This weight is proportional to the inertial mass.
In other words, the inertial mass of a body is identical to the gravitational mass. This
Ref. 156 is indeed observed in experiments, and to the highest precision achievable. Roland von
Eötvös* performed many such high-precision experiments throughout his life, without
finding any discrepancy. In these experiments, he used the connection that the inertial
mass determines centrifugal effects and the gravitational mass determines free fall. (Can
Challenge 243 ny you imagine how he tested the equality?) Recent experiments showed that the two masses
Ref. 156 agree to one part in 10−12 .
However, the mass equality is not a surprise. Remembering the definition of mass
Vol. I, page 101 ratio as negative inverse acceleration ratio, independently of the origin of the accelera-
tion, we are reminded that mass measurements cannot be used to distinguish between
inertial and gravitational mass. As we have seen, the two masses are equal by definition

Motion Mountain – The Adventure of Physics
Vol. I, page 202 in Galilean physics, and the whole discussion is a red herring. Weight is an intrinsic effect
of mass.
The equality of acceleration and gravity allows us to imagine the following. Imagine
stepping into a lift in order to move down a few stories. You push the button. The lift is
pushed upwards by the accelerating surface of the Earth somewhat less than is the build-
ing; the building overtakes the lift, which therefore remains behind. Moreover, because
of the weaker push, at the beginning everybody inside the lift feels a bit lighter. When the
contact with the building is restored, the lift is accelerated to catch up with the accelerat-
ing surface of the Earth. Therefore we all feel as if we were in a strongly accelerating car,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
pushed in the direction opposite to the acceleration: for a short while, we feel heavier,
until the lift arrives at its destination.

Why d o apples fall?

“ ”
Vires acquirit eundo.
Vergilius**

An accelerating car will soon catch up with an object thrown forward from it. For the
same reason, the surface of the Earth soon catches up with a stone thrown upwards,
because it is continually accelerating upwards. If you enjoy this way of seeing things,
imagine an apple falling from a tree. At the moment when it detaches, it stops being
accelerated upwards by the branch. The apple can now enjoy the calmness of real rest.
Because of our limited human perception, we call this state of rest free fall. Unfortunately,
the accelerating surface of the Earth approaches mercilessly and, depending on the time
for which the apple stayed at rest, the Earth hits it with a greater or lesser velocity, leading

* Roland von Eötvös (b. 1848 Budapest, d. 1919 Budapest), physicist. He performed many high-precision
gravity experiments; among other discoveries, he discovered the effect named for him. The university of
Budapest bears his name.
** ‘Going it acquires strength.’ Publius Vergilius Maro (b. 70 bce Andes, d. 19 bce Brundisium), from the
Aeneid 4, 175.
how maximum speed changes space, time and gravity 159

to more or less severe shape deformation.
Falling apples also teach us not to be disturbed any more by the statement that gravity
is the uneven running of clocks with height. In fact, this statement is equivalent to saying
that the surface of the Earth is accelerating upwards, as the discussion above shows.
Can this reasoning be continued indefinitely? We can go on for quite a while. It is
fun to show how the Earth can be of constant radius even though its surface is accel-
Challenge 244 ny erating upwards everywhere. We can thus play with the equivalence of acceleration and
gravity. However, this equivalence is only useful in situations involving only one acceler-
ating body. The equivalence between acceleration and gravity ends as soon as two falling
objects are studied. Any study of several bodies inevitably leads to the conclusion that
gravity is not acceleration; gravity is curved space-time.
Many aspects of gravity and curvature can be understood with no or only a little math-
ematics. The next section will highlight some of the differences between universal gravity
and general relativity, showing that only the latter description agrees with experiment.
After that, a few concepts relating to the measurement of curvature are introduced and
applied to the motion of objects and space-time. If the reasoning gets too involved for a

Motion Mountain – The Adventure of Physics
first reading, skip ahead. In any case, the section on the stars, cosmology and black holes
again uses little mathematics.

A summary: the implications of the invariant speed of light on
gravitation
In situations with gravity, time depends on height. The invariance of the speed of light
implies that space and space-time are curved in all regions where gravity plays a role.
Curvature of space can be visualized by threading space with lines of equal distance or
by imagining space as a mattress. In situations with gravity, these lines are curved. Masses

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
thus curve space, especially large ones. Curved space influences and determines the mo-
tion of test masses and of light.
Chapter 6

OPE N OR BI T S , BE N T L IG H T A N D
WOBBL I NG VAC U UM

“
Einstein explained his theory to me every day,
and on my arrival I was fully convinced that he

”
understood it.
Ref. 157 Chaim Weizmann, first president of Israel.

B
efore we tackle the details of general relativity, we first explore the differences

Motion Mountain – The Adventure of Physics
etween the motion of objects in general relativity and in universal gravity,
ecause the two descriptions lead to measurable differences. Since the invari-
ance of the speed of light implies that space is curved near masses, we first of all check
how weak curvature influences motion.
Gravity is strong only near horizons. Strong gravity occurs when the mass 𝑀 and the
distance scale 𝑅 obey
2𝐺𝑀
≈1. (150)
𝑅𝑐2

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Therefore, gravity is strong mainly in three situations: near black holes, near the horizon
of the universe, and at extremely high particle energies. The first two cases are explored
below, while the last will be explored in the final part of our adventure. In contrast, in
most regions of the universe, including our own planet and our solar system, there are
no nearby horizons; in these cases, gravity is a weak effect. This is the topic of the present
chapter.

Weak fields
In everyday life, despite the violence of avalanches or of falling asteroids, forces due to
gravity are much weaker than the maximum force. On the Earth, the ratio 2𝐺𝑀/𝑅𝑐2 is
only about 10−9 . Therefore, all cases of everyday life, relativistic gravitation can still be
approximated by a field, i.e., with a potential added to flat space-time, despite all what
was said above about curvature of space.
Weak gravity situations are interesting because they are simple to understand and to
describe; they mainly require for their explanation the different running of clocks at dif-
ferent heights. Weak field situations allow us to mention space-time curvature only in
passing, and allow us to continue to think of gravity as a source of acceleration. Nev-
ertheless, the change of time with height already induces many new effects that do not
occur in universal gravity. To explore these interesting effects, we just need a consistent
relativistic treatment.
open orbits, bent light and wobbling vacuum 161

𝛼

𝑏
𝑚

𝑦
light
beam
𝑥

F I G U R E 69 Calculating the bending of light by a mass.

Motion Mountain – The Adventure of Physics
Bending of light and radio waves
Gravity influences the motion of light. In particular, gravity bends light beams. Indeed, the
Page 143 detection of the bending of light beams by the Sun made Einstein famous. This happened
because the measured bending angle differed from the one predicted by universal grav-
itation and confirmed that of general relativity which takes into account the curvature of
space.
The bending of light by a mass is easy to calculate. The bending of light is observed

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
because any distant observer measures a changing value for the effective light speed 𝑣
near a mass. (Measured at a location nearby, the speed of light is of course always 𝑐.)
It turns out that a distant observer measures a lower speed, so that for him, gravity has
the same effects as a dense optical medium. It takes only a little bit of imagination to
see that this effect will thus increase the bending of light near masses already deduced
in 1801 by Soldner from universal gravity. In short, relativistic light bending differs from
non-relativistic light bending.*
Let us calculate the bending angle. As usual, we use the coordinate system of flat
space-time at spatial infinity, shown in Figure 69. The idea is to do all calculations to
first order, as the value of the bending is very small. The angle of deflection 𝛼, to first
Ref. 158 order, is simply
∞
∂𝑣
𝛼=∫ d𝑦 , (151)
−∞ ∂𝑥

Challenge 245 e where 𝑣 is the speed of light measured by a distant observer. (Can you confirm this?) For
the next step we use the Schwarzschild metric around a spherical mass

2𝐺𝑀 d𝑟2 𝑟2 2
d𝜏2 = (1 − ) d𝑡2
− − d𝜑 (152)
𝑟𝑐2 𝑐2 − 2𝐺𝑀 𝑐2
𝑟

Page 171 * In the vocabulary defined below, light bending is a pure gravitoelectric effect.
162 6 motion in general relativity

Challenge 246 ny and transform it into (𝑥, 𝑦) coordinates to first order. This gives

2𝐺𝑀 2𝐺𝑀 1
d𝜏2 = (1 − 2
) d𝑡2 − (1 + ) (d𝑥2 + d𝑦2 ) (153)
𝑟𝑐 𝑟𝑐2 𝑐2
which, again to first order, leads to

∂𝑣 2𝐺𝑀
= (1 − )𝑐 . (154)
∂𝑥 𝑟𝑐2

This expression confirms what we know already, namely that distant observers see light
slowed down when passing near a mass. Thus we can also speak of a height-dependent
index of refraction. In other words, constant local light speed leads to a global slowdown.
Challenge 247 ny Inserting the last result into expression (151) and using a clever substitution, we get a
deviation angle 𝛼 given by
4𝐺𝑀 1
𝛼= 2 (155)

Motion Mountain – The Adventure of Physics
𝑐 𝑏
where the distance 𝑏 is the so-called impact parameter of the approaching light beam.

⊳ The light deviation angle 𝛼 due to general relativity is twice the result for
Vol. I, page 201 universal gravity.

For a beam just above the surface of the Sun, the bending angle has the famous value
of 1.75 󸀠󸀠 = 8.5 μrad. This small value was spectacularly confirmed by the measurement

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 248 s expedition of 1919. (How did the astronomers measure the deviation angle?) The result
showed that universal gravity is wrong. Since then, the experiment has been repeated
hundreds of times, even by hobby astronomers.
In fact, Einstein was lucky. Two earlier expeditions organized to measure the value
had failed. In 1912, it was impossible to take data because of rain, and in 1914 in Crimea,
scientists were arrested (by mistake) as spies, because the First World War had just be-
Ref. 159 gun. But in 1911, Einstein had already published an incorrect calculation, giving only the
Soldner value with half the correct size; only in 1915, when he completed general relativ-
Vol. I, page 201 ity, did he find the correct result. Therefore Einstein became famous only because of the
failure of the two expeditions that took place before he published his correct calculation!
For high-precision experiments around the Sun, it is more effective to measure the
bending of radio waves, as they encounter fewer problems when they propagate through
the solar corona. So far, hundreds of independent experiments have done so, using radio
Ref. 151, Ref. 128 sources in the sky which lie on the path of the Sun. All the measurements have confirmed
Ref. 129 general relativity’s prediction within a few per cent or less. A beautiful example of such
Ref. 160 a measurement is shown in Figure 70. The left curve shows the measured values for ex-
pression (155); the right graph shows how the image of the radio source moves in the
sky. Note the small angles that can be measured with the method of very long baseline
interferometry nowadays.
The bending of electromagnetic beams has also been observed near Jupiter, near cer-
Page 252 tain stars, near several galaxies and near galaxy clusters. For the Earth itself, the angle
open orbits, bent light and wobbling vacuum 163

Ref. 160 F I G U R E 70 How the image of radio source 0552+398 changes in position over the course of ten years.
Left: how the deviation changes with angular distance (impact parameter) from the Sun; right: how the
position of the image in the sky changes from (0,0), the position in the sky when the quasar is far from
the Sun (large impact parameter), to cases when the quasar image approaches the Sun (smaller impact
parameter).

Motion Mountain – The Adventure of Physics
is at most 3 nrad, too small to be measured yet, even though this may be feasible in the
near future. There is a chance to detect this value if, as Andrew Gould proposes, the data
of the satellite Hipparcos, which was taking precision pictures of the night sky for many
years, are analysed properly in the future.
Page 189 By the way, the bending of light also confirms that in a triangle, the sum of the angles
does not add up to π (two right angles), as is predicted for curved space. What is the sign
Challenge 249 e of the curvature?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Time delay
The calculation of the bending of light near masses shows that for a distant observer,
light is slowed down near a mass. Constant local light speed leads to a global light speed
slowdown. If light were not slowed down near a mass, it would have to go faster than 𝑐
for an observer near the mass!*

⊳ Masses lead to a time delay of passing electromagnetic waves.

Ref. 161 In 1964, Irwin Shapiro had the idea to measure this effect. He proposed two methods.
The first was to send radar pulses to Venus, and measure the time taken for the reflection
to get back to Earth. If the signals pass near the Sun, they will be delayed. The second
method was to use a space probe communicating with Earth.
Ref. 162 The first measurement was published in 1968, and directly confirmed the prediction of
general relativity within experimental errors. All subsequent tests of the same type, such
as the one shown in Figure 71, have also confirmed the prediction within experimental

* A nice exercise is to show that the bending of a slow particle gives the Soldner value, whereas with increas-
Challenge 250 e ing speed, the value of the bending approaches twice that value. In all these considerations, the rotation of
the mass has been neglected. As the effect of frame dragging shows, rotation also changes the deviation
angle; however, in all cases studied so far, the influence is below the detection threshold.
164 6 motion in general relativity

10 May 1970
Earth orbit
31 March 1970 periastron
(e.g. perihelion,
Sun Mariner 6 perigee)
a: semimajor
orbit
axis
a

240
Time delay (μs)

180

120

Motion Mountain – The Adventure of Physics
60

0
Jan Feb Mar Apr May Jun
1970
F I G U R E 71 Time delay in radio signals – one of F I G U R E 72 The orbit around a central body in
the experiments by Irwin Shapiro. general relativity.

errors, which nowadays are of the order of one part in a thousand. The delay has also
Ref. 163 been measured in binary pulsars, as there are a few such systems in the sky for which the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
line of sight lies almost precisely in the orbital plane.
In short, relativistic gravitation is also confirmed by time delay measurements; in con-
trast, universal gravitation predicts no such effect. The simple calculations presented here
suggest a challenge: Is it also possible to describe full general relativity – thus gravitation
in strong fields – as a change of the speed of light with position and time induced by mass
Challenge 251 ny and energy?

R elativistic effects on orbits
Astronomy allows the most precise measurements of motions known. This is especially
valid for planet motion. So, Einstein first of all tried to apply his results on relativistic
gravitation to the motion of planets. He looked for deviations of their motions from the
predictions of universal gravity. Einstein found such a deviation: the precession of the peri-
helion of Mercury. The effect is shown in Figure 72. Einstein said later that the moment
he found out that his calculation for the precession of Mercury matched observations
was one of the happiest moments of his life.
The calculation is not difficult. In universal gravity, orbits are calculated by setting
𝑎grav = 𝑎centri , in other words, by setting 𝐺𝑀/𝑟2 = 𝜔2 𝑟 and fixing energy and angular
momentum. The mass of the orbiting satellite does not appear explicitly. In general re-
lativity, the mass of the orbiting satellite is made to disappear by rescaling energy and
Ref. 128, Ref. 129 angular momentum as 𝑒 = 𝐸/𝑐2 𝑚 and 𝑗 = 𝐽/𝑚. Next, we include space curvature. We
open orbits, bent light and wobbling vacuum 165

Page 145 use the Schwarzschild metric (152) mentioned above to deduce that the initial condition
for the energy 𝑒, together with its conservation, leads to a relation between proper time
Challenge 252 e 𝜏 and time 𝑡 at infinity:
d𝑡 𝑒
= , (156)
d𝜏 1 − 2𝐺𝑀/𝑟𝑐2

whereas the initial condition on the angular momentum 𝑗 and its conservation imply
that
d𝜑 𝑗
= 2 . (157)
d𝜏 𝑟
These relations are valid for any particle, whatever its mass 𝑚. Inserting all this into the
Schwarzschild metric, we find that the motion of a particle follows

d𝑟 2
( ) + 𝑉2 (𝑗, 𝑟) = 𝑒2 (158)
𝑐d𝜏

Motion Mountain – The Adventure of Physics
where the effective potential 𝑉 is given by

2𝐺𝑀 𝑗2
𝑉2 (𝐽, 𝑟) = (1 − ) (1 + ) . (159)
𝑟𝑐2 𝑟2 𝑐2

Challenge 253 e The expression differs slightly from the one in universal gravity, as you might want to
Challenge 254 e check. We now need to solve for 𝑟(𝜑). For circular orbits we get two possibilities

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
6𝐺𝑀/𝑐2
𝑟± = (160)
1 ± √1 − 12( 𝐺𝑀
𝑐𝑗
)2

where the minus sign gives a stable and the plus sign an unstable orbit. If 𝑐𝑗/𝐺𝑀 < 2√3 ,
no stable orbit exists; the object will impact the surface or, for a black hole, be swallowed.
There is a stable circular orbit only if the angular momentum 𝑗 is larger than 2√3 𝐺𝑀/𝑐.
We thus find that in general relativity, in contrast to universal gravity, there is a smallest
stable circular orbit. The radius of this smallest stable circular orbit is 6𝐺𝑀/𝑐2 = 3𝑅S .
What is the situation for elliptical orbits? Setting 𝑢 = 1/𝑟 in (158) and differentiating,
the equation for 𝑢(𝜑) becomes

𝐺𝑀 3𝐺𝑀 2
𝑢󸀠 + 𝑢 = + 2 𝑢 . (161)
𝑗2 𝑐

Without the nonlinear correction due to general relativity on the far right, the solutions
Challenge 255 e are the famous conic sections

𝐺𝑀
𝑢0 (𝜑) = (1 + 𝜀 cos 𝜑) , (162)
𝑗2
166 6 motion in general relativity

i.e., ellipses, parabolas or hyperbolas. The type of conic section depends on the value
of the parameter 𝜀, the so-called eccentricity. We know the shapes of these curves from
Vol. I, page 193 universal gravity. Now, general relativity introduces the nonlinear term on the right-hand
side of equation (161). Thus the solutions are not conic sections any more; however, as
Challenge 256 e the correction is small, a good approximation is given by

𝐺𝑀 3𝐺2 𝑀2
𝑢1 (𝜑) = (1 + 𝜀 cos(𝜑 − 𝜑)) . (163)
𝑗2 𝑗2 𝑐2

The hyperbolas and parabolas of universal gravity are thus slightly deformed.

⊳ Instead of elliptical orbits, general relativity leads to the famous rosetta path
shown in Figure 72.

Such a path is above all characterized by a periastron shift. The periastron, or perihelion
in the case of the Sun, is the nearest point to the central body reached by an orbiting

Motion Mountain – The Adventure of Physics
Challenge 257 e body. The periastron turns around the central body by an angle

𝐺𝑀
𝛼 ≈ 6π (164)
𝑎(1 − 𝜀2 )𝑐2

for every orbit, where 𝑎 is the semimajor axis. For Mercury, the value is 43 󸀠󸀠 = 0.21 mrad
per century. Around 1900, this was the only known effect that was unexplained by univer-
sal gravity; when Einstein’s calculation led him to exactly that value, he was overflowing
with joy for many days.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
To be sure about the equality between calculation and experiment, all other effects
leading to rosetta paths must be eliminated. For some time, it was thought that the quad-
rupole moment of the Sun could be an alternative source of this effect; later measure-
ments ruled out this possibility.
In the past century, the perihelion shift has been measured also for the orbits of Icarus,
Venus and Mars around the Sun, as well as for several binary star systems. In binary
Ref. 163 pulsars, the periastron shift can be as large as several degrees per year. In all cases, ex-
pression (164) describes the motion within experimental errors.
We note that even the rosetta orbit itself is not really stable, due to the emission of
gravitational waves. But in the solar system, the power lost this way is completely negli-
Page 179 gible even over thousands of millions of years, as we saw above, so that the rosetta path
remains an excellent description of observations.

The geodesic effect
Relativistic gravitation has a further effect on orbiting bodies, predicted in 1916 by
Willem de Sitter.* When a pointed body orbits a central mass 𝑚 at distance 𝑟, the dir-
ection of the tip will change after a full orbit. This effect, shown in Figure 73, exists only
in general relativity. The angle 𝛼 describing the direction change after one orbit is given

* Willem de Sitter (b. 1872 Sneek, d. 1934 Leiden) was mathematician, physicist and astronomer.
open orbits, bent light and wobbling vacuum 167

geodesic
precession
Earth
start
Lense– S
after one
Thirring
orbit
precession

F I G U R E 73 The geodesic
effect.

Motion Mountain – The Adventure of Physics
by
3𝐺𝑚 3π𝐺𝑚
𝛼 = 2π (1 − √1 − 2
)≈ . (165)
𝑟𝑐 𝑟𝑐2

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
This angle change is called the geodesic effect – ‘geodetic’ in other languages. It is a further
consequence of the split into gravitoelectric and gravitomagnetic fields, as you may want
Challenge 258 e to show. Obviously, it does not exist in universal gravity.
In cases where the pointing of the orbiting body is realized by an intrinsic rotation,
such as a spinning satellite, the geodesic effect produces a geodesic precession of the axis.
Thus the effect is comparable to spin–orbit coupling in atomic theory. (The Thirring–
Lense effect mentioned below is analogous to spin–spin coupling.)
When Willem de Sitter predicted the geodesic effect, or geodesic precession, he pro-
posed detecting that the Earth–Moon system would change its pointing direction in its
fall around the Sun. The effect is tiny; for the axis of the Moon the precession angle is
Ref. 164 about 0.019 arcsec per year. The effect was first measured in 1987 by an Italian team for
the Earth–Moon system, through a combination of radio-interferometry and lunar ran-
ging, making use of the Cat’s-eyes, shown in Figure 74, deposited by Lunokhod and
Apollo on the Moon. In 2005, the geodesic effect was confirmed to high precision with
Ref. 169 the help of an artificial satellite around the Earth that contained a number of high preci-
sion gyroscopes.
At first sight, geodesic precession is similar to the Thomas precession found in special
Page 62 relativity. In both cases, a transport along a closed line results in the loss of the original
direction. However, a careful investigation shows that Thomas precession can be added
to geodesic precession by applying some additional, non-gravitational interaction, so the
analogy is shaky.
168 6 motion in general relativity

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 74 The lunar retroreﬂectors deposited by Apollo 11 (top right), the two Lunokhods (right),
Apollo 14 (bottom right) and Apollo 15 (bottom left), their locations on the Moon (top left) and a
telescope performing a laser distance measurement (© NASA, Wikimedia, Observatoire de la Côte
d’Azur).
open orbits, bent light and wobbling vacuum 169

Thirring effect

universal gravity prediction relativistic prediction

Moon a
m

Earth M

universe or mass shell

Thirring–Lense effect
universal gravity prediction relativistic prediction

Motion Mountain – The Adventure of Physics
Foucault's pendulum
or
orbiting satellite

Earth
Earth
universe or mass shell F I G U R E 75 The Thirring and
the Thirring–Lense effects.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The Thirring effects
In 1918, the Austrian physicist Hans Thirring published two new, simple and beautiful
predictions of motions, one of them with his collaborator Josef Lense. Neither motion
Ref. 165, Ref. 166 appears in universal gravity, but they both appear in general relativity. Figure 75 illus-
trates these predictions.
The first example, nowadays called the Thirring effect, predicts centrifugal accelera-
tions and Coriolis accelerations for masses in the interior of a rotating mass shell. Thirr-
ing showed that if an enclosing mass shell rotates, masses inside it are attracted towards
the shell. The effect is very small; however, this prediction is in stark contrast to that
of universal gravity, where a spherical mass shell – rotating or not – has no effect at all
on masses in its interior. Can you explain this effect using the figure and the mattress
Challenge 259 e analogy?
The second effect, the Thirring–Lense effect,* is more famous. General relativity pre-
dicts that an oscillating Foucault pendulum, or a satellite circling the Earth in a polar
orbit, does not stay precisely in a fixed plane relative to the rest of the universe, but that
the rotation of the Earth drags the plane along a tiny bit. This frame-dragging, as the ef-
fect is also called, appears because the Earth in vacuum behaves like a rotating ball in a
foamy mattress. When a ball or a shell rotates inside the foam, it partly drags the foam

* Even though the order of the authors is Lense and Thirring, it is customary (but not universal) to stress
the idea of Hans Thirring by placing him first.
170 6 motion in general relativity

F I G U R E 76 The LAGEOS satellites: metal spheres with a
diameter of 60 cm, a mass of 407 kg, and covered with 426
retroreﬂectors (courtesy NASA).

along with it. Similarly, the Earth drags some vacuum with it, and thus turns the plane of
the pendulum. For the same reason, the Earth’s rotation turns the plane of an orbiting
satellite.

Motion Mountain – The Adventure of Physics
The Thirring–Lense or frame-dragging effect is extremely small. It might be that it
was measured for the first time in 1998 by an Italian group led by Ignazio Ciufolini, and
then again by the same group in the years up to 2004. The group followed the motion of
two special artificial satellites – shown in Figure 76 – consisting only of a body of steel
and some Cat’s-eyes. The group measured the satellite’s motion around the Earth with
Ref. 167 extremely high precision, making use of reflected laser pulses. This method allowed this
experiment to be comparatively cheap and quick. Unfortunately, the size of the system-
Ref. 168 atic effects and other reasons imply that the published results cannot be trusted.
So far, only one other group tried the experiment around Earth. The satellite for the
so-called Gravity Probe B experiment was put in orbit in 2005, after over 30 years of

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 169 planning. These satellites were extremely involved and were carrying rapidly rotating
superconducting spheres. Despite several broken systems, in 2009 the experiment con-
firmed the existence of frame dragging around Earth. The evaluation confirmed the pre-
dictions of general relativity within about 25 %.
In the meantime, frame dragging effects have also been measured in various other
astronomical systems. The best confirmations have come from pulsars. Pulsars send out
regular radio pulses, e.g. every millisecond, with extremely high precision. By measuring
the exact times when the pulses arrive on Earth, one can deduce the details of the motion
Ref. 170 of these stars and confirm that such subtle effects as frame dragging do indeed take place.

Gravitomagnetism*
Frame-dragging, the geodesic effect and the Thirring effects can be seen as special cases
of gravitomagnetism. (We will show the connection below.) This approach to gravity was
already studied in the nineteenth century by Holzmüller and by Tisserand, long before
Ref. 171 general relativity was discovered. The approach has become popular again in recent years
because it is simple to understand. As mentioned above, talking about a gravitational field
is always an approximation. In the case of weak gravity, such as occurs in everyday life,
the approximation is very good. Many relativistic effects can be described in terms of the

* This section can be skipped at first reading.
open orbits, bent light and wobbling vacuum 171

gravitational field, without using the concept of space curvature or the metric tensor. In-
stead of describing the complete space-time mattress, the gravitational-field model only
describes the deviation of the mattress from the flat state, by pretending that the de-
viation is a separate entity, called the gravitational field. But what is the relativistically
correct way to describe the gravitational field?
We can compare the situation to electromagnetism. In a relativistic description of
electrodynamics, the electromagnetic field has an electric and a magnetic component.
Vol. III, page 53 The electric field is responsible for the inverse-square Coulomb force. In the same way,
in a relativistic description of (weak) gravity,* the gravitational field has an gravitoelec-
tric and a gravitomagnetic component. The gravitoelectric field is responsible for the
inverse square acceleration of gravity; what we call the gravitational field in everyday life
Ref. 172, Ref. 173 is simply the gravitoelectric part of the full relativistic (weak) gravitational field.
What is the gravitomagnetic field? In electrodynamics, electric charge produces an
electric field, and a moving charge, i.e., a current, produces a magnetic field. Simil-
arly, in relativistic weak-field gravitation, mass–energy produces the gravitoelectric field,
and moving mass–energy produces the gravitomagnetic field. In other words, frame-

Motion Mountain – The Adventure of Physics
dragging is due to a gravitomagnetic effect and is due to mass currents.
In the case of electromagnetism, the distinction between magnetic and electric field
depends on the observer; each of the two can (partly) be transformed into the other. The
Ref. 172 same happens in the case of gravitation. Electromagnetism provides a good indication
as to how the two types of gravitational fields behave; this intuition can be directly trans-
Vol. III, page 48 ferred to gravity. In electrodynamics, the motion 𝑥(𝑡) of a charged particle is described
by the Lorentz equation
𝑚𝑥̈ = 𝑞𝐸 + 𝑞𝑥̇ × 𝐵 , (166)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
where the dot denotes the derivative with respect to time. In other words, the change
of speed 𝑥̈ is due to electric field 𝐸, whereas the magnetic field 𝐵 produces a velocity-
dependent change of the direction of velocity, without changing the speed itself. Both
changes depend on the value of the electric charge 𝑞. In the case of gravity this expression
becomes
𝑚𝑥̈ = 𝑚𝐺 + 𝑚𝑥̇ × 𝐻 . (167)

The role of charge is taken by mass. The role of the electric field is taken by the gravito-
electric field 𝐺 – which we simply call gravitational field in everyday life – and the role
of the magnetic field is taken by the gravitomagnetic field 𝐻. In this expression for the
motion we already know the gravitoelectric field 𝐺; it is given by

𝐺𝑀 𝐺𝑀𝑥
𝐺 = ∇𝜑 = ∇ =− 3 . (168)
𝑟 𝑟
As usual, the quantity 𝜑 is the (scalar) potential. The field 𝐺 is the usual gravitational field
of universal gravity, produced by every mass, and has the dimension of an acceleration.
Masses are the sources of the gravitoelectric field. The gravitoelectric field obeys ∇𝐺 =
−4π𝐺𝜌, where 𝜌 is the mass density. A static field 𝐺 has no vortices; it obeys ∇ × 𝐺 = 0.

* The approximation requires low velocities, weak fields, and localized and stationary mass–energy distri-
172 6 motion in general relativity

𝑚 particle

free
fall

𝑀 rod
𝑣

F I G U R E 77 The reality of gravitomagnetism.

It is not hard to show that if gravitoelectric fields exist, relativity requires that
Ref. 174 gravitomagnetic fields must exist as well. The latter appear whenever we change from an
observer at rest to a moving one. (We will use the same argument in electrodynamics.)

Motion Mountain – The Adventure of Physics
Vol. III, page 53 A particle falling perpendicularly towards an infinitely long rod illustrates the point, as
shown in Figure 77. An observer at rest with respect to the rod can describe the whole
situation with gravitoelectric forces alone. A second observer, moving along the rod with
constant speed, observes that the momentum of the particle along the rod also increases.
This observer will thus not only measure a gravitoelectric field; he also measures a grav-
itomagnetic field. Indeed, a mass moving with velocity 𝑣 produces a gravitomagnetic (3-)
acceleration on a test mass 𝑚 given by

𝑚𝑎 = 𝑚𝑣 × 𝐻 (169)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 260 ny where, almost as in electrodynamics, the static gravitomagnetic field 𝐻 obeys

𝐻 = 16π𝑁𝜌𝑣 (170)

where 𝜌 is mass density of the source of the field and 𝑁 is a proportionality constant.
In nature, there are no sources for the gravitomagnetic field; it thus obeys ∇𝐻 = 0. The
gravitomagnetic field has dimension of inverse time, like an angular velocity.
Challenge 261 ny When the situation in Figure 77 is evaluated, we find that the proportionality constant
𝑁 is given by
𝐺
𝑁 = 2 = 7.4 ⋅ 10−28 m/kg , (171)
𝑐
an extremely small value. We thus find that as in the electrodynamic case, the gravito-
magnetic field is weaker than the gravitoelectric field by a factor of 𝑐2 . It is thus hard to
observe. In addition, a second aspect renders the observation of gravitomagnetism even
more difficult. In contrast to electromagnetism, in the case of gravity there is no way to
Challenge 262 s observe pure gravitomagnetic fields (why?); they are always mixed with the usual, grav-
itoelectric ones. For these reasons, gravitomagnetic effects were measured for the first

butions.
open orbits, bent light and wobbling vacuum 173

time only in the 1990s. In other words, universal gravity is the weak-field approximation
of general relativity that arises when all gravitomagnetic effects are neglected.
In summary, if a mass moves, it also produces a gravitomagnetic field. How can we
imagine gravitomagnetism? Let’s have a look at its effects. The experiment of Figure 77
showed that a moving rod has the effect to slightly accelerate a test mass in the same
direction as its motion. In our metaphor of the vacuum as a mattress, it looks as if a
moving rod drags the vacuum along with it, as well as any test mass that happens to be
in that region. Gravitomagnetism appears as vacuum dragging. Because of a widespread
reluctance to think of the vacuum as a mattress, the expression frame dragging is used
instead.
In this description, all frame dragging effects are gravitomagnetic effects. In particular,
a gravitomagnetic field also appears when a large mass rotates, as in the Thirring–Lense
effect of Figure 75. For an angular momentum 𝐽 the gravitomagnetic field 𝐻 is a dipole
field; it is given by
𝐽×𝑥
𝐻 = ∇ × (−2 3 ) (172)
𝑟

Motion Mountain – The Adventure of Physics
exactly as in the electrodynamic case. The gravitomagnetic field around a spinning mass
has three main effects.
First of all, as in electromagnetism, a spinning test particle with angular momentum
𝑆 feels a torque if it is near a large spinning mass with angular momentum 𝐽. This torque
𝑇 is given by
d𝑆 1
𝑇= = 𝑆×𝐻. (173)
d𝑡 2
The torque leads to the mentioned precession of gyroscopes or geodesic precession. For the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Earth, this effect is extremely small: at the North Pole, the precession has a conic angle
of 0.6 milli-arcseconds and a rotation rate of the order of 10−10 times that of the Earth.
A second effect of gravitomagnetism is the following. Since for a torque we have 𝑇 =
Ω̇ × 𝑆, the dipole field of a large rotating mass with angular momentum 𝐽 has an effect
on orbiting masses. An orbiting mass will experience precession of its orbital plane. Seen
Challenge 263 ny from infinity we get, for an orbit with semimajor axis 𝑎 and eccentricity 𝑒,

𝐻 𝐺 𝐽 𝐺 3(𝐽𝑥)𝑥 𝐺 2𝐽
Ω̇ = − = − 2 3 + 2 5
= 2 3 (174)
2 𝑐 |𝑥| 𝑐 |𝑥| 𝑐 𝑎 (1 − 𝑒2 )3/2

which is the prediction of Lense and Thirring.* The effect – analogous to spin–spin coup-
ling in atoms – is extremely small, giving an angle change of only 8 󸀠󸀠 per orbit for a satel-
lite near the surface of the Earth. This explains the difficulties and controversies around
such Earth-bound experiments. As mentioned above, the effect is much larger in pulsar
systems.
As a third effect of gravitomagnetism, not mentioned yet, a rotating mass leads to an
additional precession of the periastron. This is a similar effect to the one produced by space
curvature on orbiting masses even if the central body does not rotate. The rotation just
reduces the precession due to space-time curvature. This effect has been fully confirmed
Challenge 264 ny * A homogeneous spinning sphere has an angular momentum given by 𝐽 = 25 𝑀𝜔𝑅2 .
174 6 motion in general relativity

F I G U R E 78 A Gedanken
experiment showing the
necessity of gravitational waves.

for the famous binary pulsar PSR 1913+16, discovered in 1974, as well as for the ‘real’ double
pulsar PSR J0737-3039, discovered in 2003. This latter system shows a periastron precession
Ref. 175 of 16.9°/a, the largest value observed so far.
The split into gravitoelectric and gravitomagnetic effects is thus a useful approxima-
tion to the description of gravity. The split also helps to answer questions such as: How

Motion Mountain – The Adventure of Physics
can gravity keep the Earth orbiting around the Sun, if gravity needs 8 minutes to get
Challenge 265 s from the Sun to us? Above all, the split of the gravitational field into gravitoelectric and
gravitomagnetic components allows a simple description of gravitational waves.

Gravitational waves
One of the most fantastic predictions of physics is the existence of gravitational waves.
Gravity waves* prove that empty space itself has the ability to move and vibrate. The basic
idea is simple. Since space is elastic, like a large mattress in which we live, space should
be able to oscillate in the form of propagating waves, like a mattress or any other elastic

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
medium.
Gravitational waves were predicted by Poincaré in 1905.** The waves were deduced
from an approximation of general relativity by Einstein in 1916. For a certain time period,
Ref. 176 Einstein – and many others – believed that his calculation was mistaken. He was con-
vinced about the existence of gravitational waves only in 1937, when several people poin-
ted out errors to him in his draft paper with Nathan Rosen on how to deduce waves from
general relativity without any approximation. He then revised the manuscript. Therefore,
only the paper published in 1937 showed unambiguously, for the first time, that gravita-
tional waves exist in general relativity. A number of side issues had to be clarified even
Ref. 177 after this paper; in the 1950s the issue was definitively settled.
Starting from the existence of a maximum energy speed, Jørgen Kalckar and Ole Ulf-
Ref. 178 beck have given a simple argument for the necessity of gravitational waves. They studied
two equal masses falling towards each other under the effect of gravitational attraction,
and imagined a spring between them. The situation is illustrated in Figure 78. Such a
spring will make the masses bounce towards each other again and again. The central

* To be strict, the term ‘gravity wave’ has a special meaning: gravity waves are the surface waves of the sea,
where gravity is the restoring force. However, in general relativity, the term is used interchangeably with
‘gravitational wave’.
** In fact, the question of the speed of gravity was discussed long before him, by Laplace, for example.
However, these discussions did not envisage the existence of waves.
open orbits, bent light and wobbling vacuum 175

TA B L E 4 The predicted spectrum of gravitational waves.

Frequency Wa v e l e n g t h Name Expected
appearance
< 10−4 Hz > 3 Tm extremely low slow binary star systems,
frequencies supermassive black holes
10−4 Hz–10−1 Hz 3 Tm–3 Gm very low frequencies fast binary star systems,
massive black holes, white
dwarf vibrations
10−1 Hz–102 Hz 3 Gm–3 Mm low frequencies binary pulsars, medium and
light black holes
102 Hz–105 Hz 3 Mm–3 km medium frequencies supernovae, pulsar
vibrations
105 Hz–108 Hz 3 km–3 m high frequencies unknown; maybe future
human-made sources
> 108 Hz < 3m maybe unknown

Motion Mountain – The Adventure of Physics
cosmological sources

spring stores the kinetic energy from the falling masses. The energy value can be meas-
ured by determining the length by which the spring is compressed. When the spring ex-
pands again and hurls the masses back into space, the gravitational attraction will gradu-
ally slow down the masses, until they again fall towards each other, thus starting the same
cycle again.
However, the energy stored in the spring must get smaller with each cycle. Whenever

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
a sphere detaches from the spring, it is decelerated by the gravitational pull of the other
sphere. Now, the value of this deceleration depends on the distance to the other mass;
but since there is a maximal propagation velocity, the effective deceleration is given by
the distance the other mass had when its gravity effect started out towards the second
mass. For two masses departing from each other, the effective distance is thus somewhat
smaller than the actual distance. In short, while departing, the real deceleration is larger
than the one calculated without taking the time delay into account.
Similarly, when one mass falls back towards the other, it is accelerated by the other
mass according to the distance it had when the gravity effect started moving towards it.
Therefore, while approaching, the acceleration is smaller than the one calculated without
time delay.
Therefore, the masses arrive with a smaller energy than they departed with. At every
bounce, the spring is compressed a little less. The difference between these two energies
is lost by each mass: the energy is taken away by space-time. In other words, the energy
difference is radiated away as gravitational radiation. The same thing happens with mat-
tresses. Remember that a mass deforms the space around it as a metal ball on a mattress
deforms the surface around it. (However, in contrast to actual mattresses, there is no fric-
tion between the ball and the mattress.) If two metal balls repeatedly bang against each
other and then depart again, until they come back together, they will send out surface
waves on the mattress. Over time, this effect will reduce the distance that the two balls
depart from each other after each bang. As we will see shortly, a similar effect has already
176 6 motion in general relativity

been measured; the two masses, instead of being repelled by a spring, were orbiting each
other.
A simple mathematical description of gravity waves follows from the split into gravito-
Ref. 179 magnetic and gravitoelectric effects. It does not take much effort to extend gravitomag-
netostatics and gravitoelectrostatics to gravitodynamics. Just as electrodynamics can be
deduced from Coulomb’s attraction by boosting to all possible inertial observers, grav-
Challenge 266 ny itodynamics can be deduced from universal gravity by boosting to other observers. One
gets the four equations

1 ∂𝐻
∇⋅𝐺 = −4π𝐺𝜌 , ∇×𝐺=−
4 ∂𝑡
𝑁 ∂𝐺
∇⋅𝐻 = 0 , ∇ × 𝐻 = −16π𝑁𝜌𝑣 + 4 . (175)
𝐺 ∂𝑡
We have met two of these equations already. The two other equations are expanded ver-
sions of what we have encountered, taking time-dependence into account. Except for the

Motion Mountain – The Adventure of Physics
various factors of 4, the equations for gravitodynamics are the same as Maxwell’s equa-
tions for electrodynamics. The additional factors of 4 appear because the ratio between
angular momentum 𝐿 and energy 𝐸 of gravity waves is different from that of electromag-
netic waves. The ratio determines the spin of a wave. For gravity waves

2
𝐿= 𝐸, (176)
𝜔
whereas for electromagnetic waves the factor is 1/𝜔. It is worth recalling that the spin of
radiation is a classical property. The spin of a wave is defined as the ratio 𝐸/𝐿𝜔, where 𝐸

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
is the energy, 𝐿 the angular momentum, and 𝜔 is the angular frequency. For electromag-
netic waves, the spin is equal to 1; for gravitational waves, it is 2.
The spin is, of course, also a property of the – so far undetected – quantum particle
that makes up gravitational waves. Interestingly, since gravity is universal, there can exist
only a single kind of spin 2 radiation particle in nature. This is in strong contrast to the
spin 1 case, of which there are several examples in nature: photons, weak bosons and
gluons.
The equations of gravitodynamics must be complemented by the definition of the
fields through the acceleration they produce:

𝑚𝑥̈ = 𝑚𝐺 + 𝑚𝑥̇ × 𝐻 . (177)

Definitions with different numerical factors are also common and then lead to different
numerical factors in the equations of gravitodynamics.
The equations of gravitodynamics have a simple property: in vacuum, we can deduce
from them a wave equation for the gravitoelectric and the gravitomagnetic fields 𝐺 and
Challenge 267 e 𝐻. (It is not hard: try!) In other words, gravity can behave like a wave: gravity can radiate.
All this follows from the expression of universal gravity when applied to moving observ-
ers, with the requirement that neither observers nor energy can move faster than 𝑐. Both
the above argument involving the spring and the present mathematical argument use the
open orbits, bent light and wobbling vacuum 177

same assumptions and arrive at the same conclusion.
Challenge 268 e A few manipulations show that the speed of gravitational waves is given by

𝐺
𝑐=√ . (178)
𝑁

Vol. III, page 106 This result corresponds to the electromagnetic expression

1
𝑐= . (179)
√𝜀0 𝜇0

The same letter has been used for the two speeds, as they are identical. Both influences
travel with the speed common to all energy with vanishing rest mass. We note that this
is, strictly speaking, a prediction: the value of the speed of gravitational waves has been
Ref. 180 confirmed directly, despite claims to the contrary, only in 2016.
Ref. 181 How should we imagine gravitational waves? We sloppily said above that a gravita-

Motion Mountain – The Adventure of Physics
tional wave corresponds to a surface wave of a mattress; now we have to do better and
imagine that we live inside the mattress. Gravitational waves are thus moving and oscil-
lating deformations of the mattress, i.e., of space. Like (certain) mattress waves, it turns
out that gravity waves are transverse. Thus they can be polarized. In fact, gravity waves
can be polarized in two ways. The effects of a gravitational wave are shown in Figure 79,
for both linear and circular polarization.* We note that the waves are invariant under
a rotation by π and that the two linear polarizations differ by an angle π/4; this shows
that the particles corresponding to the waves, the gravitons, are of spin 2. (In general,
the classical radiation field for a spin 𝑆 particle is invariant under a rotation by 2π/𝑆. In

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
* A (small amplitude) plane gravity wave travelling in the 𝑧-direction is described by a metric 𝑔 given by

1 0 0 0
0 −1 + ℎ𝑥𝑥 ℎ𝑥𝑦 0
𝑔=( ) (180)
0 ℎ𝑥𝑦 −1 + ℎ𝑥𝑥 0
0 0 0 −1

where its two components, whose amplitude ratio determine the polarization, are given by

ℎ𝑎𝑏 = 𝐵𝑎𝑏 sin(𝑘𝑧 − 𝜔𝑡 + 𝜑𝑎𝑏 ) (181)

as in all plane harmonic waves. The amplitudes 𝐵𝑎𝑏 , the frequency 𝜔 and the phase 𝜑 are determined by
the specific physical system. The general dispersion relation for the wave number 𝑘 resulting from the wave
equation is
𝜔
=𝑐 (182)
𝑘
and shows that the waves move with the speed of light.
In another gauge, a plane wave can be written as

𝑐2 (1 + 2𝜑) 𝐴1 𝐴2 𝐴3
𝐴1 −1 + 2𝜑 ℎ𝑥𝑦 0
𝑔=( ) (183)
𝐴2 ℎ𝑥𝑦 −1 + ℎ𝑥𝑥 0
𝐴3 0 0 −1
∂𝐴
where 𝜑 and 𝐴 are the potentials such that 𝐺 = ∇𝜑 − 𝑐∂𝑡
and 𝐻 = ∇ × 𝐴.
178 6 motion in general relativity

No wave Four gravitational waves, all moving perpendicularly to the page
(all times)
t1 t2 t3 t4 t5

test
body

linear polarization in + direction

linear polarization in x direction

Motion Mountain – The Adventure of Physics
circular polarization in R sense

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
circular polarization in L sense

F I G U R E 79 Effects on a circular or spherical body due to a plane gravitational wave moving in a
direction perpendicular to the page.

addition, the two orthogonal linear polarizations of a spin 𝑆 particle form an angle π/2𝑆.
For the photon, for example, the spin is 1; indeed, its invariant rotation angle is 2π and
the angle formed by the two polarizations is π/2.)
If we image empty space as a mattress that fills space, gravitational waves are wobbling
deformations of the mattress. More precisely, Figure 79 shows that a wave of circular
polarization has the same properties as a corkscrew advancing through the mattress. We
will discover later on why the analogy between a corkscrew and a gravity wave with
circular polarization works so well. Indeed, in the last part of our adventure we will find
a specific model of the space-time mattress that automatically incorporates corkscrew
Vol. VI, page 298 waves (instead of the spin 1 waves shown by ordinary latex mattresses).
open orbits, bent light and wobbling vacuum 179

Production and detection of gravitational waves
How does one produce gravitational waves? Obviously, masses must be accelerated.
But how exactly? The conservation of energy forbids mass monopoles from varying in
strength. We also know from universal gravity that a spherical mass whose radius os-
cillates would not emit gravitational waves. In addition, the conservation of momentum
Challenge 269 ny forbids mass dipoles from changing.
As a result, only changing quadrupoles can emit gravitational waves.* For example,
two masses in orbit around each other will emit gravitational waves. Also, any rotating
object that is not cylindrically symmetric around its rotation axis will do so. As a result,
rotating an arm leads to gravitational wave emission. Most of these statements also apply
Challenge 270 ny to masses in mattresses. Can you point out the differences?
Einstein found that the amplitude ℎ of waves at a distance 𝑟 from a source is given, to
Ref. 182 a good approximation, by the second derivative of the retarded quadrupole moment 𝑄:

2𝐺 1 2𝐺 1
ℎ𝑎𝑏 = d𝑡𝑡 𝑄ret
𝑎𝑏 = 4 d𝑡𝑡 𝑄𝑎𝑏 (𝑡 − 𝑟/𝑐) . (184)

Motion Mountain – The Adventure of Physics
4
𝑐 𝑟 𝑐 𝑟
This expression shows that the amplitude of gravity waves decreases only with 1/𝑟, in
contrast to naive expectations. This feature is the same as for electromagnetic waves. In
addition, the small value of the prefactor, 1.6 ⋅ 10−44 Wm/s, shows that truly gigantic sys-
tems are needed to produce quadrupole moment changes that yield any detectable length
Challenge 271 e variations in bodies. To be convinced, just insert a few numbers, keeping in mind that
the best present detectors are able to measure length changes down to ℎ = 𝛿𝑙/𝑙 = 10−21 .
The production by humans of detectable gravitational waves is probably impossible.
Gravitational waves, like all other waves, transport energy.** If we apply the general

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
formula for the emitted power 𝑃 to the case of two masses 𝑚1 and 𝑚2 in circular orbits
Ref. 129 around each other at distance 𝑙 and get

2
𝑃=−
d𝐸
=
𝐺
𝑄⃛ ret 𝑄⃛ ret = 32 𝐺 ( 𝑚1 𝑚2 ) 𝑙4 𝜔6 (185)
d𝑡 45𝑐5 𝑎𝑏 𝑎𝑏 5 𝑐5 𝑚1 + 𝑚2

which, using Kepler’s relation 4π2 𝑟3 /𝑇2 = 𝐺(𝑚1 + 𝑚2 ), becomes

32 𝐺4 (𝑚1 𝑚2 )2 (𝑚1 + 𝑚2 )
𝑃= . (186)
5 𝑐5 𝑙5
For elliptical orbits, the rate increases with the ellipticity, as explained in the text by
Ref. 129 Goenner. Inserting the values for the case of the Earth and the Sun, we get a power of
about 200 W, and a value of 400 W for the Jupiter–Sun system. These values are so small

* A quadrupole is a symmetrical arrangement, on the four sides of a square, of four alternating poles. In
gravitation, a monopole is a point-like or spherical mass, and, since masses cannot be negative, a quadrupole
is formed by two monopoles. A flattened sphere, such as the Earth, can be approximated by the sum of a
monopole and a quadrupole. The same is valid for an elongated sphere.
Vol. III, page 89 ** Gravitoelectromagnetism allows defining the gravitational Poynting vector. It is as easy to define and use
Ref. 174 as in the case of electrodynamics.
180 6 motion in general relativity

time
shift
(s) 0

data
points
5

prediction
20 by general
relativity

25
F I G U R E 80 Comparison between measured time

Motion Mountain – The Adventure of Physics
year delay for the periastron of the binary pulsar PSR
30 1913+16 and the prediction due to energy loss by
1975 1980 1985 1990 1995 2000
gravitational radiation.

that their effect cannot be detected at all.
For all orbiting systems, the frequency of the waves is twice the orbital frequency, as
Challenge 272 ny you might want to check. These low frequencies make it even more difficult to detect
them.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
As a result of the usually low power of gravitational wave emission, the first obser-
vation of their effects was in binary pulsars. Pulsars are small but extremely dense stars;
even with a mass equal to that of the Sun, their diameter is only about 10 km. Therefore
they can orbit each other at small distances and high speeds. Indeed, in the most fam-
ous binary pulsar system, PSR 1913+16, the two stars orbit each other in an amazing 7.8 h,
even though their semimajor axis is about 700 Mm, just less than twice the Earth–Moon
distance. Since their orbital speed is up to 400 km/s, the system is noticeably relativistic.
Pulsars have a useful property: because of their rotation, they emit extremely regular
radio pulses (hence their name), often in millisecond periods. Therefore it is easy to fol-
low their orbit by measuring the change of pulse arrival time. In a famous experiment,
a team of astrophysicists led by Joseph Taylor* measured the speed decrease of the bin-
Ref. 183 ary pulsar system just mentioned. Eliminating all other effects and collecting data for 20
Ref. 184 years, they found a decrease in the orbital frequency, shown in Figure 80. The slowdown
is due to gravity wave emission. The results exactly fit the prediction by general relativity,
without any adjustable parameter. (You might want to check that the effect must be quad-
Challenge 273 ny ratic in time.) This was the first case in which general relativity was tested up to (𝑣/𝑐)5
Page 156 precision. To get an idea of the precision, consider that this experiment detected a reduc-
Ref. 183 tion of the orbital diameter of 3.1 mm per orbit, or 3.5 m per year! The measurements
were possible only because the two stars in this system are neutron stars with small size,

* In 1993 he shared the Nobel Prize in Physics for his life’s work.
open orbits, bent light and wobbling vacuum 181

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 81 Detection of gravitational waves: an illustration of the merger of two black holes emitting
such waves (top left). The other images show the VIRGO detector in Cascina, Italy, with one of its huge
mirror suspensions, the mirror suspension details, and two drawings of the laser interferometer (© INFN).

large velocities and purely gravitational interactions. The pulsar rotation period around
its axis, about 59 ms, is known to eleven digits of precision, the orbital time of 7.8 h is
Ref. 129 known to ten digits and the eccentricity of the orbit to six digits. Radio astronomy can
182 6 motion in general relativity

be spectacular.
The direct detection of gravitational waves was one of the long-term aims of experi-
mental general relativity. The race has been on since the 1990s. The basic idea is simple,
as shown in Figure 81: take four bodies, usually four mirrors, for which the line connect-
ing one pair is perpendicular to the line connecting the other pair. Then measure the
distance changes of each pair. If a gravitational wave comes by, one pair will increase in
distance and the other will decrease, at the same time.
Since detectable gravitational waves cannot be produced by humans, wave detection
first of all requires the patience to wait for a strong enough wave to come by. It turns
out that even for a body around a black hole, only about 6 % of the rest mass can be
radiated away as gravitational waves; furthermore, most of the energy is radiated during
the final fall into the black hole, so that only quite violent processes, such as neutron star
collisions or black hole mergers, are good candidates for detectable gravity wave sources.
The waves produced by a black hole merger are shown in Figure 81.
In addition, a measurement system able to detect length changes of the order of 10−22
or better is needed – in other words, a lot of money. For mirrors spaced 4 km apart, the

Motion Mountain – The Adventure of Physics
detectable distance change must be less than one thousandth of the diameter of a proton.
Essential for a successful detection are the techniques to eliminate noise in the detection
signal. Since decades, worlds’s best noise reduction experts are all working on gravita-
tional wave detectors. Understanding the noise mechanisms has become a research filed
in its own.
Until 2015, gravitational waves had not been detected. The sensitivity of the detectors
was not sufficient. In fact, the race to increase the sensitivity is still ongoing across the
Ref. 112 world. After over twenty years of constant improvements, finally, in 2016, a signal with
a duration of 0.2 s – shown in Figure 82 – was published: it corresponds precisely to

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the signal expected from the merger of two black holes of 29 and 36 solar masses. The
result of the merger is a black hole of 62 solar masses, and the 3 lost solar masses were
radiated away, in large part as gravitational waves. This happened between 600 and 1800
million light-years away. The clarity of the signal, measured at two different locations,
convinced everybody of the correctness of the interpretation. The astonishingly small
peak length variation Δ𝑙/𝑙 of below 10−21 remains a fascinating experimental feat, even
when the large financial budget is taken into account. Several additional merger events
have been measured after the first one.
Challenge 274 r Gravitational waves are a fascinating topic. Can you find a cheap method to meas-
ure their speed? A few astrophysical experiments had deduced bounds on the mass of
the graviton before, and had confirmed the speed of gravity in an indirect way. The first
direct measurement was the discovery of 2016; the result is the speed of light, within
Ref. 112 measurement precision. The observation of a candidate light flash that accompanied the
black hole merger would, if confirmed in this or in a future observation, show that grav-
itational waves travel with the same speed as light waves to within one part in 1016 .
Another question on gravitational waves remains open at this point: If all change in
nature is due to motion of particles, as the Greeks maintained, how do gravity waves
fit into the picture? Quantum theory requires that gravitational waves must be made of
particles. (These hypothetical particles are called gravitons.) Now, there is no real differ-
ence between empty space at rest and wobbling empty space. If gravitational waves were
made of particles, space-time would also have to be! How can this be the case? We have
open orbits, bent light and wobbling vacuum 183

Motion Mountain – The Adventure of Physics
F I G U R E 82 The ﬁrst direct detection of gravitational waves through deformation of space, with a strain

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
of the order of 10−21 , by two detectors spaced three thousand kilometres apart (© LIGO/Physical Review
Letters).

to wait until the final part of our adventure to say more.

Curiosities and fun challenges ab ou t weak fields
Challenge 275 s Is there a static gravitational field that oscillates in space?
∗∗
If we explore the options for the speed of gravitational waves, an interesting connection
appears. If the speed of gravitational waves were smaller than the speed of light, mov-
ing bodies that move almost as rapidly as the speed of light, like cosmic ray particles,
Page 28 would be slowed down by emitting Vavilov–Čerenkov radiation, until they reach the
lower speed. This is not observed.
If on the other hand, the speed of gravitational waves were larger than that of light,
the waves would not obey causality or the second principle of thermodynamics. In short,
gravitational waves, if they exist, must propagate with the speed of light. (A speed very
near to the speed of light might also be possible.)
∗∗
184 6 motion in general relativity

One effect that disturbs gravitational wave detectors are the tides. On the GEO600 de-
tector in Hannover, tides change the distance of the mirrors, around 600 m, by 2 μm.
∗∗
Are narrow beams of gravitational waves, analogous to beams of light, possible? Would
Challenge 276 ny two parallel beams of gravitational waves attract each other?
∗∗
As predicted in earlier editions of this book, the discovery of gravitational waves was
announced in television and radio. Does the discovery help to improve the quality of
life across the planet? Except for a number of scientists, other humans will almost surely
not benefit at all. This situation is in stark contrast to scientific discoveries made in the
Challenge 277 e twentieth century. What is the reason for this contrast?
∗∗
Ref. 185 Can gravity waves be used to power a rocket? Yes, maintain Bonnor and Piper. You might

Motion Mountain – The Adventure of Physics
Challenge 278 e ponder the possibility yourself.
∗∗
Electromagnetism and gravity differ in one aspect: two equal charges repel, two equal
masses attract. In more elaborate terms: for the exchange of spin 2 particles – gravitons
– the effect of mass can be depicted with the mattress model. This is possible because
the sign of the effect in the mattress is independent of other masses. In contrast, for
electromagnetism, the sign of the potential depends on the other electric charges.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
A summary on orbits and waves
In summary, the curvature of space and space-time implies:
— in contrast to universal gravity, masses deflect light more strongly;
— in contrast to universal gravity, light is effectively slowed down near masses;
— in contrast to universal gravity, elliptical orbits are not closed;
— in contrast to universal gravity, orbiting objects change their orientation in space;
— in contrast to universal gravity, empty vacuum can propagate gravitational waves that
travel with the speed of light.
All experiments ever performed confirm these conclusions and verify the numerical pre-
dictions within measurement precision. Both the numerous experiments in weak grav-
itational fields and the less common experiments in strong fields fully confirm general
relativity. All experiments also confirm the force and power limits.
Chapter 7

F ROM C U RVAT U R E TO MOT ION

I
n the precise description of gravity, motion depends on space-time curvature.
n order to quantify this idea, we first of all need to accurately describe curvature
tself. To clarify the issue, we will start the discussion in two dimensions, and then
move to three and four dimensions. Once we are able to explore curvature, we explore
the precise relation between curvature and motion.

Motion Mountain – The Adventure of Physics
How to measure curvature in t wo dimensions
Obviously, a flat sheet of paper has no curvature. If we roll it into a cone or a cylinder,
it gets what is called extrinsic curvature; however, the sheet of paper still looks flat for
any two-dimensional animal living on it – as approximated by an ant walking over it.
In other words, the intrinsic curvature of the sheet of paper is zero even if the sheet as a
whole is extrinsically curved.
Intrinsic curvature is thus the stronger concept, measuring the curvature which can
be observed even by an ant. We note that all intrinsically curved surfaces are also ex-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
trinsically curved. The surface of the Earth, the surface of an island, or the slopes of a
mountain** are intrinsically curved. Whenever we talk about curvature in general re-
lativity, we always mean intrinsic curvature, since any observer in nature is by definition
in the same situation as an ant on a surface: their experience, their actions and plans
always only concern their closest neighbourhood in space and time.
But how can an ant determine whether it lives on an intrinsically curved surface?***
One way is shown in Figure 83. The ant can check whether either the circumference of
a circle bears a Euclidean relation to the measured radius. She can even use the differ-
ence between the measured and the Euclidean values as a measure for the local intrinsic
curvature, if she takes the limit for vanishingly small circles and if she normalizes the val-
ues correctly. In other words, the ant can imagine to cut out a little disc around the point
she is on, to iron it flat and to check whether the disc would tear or produce folds. Any
two-dimensional surface is intrinsically curved whenever ironing is not able to make a
flat street map out of it. The ‘density’ of folds or tears is related to the curvature. Folds
imply negative intrinsic curvature, tears positive curvature.

Challenge 279 e ** Unless the mountain has the shape of a perfect cone. Can you confirm this?
*** Note that the answer to this question also tells us how to distinguish real curvature from curved co-
ordinate systems on a flat space. This question is often asked by those approaching general relativity for the
first time.
186 7 from curvature to motion

𝑎

F I G U R E 83
Positive,
vanishing and
negative
curvature in two
dimensions.

Motion Mountain – The Adventure of Physics
Check your understanding: Can a one-dimensional space have intrinsic curvature? Is
Challenge 280 s a torus intrinsically curved?
Alternatively, we can recognize intrinsic curvature also by checking whether two par-
allel lines that are locally straight stay parallel, approach each other, or depart from each
other. On a paper cylinder, parallel lines remain parallel; in this case, the surface is said
to have vanishing intrinsic curvature. A surface with approaching parallels, such as the
Earth, is said to have positive intrinsic curvature, and a surface with diverging parallels,
such as a saddle, is said to have negative intrinsic curvature. Speaking simply, positive

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
curvature means that we are more restricted in our movements, negative that we are
less restricted. A constant curvature even implies being locked in a finite space. You
Page 188 might want to check this with Figure 83 and Figure 85. We can even measure instrinsic
curvature by determining how rapidly to parallel lines depart or converge.
A third way to measure intrinsic curvature of surfaces uses triangles. On curved sur-
faces the sum of angles in a triangle is larger than π, i.e., larger than two right angles, for
positive curvature, and smaller than π for negative curvature.
Ref. 186 Let us see in detail how we can quantify and measure the curvature of surfaces. First a
question of vocabulary: a sphere with radius 𝑎 is said, by definition, to have an intrinsic
curvature 𝐾 = 1/𝑎2 . Therefore a plane has zero curvature. You might check that for a
Challenge 281 e circle on a sphere, the measured radius 𝑟, circumference 𝐶, and area 𝐴 are related by

𝐾 2 𝐾 2
𝐶 = 2π𝑟 (1 − 𝑟 + ...) and 𝐴 = π𝑟2 (1 − 𝑟 + ...) (187)
6 12

where the dots imply higher-order terms. This allows us to define the intrinsic curvature
𝐾, also called the Gaussian curvature, for a general point on a two-dimensional surface
in either of the following two equivalent ways:

𝐶 1 𝐴 1
𝐾 = 6 lim (1 − ) or 𝐾 = 12 lim (1 − ) . (188)
𝑟→0 2π𝑟 𝑟2 𝑟→0 π𝑟2 𝑟2
from curvature to motion 187

direction of point of interest
minimal curvature

right
angle ! direction of F I G U R E 84 The maximum and
maximal curvature minimum curvature of a surface are
always at a right angle to each other.

These expressions allow an ant to measure the intrinsic curvature at each point for any
smooth surface.*
From now on in this text, curvature will always mean intrinsic curvature, i.e., Gaussian
curvature and its higher-dimensional analogs. Like an ant on a surface, also an observer

Motion Mountain – The Adventure of Physics
in space can only detect intrinsic curvature. Therefore, only intrinsic curvature is of in-
terest in the description of nature.
We note that the curvature of a surface can be different from place to place, and that
it can be positive, as for an egg, or negative, as for the part of a torus nearest to the hole.
A saddle is another example of negative curvature, but, unlike the torus, its curvature
changes along all directions. In fact, it is not possible at all to fit a two-dimensional sur-
face of constant negative curvature inside three-dimensional space; we need at least four
Challenge 283 e dimensions to do so, as you can find out if you try to imagine the situation.
For any surface, at every point, the direction of maximum curvature and the direc-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
tion of minimum curvature are perpendicular to each other. This relationship, shown in
Figure 84, was discovered by Leonhard Euler in the eighteenth century. You might want
to check this with a tea cup, with a sculpture by Henry Moore, or with any other curved
Challenge 284 e object from your surroundings, such as a Volkswagen Beetle. The Gaussian curvature 𝐾
defined in (188) is in fact the product of the two corresponding inverse curvature radii.
Thus, even though line curvature is not an intrinsic property, the Gaussian curvature is.
The Gaussian curvature is an intrinsic property of a surface at each point. This means,
as just explained, that bending the surface does not change its value at each point. For
example, a flat sheet of paper, a paper rolled up into a cylinder and a folded paper all
have zero intrinsic curvature. Because the intrinsic, Gaussian curvature of a flat sheet is
zero, for every bent sheet, at every point, there is always a line with zero curvature. Bent
sheets are made up of straight lines. This property follows from the shape-independence
of the Gaussian curvature. The property makes bent sheets – but not flat sheets – stiff
against bending attempts that try to bend the straight line. This property is the reason

* If the 𝑛-dimensional volume of a sphere is written as 𝑉𝑛 = 𝐶𝑛 𝑟𝑛 and its (𝑛 − 1)-dimensional ‘surface’ as
Ref. 187 𝑂𝑛 = 𝑛𝐶𝑛 𝑟𝑛−1 , we can generalize the expressions for curvature to

𝑉𝑛 1 𝑂𝑛 1
𝐾 = 3(𝑛 + 2) lim (1 − ) or 𝐾 = 3𝑛 lim (1 − ) , (189)
𝑟→0 𝐶𝑛 𝑟 𝑛 𝑟 2 𝑟→0 𝑛𝐶𝑛 𝑟𝑛−1 𝑟2

Challenge 282 ny as shown by Vermeil. A famous riddle is to determine the number 𝐶𝑛 .
188 7 from curvature to motion

F I G U R E 85
Positive,
vanishing and
negative
curvature (in
two dimensions)
illustrated with
the
Σα> π Σα=π Σα<π corresponding
geodesic
behaviour and

Motion Mountain – The Adventure of Physics
the sum of
angles in a
triangle.

that straight tubes, cones and folded paper are particularly stiff and light structures. For
the same reason, the best way to hold a pizza slice is to fold it along the central radius.
In this case, intrinsic curvature prevents that the tip bends down.
Also roofs in the shape of a circular hyperboloid or of a hyperbolic paraboloid are

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
stiff and have two straight lines through every point on their surface. Are these surfaces
Challenge 285 s made of a bent flat sheet?
In summary, Gaussian curvature is a measure of the intrinsic curvature of two-
dimensional surfaces. Such an intrinsic measure of curvature is needed if we are forced
to stay and move inside the surface or inside the space that we are exploring. Because this
applies to all humans, physicists are particularly interested in intrinsic curvature, though
for more than two dimensions.

Three dimensions: curvature of space
For three-dimensional space, describing intrinsic curvature is a bit more involved. To
start with, we have difficulties imagining the situation, because we usually associate
curvature with extrinsic curvature. In fact, the only way to explore three-dimensional
curvature of space is to think like the ant on a surface, and to concentrate on in-
trinsic curvature. Therefore we will describe three-dimensional curvature using two-
dimensional curvature.
In curved three-dimensional space, the Gaussian curvature of an arbitrary, small two-
dimensional disc around a general point will depend on the orientation of the disc. Let
us first look at the simplest case. If the Gaussian curvature at a point is the same for
all orientations of the disc, the point is called isotropic. We can imagine a small sphere
around that point. In this special case, in three dimensions, the relation between the
from curvature to motion 189

Challenge 286 ny measured radius 𝑟 and the measured surface area 𝐴 and volume 𝑉 of the sphere lead to

𝐾 2 4π 3 𝐾
𝐴 = 4π𝑟2 (1 − 𝑟 + ...) and 𝑉 = 𝑟 (1 − 𝑟2 + ...) , (190)
3 3 5
where 𝐾 is the curvature for an isotropic point. This leads to

𝐴 1 𝑟 − √𝐴/4π 𝑟
𝐾 = 3 lim (1 − 2
) 2 = 6 lim 3
= 6 lim excess , (191)
𝑟→0 4π𝑟 𝑟 𝑟→0 𝑟 𝑟→0 𝑟3

where we defined the excess radius as 𝑟excess = 𝑟 − √𝐴/4π . We thus find that

⊳ For a three-dimensional space, the average curvature is six times the excess
radius of a small sphere divided by the cube of the radius.

A positive curvature is equivalent to a positive excess radius, and similarly for vanish-

Motion Mountain – The Adventure of Physics
ing and negative cases. The average curvature at a point is the curvature calculated by
applying the definition with a small sphere to an arbitrary, non-isotropic point.
For a non-isotropic point in three-dimensional space, the Gaussian curvature value
determined with a two-dimensional disc will depend on the orientation of the disc. In
fact, there is a relationship between all possible disc curvatures at a given point; taken
Challenge 287 ny together, they must form a tensor. (Why?) In other words, the Gaussian curvature values
define an ellipsoid at each point. For a full description of curvature, we thus have to
specify, as for any tensor in three dimensions, the main Gaussian curvature values in
three orthogonal directions, corresponding to the thee main axes of the ellipsoid.*

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
What are the curvature values for the three-dimensional space around us? Already
in 1827, the mathematician and physicist Carl-Friedrich Gauß** is said to have checked
whether the three angles formed by three mountain peaks near his place of residence
added up to π. Nowadays we know that the deviation 𝛿 from the angle π on the surface

* These three disc values are not independent however, since together, they must yield the just-mentioned
average volume curvature 𝐾. In total, there are thus three independent scalars describing the curvature in
three dimensions (at each point). Using the metric tensor 𝑔𝑎𝑏 and the Ricci tensor 𝑅𝑎𝑏 to be introduced be-
low, one possibility is to take for the three independent numbers the values 𝑅 = −2𝐾, 𝑅𝑎𝑏 𝑅𝑎𝑏 and det𝑅/det𝑔.
** Carl-Friedrich Gauß (b. 1777 Braunschweig, d. 1855 Göttingen), together with Leonhard Euler, was the
most important mathematician of all times. (His name is written ‘Gauss’ in English texts.) A famous child
prodigy, when he was 19 years old, he constructed the regular heptadecagon with compass and ruler (see
www.mathworld.wolfram.com/Heptadecagon.html). He was so proud of this result that he put a drawing of
the figure on his tomb. Gauss produced many results in number theory, topology, statistics, algebra, complex
numbers and differential geometry which are part of modern mathematics and bear his name. Among his
many accomplishments, he produced a theory of curvature and developed non-Euclidean geometry. He
also worked on electromagnetism and astronomy.
Gauss was a difficult character, worked always for himself, and did not found a school. He published
little, as his motto was: pauca sed matura. As a consequence, when another mathematician published a new
result, he regularly produced a notebook in which he had noted the very same result already years before.
These famous notebooks are now available online at www.sub.uni-goettingen.de.
190 7 from curvature to motion

of a body of mass 𝑀 and radius 𝑟 is given by

𝐺𝑀
𝛿 = π − (𝛼 + 𝛽 + 𝛾) ≈ −𝐴 triangle𝐾 = 𝐴 triangle . (192)
𝑟3 𝑐2
This expression is typical for hyperbolic geometries. For the case of mathematical neg-
ative curvature 𝐾, the first equality was deduced by Johann Lambert.* The last equation
came only one and a half century later, and is due to Einstein, who made clear that the
negative curvature 𝐾 of the space around us is related to the mass and gravitation of a
body. For the case of the Earth and typical mountain distances, the angle 𝛿 is of the order
of 10−14 rad. Gauss had no chance to detect any deviation, and in fact he detected none.
Even today, studies with lasers and high-precision apparatus have detected no deviation
yet – on Earth. The proportionality factor that determines the curvature of space-time
on the surface of the Earth is simply too small. But Gauss did not know, as we do today,
that gravity and curvature go hand in hand.

Motion Mountain – The Adventure of Physics
Curvature in space-time

“
Notre tête est ronde pour permettre à la pensée

”
de changer de direction.**
Francis Picabia

In nature, with four space-time dimensions, specifying curvature requires a more in-
volved approach. First of all, the use of space-time coordinates automatically introduces
the speed of light 𝑐 as limit speed. Furthermore, the number of dimensions being four, we
expect several types of curvature: We expect a value for an average curvature at a point,
defined by comparing the 4-volume of a 4-sphere in space-time with the one deduced

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from the measured radius; then we expect a set of ‘almost average’ curvatures defined
by 3-volumes of 3-spheres in various orientations, plus a set of ‘low-level’ curvatures
defined by usual 2-areas of usual 2-discs in even more orientations. Obviously, we need
to bring some order to bear on this set.
Fortunately, physics can help to make the mathematics easier. We start by defin-
ing what we mean by curvature in space-time. To achieve this, we use the definition
of curvature of Figure 85. As shown in the figure, the curvature 𝐾 also describes how
geodesics diverge or converge.
Geodesics are the straightest paths on a surface, i.e., those paths that a tiny car or tri-
cycle would follow if it drove on the surface keeping the steering wheel straight. Locally,
nearby geodesics are parallel lines. If two nearby geodesics are in a curved space, their
Challenge 288 e separation 𝑠 will change along the geodesics. This happens as

d2 𝑠
= −𝐾𝑠 + higher orders (193)
d𝑙2

* Johann Lambert (1728–1777), Swiss mathematician, physicist and philosopher. Among many achieve-
ments, he proved the irrationality of π; also several laws of optics are named after him.
** ‘Our head is round in order to allow our thougths to change direction.’ Francis Picabia (b. 1879 Paris,
d. 1953 Paris) dadaist and surrealist painter.
from curvature to motion 191

before

after

F I G U R E 86 Tidal effects measure the curvature of space-time.

where 𝑙 measures the length along the geodesic. Here, 𝐾 is the local curvature, in other

Motion Mountain – The Adventure of Physics
words, the inverse squared curvature radius. In the case of space-time, this relation is
extended by substituting proper time 𝜏 (times the speed of light) for proper length. Thus
separation and curvature are related by

d2 𝑠
= −𝐾𝑐2 𝑠 + higher orders . (194)
d𝜏2
But this is the definition of an acceleration! In space-time, geodesics are the paths fol-
lowed by freely falling particles. In other words, what in the purely spatial case is de-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
scribed by curvature, in the case of space-time becomes the relative acceleration of two
nearby, freely falling particles. Indeed, we have encountered these accelerations already:
Vol. I, page 197 they describe tidal effects. In short, space-time curvature and tidal effects are precisely
the same.
Obviously, the magnitude of tidal effects, and thus of curvature, will depend on the
orientation – more precisely on the orientation of the space-time plane formed by the
two particle velocities. Figure 86 shows that the sign of tidal effects, and thus the sign
of curvature, depends on the orientation: particles above each other diverge, particles
side-by-side converge.
The definition of curvature also implies that 𝐾 is a tensor, so that later on we will have
Challenge 289 ny to add indices to it. (How many?) The fun is that we can avoid indices for a while by
Ref. 188 looking at a special combination of spatial curvatures. If we take three planes in space,
all orthogonal to each other and intersecting at a given point, the sum of these three so-
called sectional curvatures does not depend on the observer. (This corresponds to the
Challenge 290 ny tensor trace.) Can you confirm this, by using the definition of the curvature just given?
The sum of the three sectional curvatures defined for mutually orthogonal planes
𝐾(12) , 𝐾(23) and 𝐾(31) , is related to the excess radius defined above. Can you find out
Challenge 291 ny how?
If a surface has constant curvature, i.e., the same curvature at all locations, geometrical
Challenge 292 e objects can be moved around without deforming them. Can you picture this?
In summary, space-time curvature is an intuitive concept that describes how space-
192 7 from curvature to motion

time is deformed. The local curvature of space-time is determined by following the mo-
Ref. 189 tion of nearby, freely falling particles. If we imagine space (-time) as a mattress, a big
blob of rubber inside which we live, the curvature at a point describes how this mat-
tress is squeezed at that point. Since we live inside the mattress, we need to use ‘insider’
methods, such as excess radii and sectional curvatures, to describe the deformation.
General relativity often seems difficult to learn because people do not like to think
about the vacuum as a mattress, and even less to explain it in this way. We recall that for
a hundred years it is an article of faith for every physicist to say that the vacuum is empty.
This remains true. Nevertheless, picturing vacuum as a mattress, or as a substance, helps
in many ways to understand general relativity.

Average curvature and motion in general relativity
One half of general relativity is the statement that any object moves along geodesics, i.e.,
along paths of maximum proper time. The other half is contained in a single expression:
for every observer, the sum of all three proper sectional spatial curvatures at a point, the

Motion Mountain – The Adventure of Physics
average curvature, is given by

8π𝐺 (0)
𝐾(12) + 𝐾(23) + 𝐾(31) = 𝑊 (195)
𝑐4

where 𝑊(0) is the proper energy density at the point. The lower indices indicate the mixed
curvatures defined by the three orthogonal directions 1, 2 and 3. This is all of general
relativity in one paragraph.
We know that space-time is curved around mass and energy. Expression (195) spe-
cifies how much mass and energy curve space. We note that the factor on the right side

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is 2π divided by the maximum force.
Challenge 293 e An equivalent description is easily found using the excess radius defined above, by
introducing the mass 𝑀 = 𝑉𝑊(0) /𝑐2 . For the surface area 𝐴 of the spherical volume 𝑉
containing the mass, we get

𝐺
𝑟excess = 𝑟 − √𝐴/4π = 𝑀. (196)
3𝑐2
In short, general relativity affirms that for every observer, the excess radius of a small
sphere is given by the mass inside the sphere.*
Note that both descriptions imply that the average space curvature at a point in empty
space vanishes. As we will see shortly, this means that near a spherical mass the negative
of the curvature towards the mass is equal to twice the curvature around the mass; the
total sum is thus zero.
Curvature differs from point to point. In particular, the two descriptions imply that if

Ref. 190 * Another, equivalent formulation is that for small radii the area 𝐴 is given by

1
𝐴 = 4π𝑟2 (1 + 𝑟2 𝑅) (197)
9
where 𝑅 is the Ricci scalar, to be introduced later on.
from curvature to motion 193

energy moves, curvature will move with it. In short, both space curvature and, as we will
see shortly, space-time curvature change over space and time.
We note in passing that curvature has an annoying effect: the relative velocity of dis-
Challenge 294 ny tant observers is undefined. Can you provide the argument? In curved space, relative
velocity is defined only for nearby objects – in fact only for objects at no distance at all.
Relative velocities of distant objects are well defined only in flat space.
The quantities appearing in expression (195) are independent of the observer. But often
people want to use observer-dependent quantities. The relation then gets more involved;
the single equation (195) must be expanded to ten equations, called Einstein’s field equa-
tions. They will be introduced below. But before we do that, we will check that general
relativity makes sense. We will skip the check that it contains special relativity as a lim-
iting case, and go directly to the main test.

Universal gravit y

“ ”
The only reason which keeps me here is gravity.
Anonymous

Motion Mountain – The Adventure of Physics
For small velocities and low curvature values, the temporal curvatures 𝐾(0𝑗) turn out to
have a special property. In this case, they can be defined as the second spatial derivatives
Challenge 295 e of a single scalar function 𝜑. In other words, in everyday situations we can write

∂2 𝜑
𝐾(0𝑗) = . (198)
∂(𝑥𝑗 )2

In everyday situations, this approximation is excellent, and the function 𝜑 turns out to

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
be the gravitational potential. Indeed, low velocities and low curvature imply that we can
set 𝑊(0) = 𝜌𝑐2 and 𝑐 → ∞, so that we get

𝐾(𝑖𝑗) = 0 and 𝐾(01) + 𝐾(02) + 𝐾(03) = Δ𝜑 = 4π𝐺𝜌 . (199)

In other words, for small speeds, space is flat and the potential 𝜑 obeys Poisson’s equa-
tion. Universal gravity is thus indeed the low speed and low curvature limit of general
relativity.
Challenge 296 ny Can you show that relation (195) between curvature and energy density indeed im-
plies, in a more precise approximation, that time near a mass depends on the height, as
Page 137 mentioned before?

The S chwarzschild metric
Ref. 188 What is the exact curvature of space-time near a spherical mass? The answer was given
in 1915 by Karl Schwarzschild, who calculated the result during his military service in the
First World War. Einstein then called the solution after him.
194 7 from curvature to motion

Page 145 In spherical coordinates the line element is

2𝐺𝑀 2 2 d𝑟2
d𝑠2 = (1 − 2
) 𝑐 d𝑡 − 2𝐺𝑀
− 𝑟2 d𝜑2 . (200)
𝑟𝑐 1 − 𝑟𝑐2

Challenge 297 ny The curvature of the Schwarzschild metric is then by

𝐺𝑀 𝐺𝑀
𝐾𝑟𝜑 = 𝐾𝑟𝜃 = − 2 3
and 𝐾𝜃𝜑 = 2 2 3
𝑐 𝑟 𝑐 𝑟
𝐺𝑀 𝐺𝑀
𝐾𝑡𝜑 = 𝐾𝑡𝜃 = 2 3 and 𝐾𝑡𝑟 = −2 2 3 (201)
𝑐 𝑟 𝑐 𝑟

Ref. 188 everywhere. The dependence on 1/𝑟3 follows from the general dependence of all tidal
Vol. I, page 197 effects; we have already calculated them in the chapter on universal gravity. The factors
𝐺/𝑐2 are due to the maximum force of gravity. Only the numerical prefactors need to be

Motion Mountain – The Adventure of Physics
calculated from general relativity. The average curvature obviously vanishes, as it does
Challenge 298 ny for all points in vacuum. As expected, the values of the curvatures near the surface of the
Earth are exceedingly small.

Curiosities and fun challenges ab ou t curvature

“ ”
Il faut suivre sa pente, surtout si elle monte.*
André Gide

A fly has landed on the outside of a cylindrical glass, 1 cm below its rim. A drop of honey

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
is located halfway around the glass, also on the outside, 2 cm below the rim. What is the
Challenge 299 e shortest distance from the fly to the drop? What is the shortest distance if the drop is on
the inside of the glass?
∗∗
Challenge 300 e Where are the points of highest and lowest Gaussian curvature on an egg?

Three-dimensional curvature: the R icci tensor**

“
Jeder Straßenjunge in unserem mathematischen
Göttingen versteht mehr von vierdimensionaler
Geometrie als Einstein. Aber trotzdem hat
Einstein die Sache gemacht, und nicht die

”
großen Mathematiker.
David Hilbert***

Now that we have a feeling for curvature, let us describe it in a way that allows any
observer to talk to any other observer. Unfortunately, this means using formulae with

* ‘One has to follow one’s inclination, especially if it climbs upwards.’
*** ‘Every street urchin in our mathematical Göttingen knows more about four-dimensional geometry than
Einstein. Nevertheless, it was Einstein who did the work, not the great mathematicians.’
*** The rest of this chapter might be skipped at first reading.
from curvature to motion 195

tensors. At fisrt, these formulae look daunting. The challenge is to see in each of the
expressions the essential point (e.g. by forgetting all indices for a while) and not to be
distracted by those small letters sprinkled all over them.
We mentioned above that a 4-dimensional space-time is described by 2-curvature,
3-curvature and 4-curvature. Many introductions to general relativity start with 3-
curvature. 3-curvature describes the distinction between the 3-volume calculated from a
radius and the actual 3-volume. The details are described by the Ricci tensor.* Exploring
geodesic deviation, it turns out that the Ricci tensor describes how the shape of a spher-
ical cloud of freely falling particles – a coffee cloud – is deformed along its path. More
precisely, the Ricci tensor 𝑅𝑎𝑏 is (the precise formulation of) the second (proper) time
derivative of the cloud volume divided by the cloud volume. In vacuum, the volume of
Ref. 191 such a falling coffee cloud always stays constant, and this despite the deformation due to
Page 191 tidal forces. Figure 86 illustrates that gravitation does not change coffee cloud volumes.
In short, the Ricci tensor is the general-relativistic version of the Laplacian of the poten-
tial Δ𝜑, or better, of the four-dimensional analogue ◻𝜑.

Motion Mountain – The Adventure of Physics
Average curvature: the R icci scalar
The most global, but least detailed, definition of curvature is the one describing the dis-
tinction between the 4-volume calculated from a measured radius and the actual 4-
volume. This is the average curvature at a space-time point and is represented by the
so-called Ricci scalar 𝑅, defined as

−2
𝑅 = −2𝐾 = 2
. (202)
𝑟curvature

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
It turns out that the Ricci scalar can be derived from the Ricci tensor by a so-called con-
traction, which is a precise averaging procedure. For tensors of rank two, contraction is
the same as taking the trace:
𝑅 = 𝑅𝜆 𝜆 = 𝑔𝜆𝜇 𝑅𝜆𝜇 . (203)

The Ricci scalar describes the curvature averaged over space and time. In the image
of a falling spherical cloud, the Ricci scalar describes the volume change of the cloud.
The Ricci scalar always vanishes in vacuum. This result allows us to relate the spatial
Challenge 301 ny curvature to the change of time with height on the surface of the Earth.

The Einstein tensor
After two years of hard work, Einstein discovered that the best quantity for the descrip-
tion of curvature in nature is not the Ricci tensor 𝑅𝑎𝑏 , but a tensor built from it. This
so-called Einstein tensor 𝐺𝑎𝑏 is defined mathematically (for vanishing cosmological con-
stant) as
1
𝐺𝑎𝑏 = 𝑅𝑎𝑏 − 𝑔𝑎𝑏 𝑅 . (204)
2

* Gregorio Ricci-Cubastro (b. 1853 Lugo, d. 1925 Bologna), mathematician. He is the father of absolute dif-
ferential calculus, also called ‘Ricci calculus’. Tullio Levi-Civita was his pupil.
196 7 from curvature to motion

It is not difficult to understand its meaning. The value 𝐺00 is the sum of sectional
curvatures in the planes orthogonal to the 0 direction and thus the sum of all spatial
sectional curvatures:
𝐺00 = 𝐾(12) + 𝐾(23) + 𝐾(31) . (205)

Similarly, for each dimension 𝑖 the diagonal element 𝐺𝑖𝑖 is the sum (taking into consid-
eration the minus signs of the metric) of sectional curvatures in the planes orthogonal to
the 𝑖 direction. For example, we have

𝐺11 = 𝐾(02) + 𝐾(03) − 𝐾(23) . (206)

The distinction between the Ricci tensor and the Einstein tensor thus lies in the way in
which the sectional curvatures are combined: discs containing the coordinate in question
for the Ricci tensor, and discs orthogonal to the coordinate for the Einstein tensor. Both
describe the curvature of space-time equally well, and fixing one means fixing the other.
Challenge 302 d (What are the trace and the determinant of the Einstein tensor?)

Motion Mountain – The Adventure of Physics
The Einstein tensor is symmetric, which means that it has ten independent compon-
ents. Most importantly, its divergence vanishes; it therefore describes a conserved quant-
ity. This was the essential property which allowed Einstein to relate it to mass and energy
in mathematical language.

The description of momentum, mass and energy
Obviously, for a complete description of gravity, the motion of momentum and energy
need to be quantified in such a way that any observer can talk to any other. We have seen
that momentum and energy always appear together in relativistic descriptions; the next

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
step is thus to find out how their motions can be quantified for general observers.
First of all, the quantity describing energy, let us call it 𝑇, must be defined using the
energy–momentum vector 𝑝 = 𝑚𝑢 = (𝛾𝑚𝑐, 𝛾𝑚𝑣) of special relativity. Furthermore, 𝑇
does not describe a single particle, but the way energy–momentum is distributed over
space and time. As a consequence, it is most practical to use 𝑇 to describe a density of
energy and momentum. 𝑇 will thus be a field, and depend on time and space, a fact
usually indicated by the notation 𝑇 = 𝑇(𝑡, 𝑥).
Since the energy–momentum density 𝑇 describes a density over space and time, it
defines, at every space-time point and for every infinitesimal surface d𝐴 around that
point, the flow of energy–momentum d𝑝 through that surface. In other words, 𝑇 is
defined by the relation
d𝑝 = 𝑇 d𝐴 . (207)

The surface is assumed to be characterized by its normal vector d𝐴. Since the energy–
momentum density is a proportionality factor between two vectors, 𝑇 is a tensor.
Of course, we are talking about 4-flows and 4-surfaces here. Therefore the energy–
from curvature to motion 197

momentum density tensor can be split in the following way:

𝑤 𝑆1 𝑆2 𝑆3 energy energy flow or
𝑆1 𝑡11 𝑡12 𝑡13 density momentum density
𝑇=( )=( ) (208)
𝑆2 𝑡21 𝑡22 𝑡23 energy flow or momentum
𝑆3 𝑡31 𝑡32 𝑡33 momentum density flow density

where 𝑤 = 𝑇00 is a 3-scalar, 𝑆 a 3-vector and 𝑡 a 3-tensor. The total quantity 𝑇 is called
the energy–momentum (density) tensor. It has two essential properties: it is symmetric
and its divergence vanishes.
The symmetry of the tensor 𝑇 is a result of the conservation of angular momentum.
The vanishing divergence of the tensor 𝑇, often written as

∂𝑎 𝑇𝑎𝑏 = 0 or abbreviated 𝑇𝑎𝑏 , 𝑎 = 0 , (209)

implies that the tensor describes a conserved quantity. In every volume, energy can

Motion Mountain – The Adventure of Physics
change only via flow through its boundary surface. Can you confirm that the description
of energy–momentum with this tensor satisfies the requirement that any two observers,
differing in position, orientation, speed and acceleration, can communicate their results
Challenge 303 ny to each other?
The energy–momentum density tensor gives a full description of the distribution of
energy, momentum and mass over space and time. As an example, let us determine the
energy–momentum density for a moving liquid. For a liquid of density 𝜌, a pressure 𝑝
and a 4-velocity 𝑢, we have

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑇𝑎𝑏 = (𝜌0 + 𝑝)𝑢𝑎 𝑢𝑏 − 𝑝𝑔𝑎𝑏 (210)

where 𝜌0 is the density measured in the comoving frame, the so-called proper density.*
Obviously, 𝜌, 𝜌0 and 𝑝 depend on space and time.
Of course, for a particular material fluid, we need to know how pressure 𝑝 and density
𝜌 are related. A full material characterization thus requires the knowledge of the relation

𝑝 = 𝑝(𝜌) . (212)

This relation is a material property and thus cannot be determined from relativity. It has
to be derived from the constituents of matter or radiation and their interactions. The
simplest possible case is dust, i.e., matter made of point particles** with no interactions

* In the comoving frame we thus have

𝜌0 𝑐2 0 0 0
0 𝑝 0 0
𝑇𝑎𝑏 = ( ) . (211)
0 0 𝑝 0
0 0 0 𝑝

** Even though general relativity expressly forbids the existence of point particles, the approximation is
useful in cases when the particle distances are large compared to their own size.
198 7 from curvature to motion

at all. Its energy–momentum tensor is given by

𝑇𝑎𝑏 = 𝜌0 𝑢𝑎 𝑢𝑏 . (213)

Challenge 304 ny Can you explain the difference from the liquid case?
The divergence of the energy–momentum tensor vanishes for all times and positions,
Challenge 305 ny as you may want to check. This property is the same as for the Einstein tensor presen-
ted above. But before we elaborate on this issue, a short remark. We did not take into
account gravitational energy. It turns out that gravitational energy cannot be defined in
general. In general, gravity does not have an associated energy. In certain special cir-
cumstances, such as weak fields, slow motion, or an asymptotically flat space-time, we
can define the integral of the 𝐺00 component of the Einstein tensor as negative gravita-
tional energy. Gravitational energy is thus only defined approximately, and only for our
everyday environment.*

Einstein ’ s field equations

Motion Mountain – The Adventure of Physics
“
[Einstein’s general theory of relativity] cloaked

”
the ghastly appearance of atheism.
A witch hunter from Boston, around 1935

“ ”
Do you believe in god? Prepaid reply 50 words.
Subsequent telegram by another witch hunter
to his hero Albert Einstein

“
I believe in Spinoza’s god, who reveals himself
in the orderly harmony of what exists, not in a

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
god who concerns himself with fates and

”
actions of human beings.
Albert Einstein’s answer
Einstein’s famous field equations were the basis of many religious worries. They contain
the full description of general relativity. The equations can be deduced in many ways. The
Page 113 simplest way to deduce them is to start from the principle of maximum force. Another
Page 201 way is to deduce the equation from the Hilbert action, as explained below. A third way
is what we are doing at present, namely to generalize the relation between curvature and
energy to general observers.
Einstein’s field equations are given by

𝐺𝑎𝑏 = −𝜅 𝑇𝑎𝑏
or, in more detail
1
𝑅𝑎𝑏 − 𝑔𝑎𝑏 𝑅 − Λ𝑔𝑎𝑏 = −𝜅 𝑇𝑎𝑏 . (214)
2

* This approximation leads to the famous speculation that the total energy of the universe is zero. Do you
Challenge 306 s agree?
from curvature to motion 199

The constant 𝜅, called the gravitational coupling constant, has been measured to be

8π𝐺
𝜅= = 2.1 ⋅ 10−43 /N (215)
𝑐4

and its small value – the value 2π divided by the maximum force 𝑐4 /4𝐺 – reflects the
weakness of gravity in everyday life, or better, the difficulty of bending space-time. The
constant Λ, the so-called cosmological constant, corresponds to a vacuum energy volume
Page 243 density, or pressure Λ/𝜅. Its low value is quite hard to measure. The currently favoured
value is
Λ ≈ 10−52 /m2 or Λ/𝜅 ≈ 0.5 nJ/m3 = 0.5 nPa . (216)

Ref. 192 Current measurements and simulations suggest that this parameter, even though it is
numerically near to the inverse square of the present radius of the universe, is a constant
of nature that does not vary with time.
In summary, the field equations state that the curvature at a point is equal to the flow of

Motion Mountain – The Adventure of Physics
energy–momentum through that point, taking into account the vacuum energy density.
In other words: Energy–momentum tells space-time how to curve, using the maximum
force as proportionality factor.*

Universal gravitation – again
The field equations of general relativity can be simplified for the case in which speeds are
small. In that case 𝑇00 = 𝑐2 𝜌 and all other components of 𝑇 vanish. Using the definition
Challenge 307 ny of the constant 𝜅 and setting 𝜑 = (𝑐2 /2)ℎ00 in 𝑔𝑎𝑏 = 𝜂𝑎𝑏 + ℎ𝑎𝑏 , we find

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
d2 𝑥
∇2 𝜑 = 4π𝜌 and = −∇𝜑 (217)
d𝑡2
* Einstein arrived at his field equations using a number of intellectual guidelines that are called principles in
the literature. Today, many of them are not seen as central any more. Nevertheless, we give a short overview.
- Principle of general relativity: all observers are equivalent; this principle, even though often stated, is
probably empty of any physical content.
- Principle of general covariance: the equations of physics must be stated in tensor form; even though it
Ref. 193 is known today that all equations can be written with tensors, even universal gravity, in many cases they
require unphysical ‘absolute’ elements, i.e., quantities which affect others but are not affected themselves.
Vol. III, page 323 This unphysical idea is in contrast with the idea of interaction, as explained later on.
- Principle of minimal coupling: the field equations of gravity are found from those of special relativity
by taking the simplest possible generalization. Of course, now that the equations are known and tested
experimentally, this principle is only of historical interest.
- Equivalence principle: acceleration is locally indistinguishable from gravitation; we used it to argue that
space-time is semi-Riemannian, and that gravity is its curvature.
- Mach’s principle: inertia is due to the interaction with the rest of the universe; this principle is correct,
even though it is often maintained that it is not fulfilled in general relativity. In any case, it is not the essence
Page 258 of general relativity.
- Identity of gravitational and inertial mass: this is included in the definition of mass from the outset, but
restated ad nauseam in general relativity texts; it is implicitly used in the definition of the Riemann tensor.
- Correspondence principle: a new, more general theory, such as general relativity, must reduce to previous
theories, in this case universal gravity or special relativity, when restricted to the domains in which those
are valid.
200 7 from curvature to motion

which we know well, since it can be restated as follows: a body of mass 𝑚 near a body of
mass 𝑀 is accelerated by
𝑀
𝑎=𝐺 2, (218)
𝑟
a value which is independent of the mass 𝑚 of the falling body. And indeed, as noted
already by Galileo, all bodies fall with the same acceleration, independently of their size,
their mass, their colour, etc. In general relativity also, gravitation is completely demo-
cratic.* The independence of free fall from the mass of the falling body follows from the
description of space-time as a bent mattress. Objects moving on a mattress also move in
the same way, independently of the mass value.

Understanding the field equations
To get a feeling for the complete field equations, we will take a short walk through their
main properties. First of all, all motion due to space-time curvature is reversible, differ-
entiable and thus deterministic. Note that only the complete motion, of space-time and

Motion Mountain – The Adventure of Physics
Challenge 308 e
matter and energy, has these properties. For particle motion only, motion is in fact irre-
versible, since some gravitational radiation is usually emitted.
By contracting the field equations we find, for vanishing cosmological constant, the
following expression for the Ricci scalar:

𝑅 = −𝜅𝑇 . (223)

This result also implies the relation between the excess radius and the mass inside a
Challenge 309 ny sphere.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The field equations are nonlinear in the metric 𝑔, meaning that sums of solutions usu-
ally are not solutions. That makes the search for solutions rather difficult. For a complete
solution of the field equations, initial and boundary conditions should be specified. The
Ref. 194 ways to do this form a specialized part of mathematical physics; it is not explored here.
Albert Einstein used to say that general relativity only provides the understanding of
one side of the field equations (214), but not of the other. Can you see which side he
Challenge 310 ny meant?

* Here is yet another way to show that general relativity fits with universal gravity. From the definition of
the Riemann tensor we know that relative acceleration 𝑏𝑎 and speed of nearby particles are related by

∇𝑒 𝑏𝑎 = 𝑅𝑐𝑒𝑑𝑎 𝑣𝑐 𝑣𝑑 . (219)

From the symmetries of 𝑅 we know there is a 𝜑 such that 𝑏𝑎 = −∇𝑎 𝜑. That means that

∇𝑒 𝑏𝑎 = ∇𝑒 ∇𝑎 𝜑 = 𝑅𝑎𝑐𝑒𝑑 𝑣𝑐 𝑣𝑑 (220)

which implies that

Δ𝜑 = ∇𝑎 ∇𝑎 𝜑 = 𝑅𝑎𝑐𝑎𝑑 𝑣𝑐 𝑣𝑑 = 𝑅𝑐𝑑 𝑣𝑐 𝑣𝑑 = 𝜅(𝑇𝑐𝑑 𝑣𝑐 𝑣𝑑 − 𝑇/2) (221)

Introducing 𝑇𝑎𝑏 = 𝜌𝑣𝑎 𝑣𝑏 we get
Δ𝜑 = 4π𝐺𝜌 (222)
as we wanted to show.
from curvature to motion 201

What can we do of interest with the field equations? In fact, to be honest, not much
that we have not done already. Very few processes require the use of the full equations.
Many textbooks on relativity even stop after writing them down! However, studying
them is worthwhile. For example, one can show that the Schwarzschild solution is the
only spherically symmetric solution. Similarly, in 1923, Birkhoff showed that every rota-
tionally symmetric vacuum solution is static. This is the case even if masses themselves
move, as for example during the collapse of a star.
Maybe the most beautiful applications of the field equations are the various films made
of relativistic processes. The worldwide web hosts several of these; they allow one to see
what happens when two black holes collide, what happens when an observer falls into
a black hole, etc. To generate these films, the field equations usually need to be solved
directly, without approximations.*
Another area of application concerns gravitational waves. The full field equations
show that gravity waves are not harmonic, but nonlinear. Sine waves exist only approxim-
ately, for small amplitudes. Even more interestingly, if two waves collide, in many cases
singularities of curvature are predicted to appear, i.e., points of infinite curvature. This

Motion Mountain – The Adventure of Physics
whole theme is still a research topic and might provide new insights for the quantization
of general relativity in the coming years.
We end this section with a side note. Usually, the field equations are read in one sense
only, as stating that energy–momentum produces curvature. One can also read them in
the other way, calculating the energy–momentum needed to produce a given curvature.
When one does this, one discovers that not all curved space-times are possible, as some
would lead to negative energy (or mass) densities. Such solutions would contradict the
mentioned limit on length-to-mass ratios for physical systems.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Hilbert ’ s action – how d oes space bend?
When Einstein discussed his research with David Hilbert, Hilbert found a way to do in
a few weeks what had taken years for Einstein. Hilbert showed that general relativity in
empty space could be described with the least action principle.
Hilbert knew that all motion minimizes action, i.e., all motion minimizes change. Hil-
bert set out to find the Lagrangian, i.e., the measure of change, for the motion of space-
time, more precisely, for the bending of space-time. Obviously, such a measure must be
observer-invariant; in particular, it must be invariant under all possible changes of view-
point.
Motion due to gravity is determined by curvature. Any curvature measure independ-
ent of the observer must be a combination of the Ricci scalar 𝑅 and the cosmological
constant Λ. In this way both the equivalence principle and general covariance are re-
spected. It thus makes sense to expect that the change of space-time is described by an
action 𝑆 given by
𝑐4
𝑆= ∫(𝑅 − 2Λ) d𝑉 . (224)
16π𝐺

The volume element d𝑉 must be specified to use this expression in calculations. The cos-
mological constant Λ (added some years after Hilbert’s work) appears as a mathematical
* See for example the www.photon.at/~werner/black-earth website.
202 7 from curvature to motion

possibility to describe the most general action that is diffeomorphism-invariant. We will
see below that its value in nature, though small, seems to be different from zero.
We can also add matter to the Hilbert action; a lengthy calculation then confirms that
the Hilbert action allows deducing Einstein’s field equations – and vice versa. Both for-
mulations are equivalent. The Hilbert action of a chunk of space-time is thus the integral
of the Ricci scalar plus twice the cosmological constant over that chunk. The principle of
least action states that space-time moves or bends in such a way that this integral changes
as little as possible.
We note that the maximum force, with its huge value, appears as a prefactor in the
action (224). A small deviation in curvature thus implies a huge observable action or
change. This reflects the extreme stiffness of space-time. Can you show that the Hilbert
Challenge 311 ny action follows from the maximum force?
In addition to the Hilbert action, for a full description of motion we need initial con-
Ref. 128 ditions. The various ways to do this define a specific research field. This topic however,
leads too far from our path. The same is valid for other, but equivalent, expressions of
the action of general relativity.

Motion Mountain – The Adventure of Physics
In summary, the question ‘how does space move?’ is answered by the least action
principle in the following way: space evolves by minimizing scalar curvature. The question
‘how do things move?’ is answered by general relativity in the same way as by special
relativity: things follow the path of maximal ageing.

The symmetries of general relativity
The main symmetry of the Lagrangian of general relativity is called diffeomorphism in-
variance or general covariance. Physically speaking, the symmetry states that motion is
independent of the coordinate system used. More precisely, the motion of matter, radi-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ation and space-time does not change under arbitrary differentiable coordinate trans-
formations, or diffeomorphisms. Diffeomorphism invariance is the essential symmetry
of the Hilbert action: motion is independent of coordinates systems.
The field equations for empty space-time also show scale symmetry. This is the in-
variance of the equations after multiplication of all coordinates by a common numerical
factor. In 1993, Torre and Anderson showed that diffeomorphism symmetry and trivial
Ref. 195 scale symmetry are the only symmetries of the vacuum field equations.
Apart from diffeomorphism invariance, full general relativity, including mass–energy,
has an additional symmetry that is not yet fully elucidated. This symmetry connects the
various possible initial conditions of the field equations; the symmetry is extremely com-
Ref. 196 plex and is still a topic of research. These fascinating investigations might give new in-
sights into the classical description of the big bang.
In summary, the symmetries of general relativity imply that also the fastest, the most
distant and the most powerful motion in nature is relative, continuous, reversible and
mirror invariant. The symmetries also confirm that the most violent motion conserves
energy–momentum and angular momentum. Finally, Hilbert’s action confirms that even
the wildest motion in nature is lazy, i.e., described by the least action principle.
In short, despite adding motion of vacuum and horizons, general relativity does not
change our everyday concept of motion. Relativity is a classical description of motion.
from curvature to motion 203

Mass in general relativit y
The diffeomorphism-invariance of general relativity makes life quite interesting. We will
Page 285 see that it allows us to say that we live on the inside of a hollow sphere. We have seen
that general relativity does not allow us to say where energy is actually located. If energy
cannot be located, what about mass? Exploring the issue shows that mass, like energy,
can be localized only if distant space-time is known to be flat. It is then possible to define
a localized mass value by making precise an intuitive idea: the mass of an unknown body
is measured by the time a probe takes to orbit the unknown body.*
The intuitive mass definition requires flat space-time at infinity; it cannot be extended
Challenge 312 ny to other situations. In short, mass can only be localized if total mass can be defined. And
total mass is defined only for asymptotically flat space-time. The only other notion of
mass that is precise in general relativity is the local mass density at a point. In contrast, it
is not well understood how to define the mass contained in a region larger than a point
but smaller than the entirety of space-time (in the case that it is not asymptotically flat).

Motion Mountain – The Adventure of Physics
The force limit and the cosmolo gical constant
When the cosmological constant is taken into the picture, the maximum force principle
requires a second look. In the case of a non-vanishing cosmological constant, the force
Ref. 199 limit makes sense only if the constant Λ is positive; this is the case for the currently
Ref. 128, Ref. 129 measured value, which is Λ ≈ 10−52 /m2 . Indeed, the radius–mass relation of black holes

Λ 2
2𝐺𝑀 = 𝑅𝑐2 (1 − 𝑅) (227)
3

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
implies that a radius-independent maximum force is valid only for positive or zero cos-
mological constant. For a negative cosmological constant the force limit would only be
valid for infinitely small black holes. In the following, we take a pragmatic approach and
note that a maximum force limit can be seen to imply a vanishing or positive cosmolo-
gical constant. Obviously, the force limit does not specify the value of the constant; to
achieve this, a second principle needs to be added. A straightforward formulation, using
Page 134 the additional principle of a minimum force in nature, was proposed above.
One might ask also whether rotating or charged black holes change the argument that
leads from maximum force to the derivation of general relativity. However, the deriva-

Ref. 197 * This definition was formalized by Arnowitt, Deser and Misner, and since then has often been called the
ADM mass. The idea is to use the metric 𝑔𝑖𝑗 and to take the integral

𝑐2
𝑚= ∫ (𝑔 𝜈 − 𝑔𝑖𝑖,𝑗 𝜈𝑗 )d𝐴 (225)
32π𝐺 𝑆𝑅 𝑖𝑗,𝑖 𝑗
where 𝑆𝑅 is the coordinate sphere of radius 𝑅, 𝜈 is the unit vector normal to the sphere and d𝐴 is the
area element on the sphere. The limit exists for large 𝑅 if space-time is asymptotically flat and if the mass
Ref. 198 distribution is sufficiently concentrated. Mathematical physicists have also shown that for any manifold
whose metric changes at infinity as

𝑔𝑖𝑗 = (1 + 𝑓/𝑟 + 𝑂(1/𝑟2 ))𝛿𝑖𝑗 (226)

the total mass is given by 𝑀 = 𝑓𝑐2 /𝐺.
204 7 from curvature to motion

tion using the Raychaudhuri equation does not change. In fact, the only change of the
argument appears with the inclusion of torsion, which changes the Raychaudhuri equa-
tion itself. As long as torsion plays no role, the derivation given above remains valid. The
inclusion of torsion is still an open research issue.

Is gravit y an interaction?
We tend to answer this question affirmatively, as in Galilean physics gravity was seen
as an influence on the motion of bodies. In Galilean physics, we described gravity by
a potential, because gravity changes motion. Indeed, a force or an interaction is what
changes the motion of objects. However, we just saw that when two bodies attract each
other through gravitation, both always remain in free fall. For example, the Moon circles
the Earth because it continuously falls around it. Since any freely falling observer con-
tinuously remains at rest, the statement that gravity changes the motion of bodies is not
correct for all observers. In fact, given that geodesics are the path of maximum straight-
ness, we can also argue that the Moon and the Earth both follow ‘straight’ paths, and

Motion Mountain – The Adventure of Physics
for all observers. But objects that follow straight paths are not under the influence of
interactions, are they?
Vol. III, page 322 Let us explore this issue in another way. The most fundamental definition of
‘interaction’ is as the difference between the whole and the sum of its parts. In the
case of gravity, an observer in free fall could indeed claim that nothing special is going
on, independently of whether the other body is present or not, and could claim that
gravity is not an interaction.
However, an interaction also transports energy between systems. Now, we have seen
Page 198 that gravity can be said to transport energy only approximately. The properties of grav-
Challenge 313 s itational energy confirm this argument. Even in its energy aspect, gravitation is an inter-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
action only approximately.
A mathematical way to look at these issue is the following. Take a satellite orbiting
Jupiter with energy–momentum 𝑝 = 𝑚𝑢. If we calculate the energy–momentum change
Challenge 314 ny along its path 𝑠, we get

d𝑝 d𝑢 d𝑢𝑎 d𝑒𝑎 𝑎 d𝑢𝑎
=𝑚 = 𝑚 (𝑒𝑎 + 𝑢 ) = 𝑚𝑒𝑎 ( + Γ𝑎 𝑏𝑑 𝑢𝑏 𝑢𝑑 ) = 0 (228)
d𝑠 d𝑠 d𝑠 d𝑠 d𝑠

where 𝑒 describes the unit vector along a coordinate axis and Γ𝑎 𝑏𝑑 is the metric connec-
tion; it is explained below. The energy–momentum change vanishes along any geodesic,
Challenge 315 ny as you might check. Therefore, the energy–momentum of this motion is conserved. In
other words, no force is acting on the satellite. We could reply that in equation (228) the
Ref. 200 second term alone is the real gravitational force. But this term can be made to vanish
Challenge 316 ny along the entirety of any given world line. In short, also the mathematics confirm that
nothing changes between two bodies in free fall around each other: gravity could be said
not to be an interaction.
Let us look at the behaviour of light. In vacuum, light is always moving freely. In a
sense, we can say that radiation always is in free fall. Strangely, since we called free fall
the same as rest, we should conclude that radiation always is at rest. This is not wrong! We
from curvature to motion 205

have already seen that light cannot be accelerated.* We have also seen that gravitational
bending is not an acceleration, since light follows straight paths in space-time in this
case as well. Even though light seems to slow down near masses for distant observers, it
always moves at the speed of light locally. In short, even gravitation doesn’t manage to
move light.
In short, if we like such intellectual games, we can argue that gravitation is not an
interaction, even though it puts objects into orbits and deflects light. For all practical
purposes, gravity remains an interaction.

How to calculate the shape of geodesics
One half of general relativity states that bodies fall along geodesics. All orbits are
geodesics, thus curves with the longest proper time. It is thus useful to be able to calcu-
late these trajectories.** To start, one needs to know the shape of space-time, the notion
of ‘shape’ being generalized from its familiar two-dimensional meaning. For a being liv-
ing on the surface, it is usually described by the metric 𝑔𝑎𝑏 , which defines the distances

Motion Mountain – The Adventure of Physics
between neighbouring points through

d𝑠2 = d𝑥𝑎 d𝑥𝑎 = 𝑔𝑎𝑏 (𝑥) d𝑥𝑎 d𝑥𝑏 . (229)

It is a famous exercise of calculus to show from this expression that a curve 𝑥𝑎 (𝑠) depend-
ing on a well behaved (affine) parameter 𝑠 is a time-like or space-like (metric) geodesic,
Challenge 317 ny i.e., the longest possible path between the two events,*** only if

d d𝑥𝑑 1 ∂𝑔𝑏𝑐 d𝑥𝑏 d𝑥𝑐
(𝑔𝑎𝑑 )= , (230)

* Refraction, the slowdown of light inside matter, is not a counter-example. Strictly speaking, light inside
matter is constantly being absorbed and re-emitted. In between these processes, light still propagates with
the speed of light in vacuum. The whole process only looks like a slowdown in the macroscopic limit. The
Vol. III, page 157 same applies to diffraction and to reflection. A full list of ways to bend light can be found elsewhere.
** This is a short section for the more curious; it can be skipped at first reading.
*** We remember that in space in everyday life, geodesics are the shortest possible paths; however, in space-
time in general relativity, geodesics are the longest possible paths. In both cases, they are the ‘straightest’
possible paths.
206 7 from curvature to motion

as long as d𝑠 is different from zero along the path.* All bodies in free fall follow such
Page 149 geodesics. We showed above that the geodesic property implies that a stone thrown in the
air falls back, unless if it is thrown with a speed larger than the escape velocity. Expression
(230) thus replaces both the expression d2 𝑥/d𝑡2 = −∇𝜑 valid for falling bodies and the
expression d2 𝑥/d𝑡2 = 0 valid for freely floating bodies in special relativity.
The path does not depend on the mass or on the material of the body. Therefore an-
Ref. 201 timatter also falls along geodesics. In other words, antimatter and matter do not repel;
they also attract each other. Interestingly, even experiments performed with normal mat-
Challenge 318 ny ter can show this, if they are carefully evaluated. Can you find out how?
For completeness, we mention that light follows lightlike or null geodesics. In other
words, there is an affine parameter 𝑢 such that the geodesics follow

d2 𝑥𝑎 𝑏
𝑎 d𝑥 d𝑥
𝑐
+ Γ 𝑏𝑐 =0 (234)
d𝑢2 d𝑢 d𝑢
with the different condition

Motion Mountain – The Adventure of Physics
d𝑥𝑎 d𝑥𝑏
𝑔𝑎𝑏 =0. (235)
d𝑢 d𝑢

Challenge 319 ny Given all these definitions of various types of geodesics, what are the lines that are drawn
in Figure 65 on page 144?

R iemann gymnastics**
Most books introduce curvature the hard way, namely historically, using the Riemann
curvature tensor. This is a short summary, so that you can understand that old stuff when

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
you come across it.
We saw above that curvature is best described by a tensor. In 4 dimensions, this
curvature tensor, usually called 𝑅, must be a quantity which allows us to calculate, among
other things, the area for any orientation of a 2-disc in space-time. Now, in 4 dimensions,
Challenge 320 e orientations of a disc are defined in terms of two 4-vectors; let us call them 𝑝 and 𝑞. And
instead of a disc, we take the parallelogram spanned by 𝑝 and 𝑞. There are several possible
definitions.
The Riemann-Christoffel curvature tensor 𝑅 is then defined as a quantity which allows

* This is often written as
d2 𝑥𝑎 𝑏
𝑎 d𝑥 d𝑥
𝑐

2
+ Γ𝑏𝑐 =0 (231)
d𝑠 d𝑠 d𝑠
where the condition
d𝑥𝑎 d𝑥𝑏
𝑔𝑎𝑏 =1 (232)
d𝑠 d𝑠
must be fulfilled, thus simply requiring that all the tangent vectors are unit vectors, and that d𝑠 ≠ 0 all along
the path. The symbols Γ appearing above are given by

𝑎 1
Γ𝑎 𝑏𝑐 = { } = 𝑔𝑎𝑑 (∂𝑏 𝑔𝑑𝑐 + ∂𝑐 𝑔𝑑𝑏 − ∂𝑑 𝑔𝑏𝑐 ) , (233)
𝑏𝑐 2

and are called Christoffel symbols of the second kind or simply the metric connection.
** This is a short section for the more curious; it can be skipped at first reading.
from curvature to motion 207

us to calculate the curvature 𝐾(𝑝, 𝑞) for the surface spanned by 𝑝 and 𝑞, with area 𝐴,
through
𝑅 𝑝𝑞𝑝𝑞 𝑅𝑎𝑏𝑐𝑑 𝑝𝑎 𝑞𝑏 𝑝𝑐 𝑞𝑑
𝐾(𝑝, 𝑞) = 2 = (236)
𝐴 (𝑝, 𝑞) (𝑔𝛼𝛿 𝑔𝛽𝛾 − 𝑔𝛼𝛾 𝑔𝛽𝛿 )𝑝𝛼 𝑞𝛽 𝑝𝛾 𝑞𝛿

where, as usual, Latin indices 𝑎, 𝑏, 𝑐, 𝑑, etc. run from 0 to 3, as do Greek indices here,
and a summation is implied when an index name appears twice. Obviously 𝑅 is a tensor,
of rank 4. This tensor thus describes only the intrinsic curvature of a space-time. In con-
trast, the metric 𝑔 describes the complete shape of the surface, not only the curvature.
The curvature is thus the physical quantity of relevance locally, and physical descriptions
therefore use only the Riemann* tensor 𝑅 or quantities derived from it.**
But we can forget the just-mentioned definition of curvature. There is a second, more
physical way to look at the Riemann tensor. We know that curvature means gravity. As
we said above, gravity means that when two nearby particles move freely with the same
Challenge 321 e velocity and the same direction, the distance between them changes. In other words, the
local effect of gravity is relative acceleration of nearby particles.

Motion Mountain – The Adventure of Physics
It turns out that the tensor 𝑅 describes precisely this relative acceleration, i.e., what
we called the tidal effects earlier on. Obviously, the relative acceleration 𝑏 increases with
Challenge 322 ny the separation 𝑑 and the square (why?) of the speed 𝑢 of the two particles. Therefore we
can also define 𝑅 as a (generalized) proportionality factor among these quantities:

𝑏=𝑅𝑢𝑢𝑑 or, more clearly, 𝑏𝑎 = 𝑅𝑎 𝑏𝑐𝑑 𝑢𝑏 𝑢𝑐 𝑑𝑑 . (239)

The components of the Riemann curvature tensor have the dimensions of inverse square
length. Since it contains all information about intrinsic curvature, we conclude that if 𝑅

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
vanishes in a region, space-time in that region is flat. This connection is easily deduced
Challenge 323 ny from this second definition.***

* Bernhard Riemann (b. 1826 Breselenz, d. 1866 Selasca), important mathematician. One among his nu-
merous important achievements is the foundation of non-Euclidean geometry.
** We showed above that space-time is curved by noting changes in clock rates, in metre bar lengths and
in light propagation. Such experiments are the easiest way to determine the metric 𝑔. We know that space-
time is described by a 4-dimensional manifold M with a metric 𝑔𝑎𝑏 that locally, at each space-time point,
is a Minkowski metric. Such a manifold is called a Riemannian manifold. Only such a metric allows one to
define a local inertial system, i.e., a local Minkowski space-time at every space-time point. In particular, we
have
𝑔𝑎𝑏 = 1/𝑔𝑎𝑏 and 𝑔𝑎 𝑏 = 𝑔𝑎 𝑏 = 𝛿𝑏𝑎 . (237)
How are curvature and metric related? The solution to this question usually occupies a large number of
pages in relativity books; just for information, the relation is

∂Γ𝑎 𝑏𝑑 ∂Γ𝑎 𝑏𝑐
𝑅𝑎 𝑏𝑐𝑑 = − + Γ𝑎 𝑒𝑐 Γ𝑒 𝑏𝑑 − Γ𝑎 𝑓𝑑 Γ𝑓 𝑏𝑐 . (238)
∂𝑥𝑐 ∂𝑥𝑑
The curvature tensor is built from the second derivatives of the metric. On the other hand, we can also
determine the metric if the curvature is known. An approximate relation is given below.
*** This second definition is also called the definition through geodesic deviation. It is of course not evident
Ref. 202 that it coincides with the first. For an explicit proof, see the literature. There is also a third way to picture
the tensor 𝑅, a more mathematical one, namely the original way Riemann introduced it. If one parallel-
transports a vector 𝑤 around a parallelogram formed by two vectors 𝑢 and 𝑣, each of length 𝜀, the vector 𝑤
208 7 from curvature to motion

A final way to define the tensor 𝑅 is the following. For a free-falling observer, the
metric 𝑔𝑎𝑏 is given by the metric 𝜂𝑎𝑏 from special relativity. In its neighbourhood, we
have
1
𝑔𝑎𝑏 = 𝜂𝑎𝑏 + 𝑅𝑎𝑐𝑏𝑑 𝑥𝑐 𝑥𝑑 + 𝑂(𝑥3 )
3
1
= (∂𝑐 ∂𝑑 𝑔𝑎𝑏 )𝑥𝑐 𝑥𝑑 + 𝑂(𝑥3 ) , (241)
2
where 𝑂 denotes terms of higher order. The curvature term thus describes the departure
of the space-time metric from that of flat space-time. The curvature tensor 𝑅 is a large
beast; it has 44 = 256 components at each point of space-time; however, its symmetry
properties reduce them to 20 independent numbers.* The actual number of importance
in physical problems is still smaller, namely only 10. These are the components of the
Ricci tensor, which can be defined with the help of the Riemann tensor by contraction,
i.e., by setting
𝑅𝑏𝑐 = 𝑅𝑎 𝑏𝑎𝑐 .

Motion Mountain – The Adventure of Physics
(244)

Its components, like those of the Riemann tensor, are inverse square lengths. The values
of the tensor 𝑅𝑏𝑐 , or those of 𝑅𝑎𝑏𝑐𝑑 , are independent of the sign convention used in the
Challenge 326 e Minkowski metric, in contrast to 𝑅𝑎𝑏𝑐𝑑 .
Challenge 327 ny Can you confirm the relation 𝑅𝑎𝑏𝑐𝑑 𝑅𝑎𝑏𝑐𝑑 = 48𝑚2 /𝑟6 for the Schwarzschild solution?

Curiosities and fun challenges ab ou t general relativity
For various years, people have speculated why the Pioneer 10 and 11 artificial satellites,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
which are now over 70 astronomical units away from the Sun, are subject to a constant
deceleration of 8 ⋅ 10−10 m/s2 , directed towards, the Sun since they passed the orbit of
Saturn. This deceleration is called the Pioneer anomaly. The origin was an intense subject
of research. Several investigations have shown that the reason of the deceleration is not a
Ref. 203 deviation from the inverse square dependence of gravitation, as was proposed by some.

is changed to 𝑤 + 𝛿𝑤. One then has

𝛿𝑤 = −𝜀2 𝑅 𝑢 𝑣 𝑤 + higher-order terms . (240)

More can be learned about the geodesic deviation by studying the behaviour of the famous south-pointing
Vol. I, page 244 carriage which we have encountered before. This device, used in China before the compass was discovered,
only works if the world is flat. Indeed, on a curved surface, after following a large closed path, it will show
Challenge 324 s a different direction than at the start of the trip. Can you explain why?
* The free-fall definition shows that the Riemann tensor is symmetric in certain indices and antisymmetric
Challenge 325 ny in others:
𝑅𝑎𝑏𝑐𝑑 = 𝑅𝑐𝑑𝑎𝑏 , 𝑅𝑎𝑏𝑐𝑑 = −𝑅𝑏𝑎𝑐𝑑 = −𝑅𝑎𝑏𝑑𝑐 . (242)
These relations also imply that many components vanish. Of importance also is the relation

𝑅𝑎𝑏𝑐𝑑 + 𝑅𝑎𝑑𝑏𝑐 + 𝑅𝑎𝑐𝑑𝑏 = 0 . (243)

Note that the order of the indices is not standardized in the literature. The list of invariants which can be
constructed from 𝑅 is long. We mention that 12 𝜀𝑎𝑏𝑐𝑑 𝑅𝑐𝑑 𝑒𝑓 𝑅𝑎𝑏𝑒𝑓 , namely the product ∗ 𝑅 𝑅 of the Riemann
tensor with its dual, is the invariant characterizing the Thirring–Lense effect.
from curvature to motion 209

The effect is electromagnetic.
There were many hints that pointed to an asymmetry in heat radiation emission of the
satellites. The on-board generators produce 2.5 kW of heat that is radiated away by the
satellite. A front-to-back asymmetry of only 80 W is sufficient to explain the measured
Ref. 204 anomaly. Recent research has shown that such an asymmetry indeed exists, so that the
issue is now resolved.
∗∗
Maximum power or force appearing on horizons is the basis for general relativity. Are
there physical systems other than space-time that can also be described in this way?
Page 36 For special relativity, we found that all its main effects – such as a limit speed, Lorentz
contraction or energy–mass equivalence – are also found for dislocations in solids. Do
systems analogous to general relativity exist? So far, attempts to find such systems have
only been partially successful.
Several equations and ideas of general relativity are applicable to deformations of
Ref. 118 solids, since general relativity describes the deformation of the space-time mattress.

Motion Mountain – The Adventure of Physics
Kröner has studied this analogy in great detail.
Other physical systems with ‘horizons’, and thus with observables analogous to
curvature, are found in certain liquids – where vortices play the role of black holes –
Ref. 205 and in certain quantum fluids for the propagation of light. Exploring such systems has
become a research topic in its own right.
A full analogy of general relativity in a macroscopic system was discovered only a few
Vol. VI, page 281 years ago. This analogy will be presented in the final part of our adventure.
∗∗

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Can the maximum force principle be used to eliminate competing theories of gravita-
tion? The most frequently discussed competitors to general relativity are scalar–tensor
theories of gravity, such as the proposal by Brans and Dicke and its generalizations.
Page 115 If a particular scalar-tensor theory obeys the general horizon equation (112) then it
must also imply a maximum force. The general horizon equation must be obeyed both
for static and for dynamic horizons. If that were the case, the specific scalar–tensor theory
would be equivalent to general relativity, because it would allow one, using the argument
of Jacobson, to deduce the usual field equations. This case can appear if the scalar field be-
haves like matter, i.e., if it has mass–energy like matter and curves space-time like matter.
On the other hand, if in the particular scalar–tensor theory the general horizon equation
is not obeyed for all moving horizons – which is the general case, as scalar–tensor the-
ories have more defining constants than general relativity – then the maximum force
does not appear and the theory is not equivalent to general relativity. This connection
also shows that an experimental test of the horizon equation for static horizons only is
not sufficient to confirm general relativity; such a test rules out only some, but not all,
scalar–tensor theories.
∗∗
One way to test general relativity would be to send three space probes through the solar
system, and measure their relative position over time, with high precision. This is best
done using frequency-stabilized lasers that send light from one satellite to the other two.
210 7 from curvature to motion

Can you summarize the main technical risks involved in such a project? Can you find
Challenge 328 s ways to reduce them?

A simple summary of the field equations
The field equations of general relativity describe motion of space, matter and energy.
They state that:
— The local curvature of space is given by the local energy density divided by the max-
imum force.
— Objects move along the geodesics defined by this local curvature.
This description is confirmed to full precision by all experiments performed so far.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Chapter 8

W H Y C A N W E SE E T H E STA R S ?
– MOT ION I N T H E U N I V E R SE

“
Zwei Dinge erfüllen das Gemüt mit immer
neuer und zunehmender Bewunderung und
Ehrfurcht, je öfter und anhaltender sich das
Nachdenken damit beschäftigt: der bestirnte
Himmel über mir und das moralische Gesetz in

”
mir.**

Motion Mountain – The Adventure of Physics
Immanuel Kant

O
n clear nights, between two and five thousand stars are visible with the naked eye.
f them, several hundred have names. Why? Because in all parts
f the world, the stars and the constellations they form are attached to myths.
Ref. 207 In all civilisations, myths are stories told to make the incomprehensible more com-
prehensible. But the simple fact that we can see the stars is the basis for a story much
more fantastic than all myths. It touches almost all aspects of modern physics and
encompasses the complete history of the universe.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Which stars d o we see?

“
Democritus says [about the Milky Way] that it
is a region of light emanating from numerous
stars small and near to each other, of which the

”
grouping produces the brightness of the whole.
Ref. 208 Aetius, Opinions.

The stars we see on a clear night are mainly the brightest of our nearest neighbours in
the surrounding region of the Milky Way. They lie at distances between four and a few
thousand light years from us. Roughly speaking, in our environment there is a star about
every 400 cubic light years. Our Sun is just one of the one hundred thousand million stars
of the Milky Way.
At night, almost all stars visible with the naked eye are from our own galaxy. The only
extragalactic object constantly visible to the naked eye in the northern hemisphere is the
so-called Andromeda nebula, shown enlarged in Figure 91. It is a whole galaxy like our
own, as Immanuel Kant had already conjectured in 1755. Several extragalactic objects are
** ‘Two things fill the mind with ever new and increasing admiration and awe, the more often and per-
Ref. 206 sistently thought considers them: the starred sky above me and the moral law inside me.’ Immanuel Kant
(1724–1804) was the most important philospher of the Enlightenment, the movement that lead to modern
science and western standard of wealth and living by pushing aside the false ideas spread by religion-based
governments.
212 8 why can we see the stars?

F I G U R E 87 A modern photograph of the visible night sky, showing a few thousand stars and the Milky

Motion Mountain – The Adventure of Physics
Way. The image is a digital composite of many photographs of cloudless night skies taken all over the
Earth. The Milky Way is positioned horizontally (© Axel Mellinger, from Ref. 209).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 88 A false colour image of how the night sky, and our galaxy in particular, looks in the near
infrared (courtesy NASA).

visible with the naked eye in the southern hemisphere: the Tarantula nebula, as well as
the large and the small Magellanic clouds. The Magellanic clouds are neighbour galaxies
to our own. Other, temporarily visible extragalactic objects are the rare novae, exploding
stars which can be seen if they appear in nearby galaxies, or the still rarer supernovae,
which can often be seen even in faraway galaxies.
In fact, the visible stars are special in other respects also. For example, telescopes show
that about half of them are in fact double: they consist of two stars circling around each
other, as in the case of Sirius. Measuring the orbits they follow around each other allows
Challenge 329 ny one to determine their masses. Can you explain how?
motion in the universe 213

Motion Mountain – The Adventure of Physics
F I G U R E 89 A false colour image of the X-ray sources observed in the night sky, for energies between 1
and 30 MeV (courtesy NASA).

F I G U R E 90 A false colour image, composed from infrared data, showing the large-scale structure of the
universe around us; the colour of each galaxy represents its distance and the numbers in parentheses
specify the red-shift; an infrared image of the Milky Way is superposed (courtesy Thomas
Jarret/IPAC/Caltech).

Vol. III, page 163 Many more extragalactic objects are visible with telescopes. Nowadays, this is one of
the main reasons to build them, and to build them as large as technically possible.
Is the universe different from our Milky Way? Yes, it is. There are several arguments to
demonstrate this. First of all, our galaxy – the word galaxy is just the original Greek term
for ‘Milky Way’ – is flattened, because of its rotation. If the galaxy rotates, there must be
other masses which determine the background with respect to which this rotation takes
214 8 why can we see the stars?

F I G U R E 91 The Andromeda nebula M31, one of
our neighbour galaxies (and the 31st member of
the Messier object listing) (NASA).

place. In fact, there is a huge number of other galaxies – about 1011 – in the universe, a
discovery dating only from the twentieth century. Some examples are shown in Figure 91,

Motion Mountain – The Adventure of Physics
Figure 92 and Figure 93. The last figure shows how galaxies usually ‘die’: by colliding with
other galaxies.
Why did our understanding of the place of our galaxy in the universe happen so late?
Well, people had the same difficulty as they had when trying to determine the shape of
the Earth. They had to understand that the galaxy is not only a milky strip seen on clear
nights, but an actual physical system, made of about 1011 stars gravitating around each
other.* Like the Earth, the Milky Way was found to have a three-dimensional shape: As
shown by the infrared photograph in Figure 88, our galaxy is a flat and circular structure,
with a spherical bulge at its centre. The diameter is 100 000 light years. It rotates about

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 330 ny once every 200 to 250 million years. (Can you guess how this is measured?) The rotation
is quite slow: since the Sun was formed, it has made only about 20 to 25 full turns around
the centre.
It is even possible to measure the mass of our galaxy.The trick is to use a binary pulsar
on its outskirts. If it is observed for many years, one can deduce its acceleration around
the galactic centre, as the pulsar reacts with a frequency shift which can be measured
on Earth. Many decades of observation are needed and many spurious effects have to
Ref. 210 be eliminated. Nevertheless, such measurements are ongoing. Present estimates put the
mass of our galaxy at 1042 kg or 5 ⋅ 1011 solar masses.

How d o we watch the stars?
The best images of the night sky are produced by the most sensitive telescopes. On
Earth, the most sensitive telescopes are the largest ones, such as those shown in Fig-
ure 96, located in Paranal in Chile. The history and the capabilities of these telescopes are
Ref. 211 fascinating. For many wavelengths that are absorbed by the atmosphere, the most sensit-
ive telescopes are satellite-bound, such as those shown in Figure 97. For each wavelength
domain, such modern systems produce fascinating images of the night sky. Figure 87 to

* The Milky Way, or galaxy in Greek, was said to have originated when Zeus, the main Greek god, tried
to let his son Heracles feed at Hera’s breast in order to make him immortal; the young Heracles, in a sign
showing his future strength, sucked so forcefully that the milk splashed all over the sky.
motion in the universe 215

Motion Mountain – The Adventure of Physics
F I G U R E 92 The elliptical galaxy NGC 205 (the 205th member of the New Galactic Catalogue) (NASA).

F I G U R E 93 The colliding galaxies M51 and M51B, 65 000 al across, 31 Mal away, show how a galaxy
‘dies’ (NASA).

Figure 90 give some examples. A beautiful website dedicated to showing how the night
sky looks at different wavelengths is www.chromoscope.net. The website allows you to
slide from one wavelength to another simply by moving a cursor; watching it and explor-
ing the beauty of the universe is worth it.
216 8 why can we see the stars?

Motion Mountain – The Adventure of Physics
F I G U R E 94 The universe is full of galaxies – this photograph shows the Perseus cluster (NASA).

F I G U R E 95 The universe contains many clouds; an example is this molecular cloud in Ophiuchus
(© ESO).

What d o we see at night?
Astrophysics leads to a strange conclusion about matter, quite different from how we are
used to thinking in classical physics: the matter observed in the sky is found in clouds.
motion in the universe 217

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 96 One of the four Very Large Telescopes (VLT) of the European Southern Observatory (ESO) in
Paranal in Chile, the most powerful telescopes in the world, each with a diameter of 8 m (© ESO).
218 8 why can we see the stars?

Motion Mountain – The Adventure of Physics
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F I G U R E 97 Top: the XMM-Newton satellite and its high-precision, onion-like mirrors that produced an
X-ray map of the night sky. Bottom: the Planck satellite and its golden-plated microwave antennas that
produced a high-resolution map of the cosmic background radiation (© ESA).
motion in the universe 219

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 98 Rotating clouds emitting jets along their axis; top row: a composite image (visible and
infrared) of the galaxy 0313-192, the galaxy 3C296, and the Vela pulsar; middle row: the star in
formation HH30, the star in formation DG Tauri B, and a black hole jet from the galaxy M87; bottom
row: the discovery of jets in our own galaxy (all NASA).

Clouds are systems in which the matter density diminishes with the distance from the
centre, with no sharp border and with no definite size. The object shown in Figure 95
Page 216 is a molecular cloud. But this is not the only case. Most astrophysical objects, including
planets and stars, are clouds.
The Earth is also a cloud, if we take its atmosphere, its magnetosphere and the dust
220 8 why can we see the stars?

ring around it as part of it. The Sun is a cloud. It is a gas ball to start with, but is even
more a cloud if we take into consideration its protuberances, its heliosphere, the solar
wind it generates and its magnetosphere. The solar system is a cloud if we consider its
comet cloud, its asteroid belt and its local interstellar gas cloud. The galaxy is a cloud if
we remember its matter distribution and the cloud of cosmic radiation it is surrounded
by. In fact, even people can be seen as clouds, as every person is surrounded by gases,
little dust particles from skin, vapour, etc.
Ref. 212 In the universe, almost all clouds are plasma clouds. A plasma is an ionized gas, such
as fire, lightning, the inside of neon tubes, or the Sun. At least 99.9 % of all matter in the
universe is in the form of plasma clouds. Only a very small percentage exists in solid or
liquid form, such as toasters, toothpicks or their users.
All clouds in the universe share a number of common properties. First, all clouds
seen in the universe – when undisturbed by collisions or other interactions from neigh-
bouring objects – are rotating. Most clouds are therefore flattened: they are in shape of
discs. Secondly, in many rotating clouds, matter is falling towards the centre: most clouds
are accretion discs. Finally, undisturbed accretion discs usually emit something along the

Motion Mountain – The Adventure of Physics
rotation axis: they possess jets. This basic cloud structure has been observed for young
stars, for pulsars, for galaxies, for quasars and for many other systems. Figure 98 gives
some examples. Finally, in 2010, jets have been found in our own galaxy, the Milky Way.
Challenge 331 r (Does the Sun have jets? So far, none has been detected.)
In summary, at night we see mostly rotating, flattened plasma clouds emitting jets
along their axes. But the night sky has many other phenomena. A large part of astronomy
Ref. 213 and astrophysics collects information about them. An overview about the observations
is given in Table 5.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
TA B L E 5 Some observations about the universe.

Aspect Main Va l u e
propertie s
Phenomena
Galaxy formation observed by Hubble several times
trigger event unknown
Galactic collisions momentum 1045 to 1047 kg m/s
Star formation cloud collapse forms stars between 0.04 and 130 solar
masses
frequency between 0 and 1000 solar masses per
year per galaxy; around 1 solar mass
per year in the Milky Way
or by star mergers up to 250 solar masses
Novae new luminous stars, 𝐿 < 1031 W
ejecting bubble 𝑅 ≈ 𝑡 ⋅ 𝑐/100
Supernovae new bright stars, 𝐿 < 1036 W
rate 1 to 5 per galaxy per 1000 a
Hypernovae optical bursts 𝐿 > 1037 W
motion in the universe 221

TA B L E 5 (Continued) Some observations about the universe.

Aspect Main Va l u e
propertie s
Gamma-ray bursts luminosity 𝐿 up to 1045 W, about 1 % of the whole
visible universe’s luminosity
energy c. 1046 J
duration c. 0.015 to 1000 s
observed number c. 2 per day
Radio sources radio emission 1033 to 1038 W
X-ray sources X-ray emission 1023 to 1034 W
Cosmic rays energy from 1 eV to 1022 eV
Gravitational lensing light bending angles down to 10−4 󸀠󸀠
Comets recurrence, typ. period 50 a, typ. visibility lifetime
evaporation 2 ka, typ. lifetime 100 ka
up to 4.57 ⋅ 109 a

Motion Mountain – The Adventure of Physics
Meteorites age
Components
Intergalactic space mass density c. 10−26 kg/m3
Quasars red-shift up to 𝑧 = 6
luminosity 𝐿 = 1040 W, about the same as one
galaxy
Galaxy superclusters number of galaxies c. 108 inside our horizon
Our own local supercluster number of galaxies about 4000
Galaxy groups size 100 Zm

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
number of galaxies between a dozen and 1000
Our local group number of galaxies 30
Galaxies size 0.5 to 2 Zm
number c. 1011 inside horizon
containing 10 to 400 globular clusters
containing typically 1011 stars each
containing typically one supermassive and several
intermediate-mass black holes
The Milky Way, our galaxy diameter 1.0(0.1) Zm
mass 1042 kg or 5 ⋅ 1011 solar masses Ref. 210
speed 600 km/s towards Hydra-Centaurus
containing about 30 000 pulsars Ref. 214
containing 100 globular clusters each with 1
million stars
Globular clusters (e.g. M15) containing thousands of stars, one
intermediate-mass black hole
age up to 12 Ga (oldest known objects)
Nebulae, clouds composition dust, oxygen, hydrogen
Our local interstellar cloud size 20 light years
222 8 why can we see the stars?

TA B L E 5 (Continued) Some observations about the universe.

Aspect Main Va l u e
propertie s
composition atomic hydrogen at 7500 K
Star systems types orbiting double stars, over 70 stars
orbited by brown dwarfs, several
planetary systems
Our solar system size 2 light years (Oort cloud)
speed 368 km/s from Aquarius towards Leo
Stars mass up to 130 solar masses (more when
stars merge) Ref. 215
giants and supergiants large size up to 1 Tm
main sequence stars
brown dwarfs low mass below 0.072 solar masses
low temperature below 2800 K Ref. 216

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L dwarfs low temperature 1200 to 2800 K
T dwarfs low temperature 900 to 1100 K
white dwarfs small radius 𝑟 ≈ 5000 km
high temperature cools from 100 000 to 5000 K
neutron stars nuclear mass density 𝜌 ≈ 1017 kg/m3
small size 𝑟 ≈ 10 km
emitters of X-ray bursts X-ray emission
pulsars periodic radio

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
emission
mass up to around 25 solar masses
magnetars high magnetic fields up to 1011 T and higher Ref. 217
some are gamma repeaters, others are anomalous X-ray pulsars
mass above 25 solar masses Ref. 218
Black holes horizon radius 𝑟 = 2𝐺𝑀/𝑐2 , observed mass range
from 3 solar masses to 1011 solar
masses
General properties
Cosmic horizon distance c. 1026 m = 100 Ym
Expansion Hubble’s constant 71(4) km s−1 Mpc−1 or 2.3(2) ⋅ 10−18 s−1
‘Age’ of the universe 13.8(1) Ga
Vacuum energy density 0.5 nJ/m3 or ΩΛ = 0.73 for 𝑘 = 0
no evidence for time-dependence
Large-scale shape space curvature 𝑘 ≈ ΩK = 0 Page 236
topology simple at all measured scales
Dimensions number 3 for space, 1 for time, at all measured
energies and scales
motion in the universe 223

TA B L E 5 (Continued) Some observations about the universe.

Aspect Main Va l u e
propertie s
Matter density 2 to 11 ⋅ 10−27 kg/m3 or 1 to 6
hydrogen atoms per cubic metre
ΩM = 0.25
Baryons density Ωb = 0.04, one sixth of the previous
(included in ΩM )
Dark matter density ΩDM = 0.21 (included in ΩM ),
unknown
Dark energy density ΩDM = 0.75, unknown
Photons number density 4 to 5 ⋅ 108 /m3
= 1.7 to 2.1 ⋅ 10−31 kg/m3
energy density ΩR = 4.6 ⋅ 10−5
Neutrinos energy density Ω𝜈 unknown

Motion Mountain – The Adventure of Physics
Average temperature photons 2.725(2) K
neutrinos not measured, predicted value is 2 K
Radiation perturbations photon anisotropy Δ𝑇/𝑇 = 1 ⋅ 10−5
density amplitude 𝐴 = 0.8(1)
spectral index 𝑛 = 0.97(3)
tensor-to-scalar ratio 𝑟 < 0.53 with 95 % confidence
Ionization optical depth 𝜏 = 0.15(7)
Decoupling 𝑧 = 1100

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
But while we are speaking of what we see in the sky, we need to clarify a general issue.

What is the universe?

“
I’m astounded by people who want to ‘know’
the universe when it’s hard enough to find your

”
way around Chinatown.
Woody Allen

The term ‘universe’ implies turning. The universe is what turns around us at night. For
a physicist, at least three definitions are possible for the term ‘universe’:
— The (observable or visible) universe is the totality of all observable mass and energy.
This includes everything inside the cosmological horizon. Since the horizon is mov-
ing away from us, the amount of observable mass and energy is constantly increas-
ing. The content of the term ‘observable universe’ is thus not fixed in time. (What
is the origin of this increase? We will come back to this issue in the final leg of our
Vol. VI, page 307 adventure.)
— The (believed) universe is the totality of all mass and energy, including any that is
not observable. Numerous books on general relativity state that there definitely exists
matter or energy beyond the observation boundaries. We will explain the origin of
Challenge 332 e this belief below. (Do you agree with it?)
224 8 why can we see the stars?

Motion Mountain – The Adventure of Physics
F I G U R E 99 The beauty of
astronomy: the Cygnus Bubble,
discovered in 2008, a nebula

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
expelled from a central star (false
colour image courtesy of T.A.
Rector, H. Schweiker).

— The (full) universe is the sum of matter and energy as well as space-time itself.
These definitions are often mixed up in physical and philosophical discussions. There is
no generally accepted consensus on the terms, so one has to be careful. In this text, when
we use the term ‘universe’, we imply the last definition only. We will discover repeatedly
that without clear distinction between the definitions we cannot complete our adventure.
(For example: Is the amount of matter and energy in the full universe the same as in the
Challenge 333 s observable universe?)
Note that the ‘size’ of the visible universe, or better, the distance to its horizon, is a
quantity which can be imagined. The value of 1026 m, or ten thousand million light years,
is not beyond imagination. If we took all the iron from the Earth’s core and made it into a
Challenge 334 s wire reaching to the edge of the observable universe, how thick would it be? The answer
might surprise you. Also, the content of the universe is clearly finite. There are about
as many visible galaxies in the universe as there are grains in a cubic metre of sand. To
expand on the comparison, can you deduce how much space you would need to contain
all the flour you would get if every little speck, with a typical size of 150 μm, represented
motion in the universe 225

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net

F I G U R E 100 An atlas of our cosmic environment: illustrations at scales up to 12.5, 50, 250, 5 000, 50 000,
500 000, 5 million, 100 million, 1 000 million and 14 000 million light years (© Richard Powell, www.
atlasoftheuniverse.com).
226 8 why can we see the stars?

Challenge 335 s one star?

The colour and the motion of the stars

“ ”
᾽Η τοι μὲν πρώτιστα Ξάος γένετ΄ ... *
Hesiod, Theogony.

Obviously, the universe is full of motion. To get to know the universe a bit, it is useful
to measure the speed and position of as many objects in it as possible. In the twenti-
eth century, a large number of such observations were obtained from stars and galaxies.
Challenge 336 s (Can you imagine how distance and velocity are determined?) This wealth of data can
be summed up in two points.
First of all, on large scales, i.e., averaged over about five hundred million light years,
the matter density in the universe is homogeneous and isotropic. Obviously, at smaller
scales inhomogeneities exist, such as galaxies or cheesecakes. Our galaxy for example is
Ref. 219 neither isotropic nor homogeneous. But at large scales the differences average out. This

Motion Mountain – The Adventure of Physics
large-scale homogeneity of matter distribution is often called the cosmological principle.
The second point about the universe is even more important. In the 1920s, independ-
Ref. 220 ently, Carl Wirtz, Knut Lundmark and Gustaf Stromberg showed that on the whole, all
galaxies move away from the Earth, and the more so, the more they were distant. There
are a few exceptions for nearby galaxies, such as the Andromeda nebula itself; but in
general, the speed of flight 𝑣 of an object increases with distance 𝑑. In 1929, the US-
American astronomer Edwin Hubble** published the first measurement of the relation
between speed and distance. Despite his use of incorrect length scales he found a relation

𝑣=𝐻𝑑, (245)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
where the proportionality constant 𝐻 is today called the Hubble constant. A modern
graph of the relation is given in Figure 101. The Hubble constant is known today to have
a value around 71 km s−1 Mpc−1 . (Hubble’s own value was so far from this value that it is
not cited any more.) For example, a star at a distance of 2 Mpc*** is moving away from
Earth with a speed of around 142 km/s, and proportionally more for stars further away.
In fact, the discovery by Wirtz, Lundmark and Stromberg implies that every galaxy
Challenge 337 s moves away from all the others. (Why?) In other words, the matter in the universe is ex-
panding. The scale of this expansion and the enormous dimensions involved are amazing.
The motion of all the thousand million galaxy groups in the sky is described by the single
equation (245)! Some deviations are observed for nearby galaxies, as mentioned above,
and for faraway galaxies, as we will see.

* ‘Verily, at first Chaos came to be ...’ The Theogony, attributed to the probably mythical Hesiodos, was
finalized around 700 bce. It can be read in English and Greek on the www.perseus.tufts.edu website. The
famous quotation here is from verse 117.
** Edwin Powell Hubble (1889–1953), important US-American astronomer. After being an athlete and tak-
ing a law degree, he returned to his childhood passion of the stars; he finally proved Immanuel Kant’s 1755
conjecture that the Andromeda nebula was a galaxy like our own. He thus showed that the Milky Way is
only a tiny part of the universe.
Page 307 *** A megaparsec or Mpc is a distance of 30.8 Zm.
motion in the universe 227

Type Ia Supernovae

0.0001 26
Supernova Cosmology Project
24
fainter High-Z Supernova Search
0.001
22 y
pt
Relative brightness

Calan/Tololo m
25 0e

density
0.01 Supernova Survey y
erg

mass
20
en
m
0.2 0.4 0.6 1.0 uu 1
18 v ac
0.1 24 ith
w y
erg
en
16 uum
1 vac
23 out
14 w ith
0.01 0.02 0.04 0.1
Magnitude

22 Accelerating

Motion Mountain – The Adventure of Physics
Universe

21 Decelerating
Universe

20
0.2 0.4 0.6 1.0
Redshift

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0.8 0.7 0.6 0.5
Scale of the Universe
relative to today's scale
F I G U R E 101 The relation between star distance and star velocity (courtesy Saul Perlmutter and the
Supernova Cosmology Project).

The cosmological principle and the expansion taken together imply that the universe
cannot have existed before time when it was of vanishing size; the universe thus has a
finite age. Together with the evolution equations, as explained in more detail below, the
Hubble constant points to an age value of around 13 800 million years. The expansion
also means that the universe has a horizon, i.e., a finite maximum distance for sources
whose signals can arrive on Earth. Signals from sources beyond the horizon cannot reach
us.
The motion of galaxies tells something important: in the past, the night sky, and thus
the universe, has been much smaller; matter has been much denser than it is now. It
Ref. 221 turns out that matter has also been much hotter. George Gamow* predicted in 1948 that
since hot objects radiate light, the sky cannot be completely black at night, but must

* George Gamow (b. 1904 Odessa, d. 1968 St. Boulder), physicist. He explained alpha decay as a tunnelling
effect and predicted the microwave background. He wrote the first successful popular physics texts, such as
1, 2, 3, infinity and the Mr. Thompkins series, which were later imitated by many other writers.
228 8 why can we see the stars?

F I G U R E 102 The measured spectrum of the
cosmic background radiation, with the error
bars multiplied by 500, compared to the
calculated Planck spectrum for 2.728 K (NASA).

be filled with black-body radiation emitted when it was ‘in heat’. That radiation, called

Motion Mountain – The Adventure of Physics
the background radiation, must have cooled down due to the expansion of the universe.
Challenge 338 ny (Can you confirm this?) Despite various similar predictions by other authors, including
Yakov Zel’dovich, in one of the most famous cases of missed scientific communication,
the radiation was found only much later, by two researchers completely unaware of all
Ref. 222 this work. A famous paper in 1964 by Doroshkevich and Novikov had even stated that
the antenna used by the (unaware) later discoverers was the best device to search for the
radiation! In any case, only in 1965 did Arno Penzias and Robert Wilson discover the
radiation. It was in one of the most beautiful discoveries of science, for which both later
Ref. 223 received the Nobel Prize in Physics. The radiation turns out to be described by the black-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
body radiation for a body with a temperature of 2.728(1) K, as illustrated in Figure 102.
In fact, the spectrum follows the black-body dependence to a precision better than 1 part
in 104 .
In summary, data show that the universe started with a hot big bang. But apart from
expansion and cooling, the past fourteen thousand million years have also produced a
few other memorable events.

Do stars shine every night?

“
Don’t the stars shine beautifully? I am the only

”
person in the world who knows why they do.
Friedrich (Fritz) Houtermans (1903–1966)

Stars seem to be there for ever. In fact, every now and then a new star appears in the
sky: a nova. The name is Latin and means ‘new’. Especially bright novae are called su-
pernovae. Novae and similar phenomena remind us that stars usually live much longer
than humans, but that like people, stars are born, shine and die.
It turns out that one can plot all stars on the so-called Hertzsprung–Russell diagram.
This diagram, central to every book on astronomy, is shown in Figure 103. It is a beautiful
example of a standard method used by astrophysicists: collecting statistics over many
examples of a type of object, one can deduce the life cycle of the object, even though their
lifetime is much longer than that of a human. For example, it is possible, by clever use of
motion in the universe 229

Motion Mountain – The Adventure of Physics
F I G U R E 103 The
Hertzsprung–Russell
diagram (© Richard
Powell).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the diagram, to estimate the age of stellar clusters, such as the M15 cluster of Figure 104,
and thus arrive at a minimum age of the universe. The result is around thirteen thousand
million years.
The finite lifetime of stars leads to restrictions on their visibility, especially for high
red-shifts. Indeed, modern telescope can look at places (and times) so far in the past that
they contained no stars yet. At those distances one only observes quasars; these light
sources are not stars, but much more massive and bright systems. Their precise structure
is still being studied by astrophysicists.
Since the stars shine, they were also formed somehow. Over millions of years, vast dust
clouds in space can contract, due to the influence of gravity, and form a dense, hot and
rotating structure: a new star. The fascinating details of their birth from dust clouds are
Ref. 224 a central part of astrophysics, but we will not explore them here. Stars differ in evolution
and lifetime. Above all, their evolution depends on their birth mass. Stars of the mass of
the Sun live 10 to 20 Ga and die as red giants. Stars with a mass that is 20 times that of
the Sun live only a few million years and die as supernovas. The most massive stars seem
to have about 130 solar masses. Exceptions are those stars that form through merging of
Ref. 225 several stars; they can be as massive as 250 solar masses.
Yet we do not have the full answer to our question. Why do stars shine at all? Clearly,
230 8 why can we see the stars?

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 104 The Messier 15 (M15) globular star cluster, with an age of thirteen thousand million years
(© ESA, NASA).

they shine because they are hot. They are hot because of nuclear reactions in their in-
Vol. V, page 209 terior. We will discuss these processes in more detail in a latter volume.

A short history of the universe

“ ”
Anima scintilla stellaris essentiae.*
Ref. 226 Heraclitus of Ephesus (c. 540 to c. 480 bce)

Not only stars are born, shine and die. Also galaxies do so. What about the universe as
a whole? The most important adventures that the matter and radiation around us have
Ref. 227 experienced are summarized in Table 6. The steps not yet discussed will be studied in

* ‘The soul is a spark of the substance of the stars.’
motion in the universe 231

the rest of our adventure. The history table is awe-inspiring. The sequence of events is
so beautiful and impressive that nowadays it is used in certain psychotherapies to point
out to people the story behind their existence, and to remind them of their own worth.
Enjoy.

TA B L E 6 A short history of the universe.

Time Time Event Te mpe r -
before from big at u r e
n o w𝑎 b a n g𝑏
c. 13.8 ⋅ 109 a ≈ 𝑡Pl 𝑏 Time, space, matter and initial conditions are 1032 K ≈ 𝑇Pl
indeterminate
9
13 ⋅ 10 a c. 1000 𝑡Pl Distinction of space-time from matter and radiation, 1030 K
−42
≈ 10 s initial conditions are determinate
10−35 s to Inflation & GUT epoch starts; strong and 5 ⋅ 1026 K
−32
10 s electroweak interactions diverge

Motion Mountain – The Adventure of Physics
−12
10 s Antiquarks annihilate; electromagnetic and weak 1015 K
interaction separate
2 ⋅ 10−6 s Quarks get confined into hadrons; universe is a 1013 K
plasma
Positrons annihilate
0.3 s Universe becomes transparent for neutrinos 1010 K
a few seconds Nucleosynthesis: D, 4 He, 3 He and 7 Li nuclei form; 109 K
radiation still dominates
2500 a Matter domination starts; density perturbations 75 000 K

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magnify
red-shift 380 000 a Recombination: during these latter stages of the big 3000 K
𝑧 = 1100 bang, H, He and Li atoms form, and the universe
becomes ‘transparent’ for light, as matter and
radiation decouple, i.e., as they acquire different
temperatures; the ‘night’ sky starts to get darker and
darker
Sky is almost black except for black-body radiation 𝑇𝛾 =
𝑇o𝛾 (1 + 𝑧)
𝑧 = 10 to 30 Galaxy formation
𝑧 = 9.6 Oldestobject seen so far
𝑧=5 Galaxy clusters form
𝑧=3 106 a First generation of stars (population II) is formed,
starting hydrogen fusion; helium fusion produces
carbon, silicon and oxygen
9
2 ⋅ 10 a First stars explode as supernovae𝑐 ; iron is produced
9
𝑧=1 3 ⋅ 10 a Second generation of stars (population I) appears,
and subsequent supernova explosions of the ageing
stars form the trace elements (Fe, Se, etc.) we are
made of and blow them into the galaxy
232 8 why can we see the stars?

TA B L E 6 (Continued) A short history of the universe.

Time Time Event Te mpe r -
before from big at u r e
n o w𝑎 b a n g𝑏
4.7 ⋅ 109 a Primitive cloud, made from such explosion
remnants, collapses; Sun forms
4.5 ⋅ 109 a Earth and other planet formation: Azoicum starts𝑑
4.5 ⋅ 109 a Moon forms from material ejected during the
collision of a large asteroid with the still-liquid Earth
4.3 ⋅ 109 a Craters form on the planets
4.0 ⋅ 109 a Archean eon (Archaeozoicum) starts: bombardment
from space stops; Earth’s crust solidifies; oldest
minerals form
3.8 ⋅ 109 a end of water collection and condensation
3.5 ⋅ 109 a Unicellular (microscopic) life appears; stromatolites

Motion Mountain – The Adventure of Physics
form
2.5 ⋅ 109 a Proterozoic eon (‘age of first life’) starts: atmosphere
becomes rich in oxygen thanks to the activity of
microorganisms Ref. 228
1.3 ⋅ 109 a Macroscopic, multicellular life appears, fungi
conquer land
800 ⋅ 106 a Earth is completely covered with ice for the first time
(reason still unknown) Ref. 229
600 to Earth is completely covered with ice for the last time
540 ⋅ 106 a

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
540(5) ⋅ 106 a Paleozoic era (Palaeozoicum, ‘age of old life’) starts,
after a gigantic ice age ends: animals appear, oldest
fossils (with 540(5) start of Cambrian, 495(5)
Ordovician, 440(5) Silurian, 417(5) Devonian, 354(5)
Carboniferous and 292(5) Permian periods)
480 − 450 ⋅ 106 a Land plants appear
400 − 370 ⋅ 106 a Wooden trees appear, flying insects appear
250(5) ⋅ 106 a Mesozoic era (Mesozoicum, ‘age of middle life’,
formerly called Secondary) starts: most insects and
other life forms are exterminated; mammals appear
(with 250(5) start of Triassic, 205(4) Jurassic and
142(3) Cretaceous periods)
150 ⋅ 106 a Continent Pangaea splits into Laurasia and
Gondwana
The star cluster of the Pleiades forms
150 ⋅ 106 a Birds appear
142(3) ⋅ 106 a Golden time of dinosaurs (Cretaceous) starts
100 ⋅ 106 a Start of formation of Alps, Andes and Rocky
Mountains
motion in the universe 233

TA B L E 6 (Continued) A short history of the universe.

Time Time Event Te mpe r -
before from big at u r e
n o w𝑎 b a n g𝑏
65.5 ⋅ 106 a Cenozoic era (Caenozoicum, ‘age of new life’) starts:
after an asteroid hits the Earth in the Yucatan,
dinosaurs become extinct, and grass and primates
appear, (with 65.5 start of Tertiary, consisting of
Paleogene period with Paleocene, 55.0 Eocene and
33.7 Oligocene epoch, and of Neogene period, with
23.8 Miocene and 5.32 Pliocene epoch; then 1.81
Quaternary period with Pleistocene (or Diluvium)
and 0.01 Holocene (or Alluvium) epoch)
50 ⋅ 106 a Large mammals appear
7(1) ⋅ 106 a Hominids appears
3 ⋅ 106 a

Motion Mountain – The Adventure of Physics
Supernova explodes, with following consequences:
more intense cosmic radiation, higher formation rate
of clouds, Earth cools down drastically, high
evolutionary pressure on the hominids and as a
result, Homo appears Ref. 230
500 000 a Formation of youngest stars in galaxy
500 000 a Homo sapiens appears
100 000 a Beginning of last ice age
90 000 a Homo sapiens sapiens appears
11 800 a End of last ice age, start of Holocene

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
6 000 a First written texts
2 500 a Physics starts
500 a Use of coffee, pencil and modern physics starts
200 a Electricity use begins
100 a Einstein publishes
10 to 120 a You were a unicellular being
Present c. 14 ⋅ 109 a You are reading this 𝑇o𝛾 = 2.73 K
𝑇o𝜈 ≈ 1.6 K
𝑇ob ≈ 0 K
Future You enjoy life; for details and reasons, see the following volumes.

𝑎. The time coordinate used here is the one given by the coordinate system defined by the microwave back-
ground radiation, as explained on page 237. A year is abbreviated ‘a’ (Latin ‘annus’). Errors in the last digits
are given between parentheses. Sometimes the red-shift 𝑧 is given instead of the time coordinate.
𝑏. This quantity is not exactly defined since the big bang is not a space-time event. This issue will be explored
Vol. VI, page 92 later on.
𝑐. The history of the atoms on Earth shows that we are made from the leftovers of a supernova. We truly are
made of stardust.
Vol. V, page 182 𝑑. Apart from the term Azoicum, all other names and dates from the geological time scale are those of the
International Commission on Stratigraphy; the dates are measured with the help of radioactive dating.
234 8 why can we see the stars?

Despite its length and its interest, the history table has its limitations: what happened
elsewhere in the last few thousand million years? There is still a story to be written of
which next to nothing is known. For obvious reasons, investigations have been rather
Earth-centred.
Discovering and understanding all phenomena observed in the skies is the aim of
astrophysics research. In our adventure we have to skip most of this fascinating topic,
because we want to focus on motion. Interestingly, general relativity allows us to explain
many of the general observations about motion across the universe in a simple manner.

The history of space-time

“
A number of rabbits run away from a central
point in various directions, all with the same
speed. While running, one rabbit turns its head,

”
Challenge 339 s and makes a startling observation. Which one?

The data showing that the universe is sprinkled with stars all over lead to a simple con-

Motion Mountain – The Adventure of Physics
Page 225
clusion: the universe cannot be static. Gravity always changes the distances between bod-
ies; the only exceptions are circular orbits. Gravity also changes the average distances
between bodies: gravity always tries to collapse clouds. The biggest cloud of all, the one
formed by all the matter in the universe, must therefore be changing: either it is col-
lapsing, or it is still expanding.
Ref. 231 The first to dare to draw this conclusion was Aleksander Friedmann.* In 1922 he de-
duced the possible evolutions of the universe in the case of homogeneous, isotropic mass
distribution. His calculation is a classic example of simple but powerful reasoning. For a
universe which is homogeneous and isotropic for every point, the line element of space-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 340 ny time is given by
d𝑠2 = 𝑐2 d𝑡2 − 𝑎2 (𝑡)(d𝑥2 + d𝑦2 + d𝑧2 ) . (246)

The quantity 𝑎(𝑡) is called the scale factor. The scale factor is often called, sloppily, the
‘radius’ or the ‘size’ of the universe. Matter is described by a density 𝜌M and a pressure
𝑝M. Inserting all this into the field equations, we get two equations that any school stu-
dent can grasp; they are

𝑎̇ 2 𝑘𝑐2 8π𝐺 Λ𝑐2
( ) + 2 = 𝜌M + (247)
𝑎 𝑎 3 3

* Aleksander Aleksandrowitsch Friedmann (1888–1925) was the first physicist who predicted the expansion
of the universe. Following his early death from typhus, his work remained almost unknown until Georges A.
Lemaître (b. 1894 Charleroi, d. 1966 Leuven), both priest and cosmologist, took it up and expanded it in
1927, focusing on solutions with an initial singularity. Lemaître was one of the propagators of the (erro-
neous!) idea that the big bang was an ‘event’ of ‘creation’ and convinced his whole religious organization
Page 248, page 248 of it. The Friedmann–Lemaître solutions are often erroneously called after two other physicists, who studied
them again much later, in 1935 and 1936, namely H.P. Robertson and A.G. Walker.
motion in the universe 235

and

𝑎̈ 𝑎̇ 2 𝑘𝑐2 8π𝐺
2 + ( ) + 2 = − 2 𝑝 + Λ𝑐2 . (248)
𝑎 𝑎 𝑎 𝑐

Together, they imply the two equations

4π𝐺 Λ𝑐2
𝑎̈ = − (𝜌M + 3𝑝M /𝑐2 ) 𝑎 + 𝑎 (249)
3 3

and

𝑎̇
̇ = −3 (𝜌M + 𝑝M /𝑐2 ) ,
𝜌M (250)
𝑎
where the dot indicates the derivative with respect to time. Equations (249) and (250)

Motion Mountain – The Adventure of Physics
depend on only three constants of nature: the gravitational constant 𝐺, related to the
maximum force or power in nature, the speed of light 𝑐, and the cosmological constant
Page 134 Λ, describing the energy density of the vacuum, or, if one prefers, the smallest force in
nature. Equation (249) expresses, in unusual form, the conservation of energy, i.e., the
Challenge 341 e first law of thermodynamics. Energy conservation is already implied in the definition of
the metric used by Friedmann. Equation (250) expresses that the cosmological constant
Λ accelerates the expansion 𝑎 ̇ and that matter, through gravity, decelerates the expansion
𝑎̇ of the universe.
Before we discuss the equations, first a few points of vocabulary. In the following,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the index 0 refers to the present time. At the present time 𝑡0 , the pressure of matter is
Challenge 342 e negligible. In this case, the expression 𝜌M 𝑎3 is constant in time. The present-time Hubble
parameter is defined by 𝐻0 = 𝑎0̇ /𝑎0 . It describes the expansion speed of the universe – if
you prefer, the rabbit speed in the puzzle above. It is customary to relate all mass densities
Challenge 343 ny to the so-called critical mass density 𝜌c given by

3𝐻02
𝜌c = ≈ (8 ± 2) ⋅ 10−27 kg/m3 (251)
8π𝐺
corresponding to about 8, give or take 2, hydrogen atoms per cubic metre. The actual
density of the universe is not far from this value. On Earth, we would call this value an
extremely good vacuum. Such are the differences between everyday life and the universe
as a whole. In any case, the critical density characterizes a matter distribution leading
to an evolution of the universe just between never-ending expansion and collapse. In
fact, this density is the critical one, leading to a so-called marginal evolution, only in the
case of vanishing cosmological constant. Despite this restriction, the term ‘critical mass
density’ is now used in all other cases as well. We can thus speak of a dimensionless mass
density ΩM defined as
ΩM = 𝜌0 /𝜌c . (252)
236 8 why can we see the stars?

no big
bang

2
experimental on
values pa nsi
ex
ed ion
1 l e rat ans
ce x p
ac it e
ted
Ω lim elera
dec ansion
eternal exp
0 limit
llapse
eventual co
clo t
fla en
se
op

du
niv

-1
un

too
er
ive

young
rse

Motion Mountain – The Adventure of Physics
0 1 2 3
ΩM
F I G U R E 105 The ranges for the Ω
parameters and their consequences.

The cosmological constant can also be related to this critical density by setting

𝜌Λ Λ𝑐2 Λ𝑐2
ΩΛ = = = . (253)

A third dimensionless parameter ΩK describes the curvature of space. It is defined in
terms of the present-day radius of the universe 𝑅0 and the curvature constant 𝑘 =
{1, −1, 0} as
−𝑘
ΩK = 2 2 (254)
𝑅0 𝐻0

and its sign is opposite to the one of the curvature 𝑘; ΩK vanishes for vanishing curvature.
Note that a positively curved universe, when homogeneous and isotropic, is necessarily
closed and of finite volume. A flat or negatively curved universe with the same matter
distribution can be open, i.e., of infinite volume, but does not need to be so. It could even
be simply or multiply connected. In these cases the topology is not completely fixed by
the curvature.
As already mentioned, the present-time Hubble parameter is defined by 𝐻0 = 𝑎0̇ /𝑎0 .
Challenge 344 ny From equation (247) we then get the central relation

ΩM + ΩΛ + ΩK = 1 . (255)

In the past, when data were lacking, cosmologists were divided into two camps: the claus-
trophobics believing that ΩK > 0 and the agoraphobics believing that ΩK < 0. More de-
motion in the universe 237

tails about the measured values of these parameters will be given shortly. The diagram
of Figure 105 shows the most interesting ranges of parameters together with the corres-
ponding behaviours of the universe. Modern measurements are consistent with a flat
universe, thus with ΩK = 0.
For the Hubble parameter, the most modern measurements give a value of

𝐻0 = 71 ± 4 km/sMpc = 2.3 ± 2 ⋅ 10−18 /s (256)

which corresponds to an age of the universe of 13.8 ± 1 thousand million years. In other
words, the age deduced from the history of space-time agrees with the age, given above,
deduced from the history of stars.
To get a feeling of how the universe evolves, it is customary to use the so-called decel-
eration parameter 𝑞0 . It is defined as

𝑎0̈ 1
𝑞0 = − 2
= ΩM − ΩΛ . (257)
𝑎0 𝐻0 2

Motion Mountain – The Adventure of Physics
The parameter 𝑞0 is positive if the expansion is slowing down, and negative if the expan-
sion is accelerating. These possibilities are also shown in the diagram of Figure 105.
An even clearer way to picture the expansion of the universe for vanishing pressure
is to rewrite equation (247) using 𝜏 = 𝑡 𝐻0 and 𝑥(𝜏) = 𝑎(𝑡)/𝑎(𝑡0 ), yielding

d𝑥 2
( ) + 𝑈(𝑥) = ΩK
d𝜏
where 𝑈(𝑥) = −ΩΛ 𝑥 − ΩΛ 𝑥2 . (258)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
This looks like the evolution equation for the motion of a particle with mass 1, with total
energy ΩK in a potential 𝑈(𝑥). The resulting evolutions are easily deduced.
For vanishing ΩΛ , the universe either expands for ever, or recollapses, depending on
the value of the mass–energy density. For non-vanishing, positive ΩΛ , the potential has
exactly one maximum; if the particle has enough energy to get over the maximum, it will
accelerate continuously. Data shows that this is the situation the universe seems to be in
today. Either case tells:

⊳ General relativity and the black night sky imply that the universe is expand-
ing.

In other words, the universe is not static. This was Friedmann’s daring conclusion. For a
certain time range, the resulting expansion is shown in Figure 106. We note that due to
its isotropic expansion, the universe has a preferred reference frame: the frame defined
by average matter. The time measured in that frame is the time listed in Table 6 and in
Figure 106, and it is time we assume when we talk about the age of the universe.

⊳ General relativity and the black night sky imply that the universe once
was extremely small and then expanded rapidly. The very early evolution
238 8 why can we see the stars?

Expansion history of the universe

ds
an r
expreve
Scale relative fo s

0.0001

0.001

0.01
0.1
1
brightness pse
a , 1.5 colla
relative
to
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scale
Scale 1.0 0
a

redshift
a(t) 0.5

ed
The expansion

rat
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0.5 ate 1
either... ler

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Quantum de
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effects
fir

0.0

Motion Mountain – The Adventure of Physics
t Planck Time t –20 –10 0 10
Time t , in Gigayears from present

F I G U R E 106 The evolution of the universe’s scale 𝑎 for different values of its mass density, as well as
the measured data (the graph on the right is courtesy of Saul Perlmutter and the Supernova Cosmology
Project).

is called the big bang.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
There are two points to be noted: first the set of possible evolution curves is described by
two parameters, not one. In addition, lines cannot be drawn down to zero size, but only
to very small sizes. There are two main reasons: we do not yet understand the behaviour
of matter at very high energy, and we do not understand the behaviour of space-time at
very high energy. We return to this important issue later on.
In summary, the main conclusion from Friedmann’s work is that a homogeneous and
isotropic universe is not static: it either expands or contracts. In either case, the universe
has a finite age. These profound ideas took many years to spread around the cosmology
community; even Einstein took a long time to get accustomed to them.
An overview of the possibilities for the long-time evolution is given in Figure 107.
The evolution can have various outcomes. In the early twentieth century, people decided
among them by personal preference. Albert Einstein first preferred the solution 𝑘 = 1
and Λ = 𝑎−2 = 4π𝐺𝜌M . It is the unstable solution found when 𝑥(𝜏) remains at the top of
the potential 𝑈(𝑥).
Willem de Sitter had found in 1917, much to Einstein’s personal dismay, that an empty
universe with 𝜌M = 𝑝M = 0 and 𝑘 = 1 is also possible. This type of universe expands for
Challenge 345 ny large times. The De Sitter universe shows that in special cases, matter is not needed for
space-time to exist!
Lemaître had found expanding universes for positive mass, and his results were also
contested by Einstein at first. When later the first measurements confirmed the calcula-
tions, the idea of a massive and expanding universe became popular. It then became the
motion in the universe 239

Λ>0 Λ=0 Λ<0

scale factor scale factor scale factor

k = –1

time t time t time t
confirmed by
modern
scale factor scale factor scale factor
data:

k=0

time t time t time t
Λ < Λc Λ = Λc Λ > Λc

Motion Mountain – The Adventure of Physics
scale factor scale factor scale factor scale factor scale factor

k = +1

time t time t time t time t time t

F I G U R E 107 The long-term evolution of the universe’s scale factor 𝑎 for various parameters.

concordance model in textbooks.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
In a sort of collective blindness that lasted from around 1950 to 1990, almost all cos-
mologists believed that Λ = 0.* Only towards the end of the twentieth century did exper-
imental progress allow cosmologists to make statements based on evidence rather than
beliefs or personal preferences, as we will find out shortly. But first of all we will settle an
old issue.

Why is the sky dark at night?

“
In der Nacht hat ein Mensch nur ein
Nachthemd an, und darunter kommt gleich der

”
Charakter.**
Rober Musil

First of all, the sky is not black at night – it is dark blue. Seen from the surface of the
Earth, it has the same blue colour as during the day, as any long-exposure photograph,
such as Figure 108, shows. The blue colour of the night sky, like the colour of the sky
during the day, is due to light from the stars that is scattered by the atmosphere. If we
want to know the real colour of the sky, we need to go above the atmosphere. There, to the

Challenge 346 ny * In this case, for ΩM ⩾ 1, the age of the universe follows 𝑡0 ⩽ 2/(3𝐻0 ), where the limits correspond. For
vanishing mass density we have 𝑡0 = 1/𝐻o .
** ‘At night, a person is dressed only with a nightgown, and directly under it there is the character.’ Robert
Musil (b. 1880 Klagenfurt, d. 1942 Geneva), writer.
240 8 why can we see the stars?

Motion Mountain – The Adventure of Physics
F I G U R E 108 All colours, such as the blue of the sky, are present also at night, as this long-time
exposure shows. On the top left, the bright object is Mars; the lower half shows a rare coloured fog bow
created by moonlight (© Wally Pacholka).

eye, the sky is pitch black. But precise measurements show that even the empty sky is not

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
completely black at night; it is filled with radiation of around 200 GHz; more precisely, it
is filled with radiation that corresponds to the thermal emission of a body at 2.73 K. This
cosmic background radiation is the thermal radiation left over from the big bang.
Ref. 232 Thus the universe is indeed colder than the stars. But why is this so? If the universe
were homogeneous on large scales and also infinitely large, it would have an infinite num-
ber of stars. Looking in any direction, we would see the surface of a star. The night sky
would be as bright as the surface of the Sun! Can you convince your grandmother about
Challenge 347 s this?
In a deep forest, we see a tree in every direction, as shown in Figure 109. Similarly,
in a ‘deep’ universe, we would see a star in every direction. Now, the average star has a
surface temperature of about 6000 K. If we lived in a deep and old universe, we would
effectively live inside an oven with a temperature of around 6000 K! Such a climate would
make it difficult to enjoy ice cream.
So why is the sky black at night, despite being filled with radiation from stars at 6000 K,
i.e., with white light? This paradox was most clearly formulated in 1823 by the astronomer
Wilhelm Olbers.* Because he extensively discussed the question, it is also called Olbers’

* Heinrich Wilhelm Matthäus Olbers (b. 1758 Arbergen, d. 1840 Bremen) was an important astronomer.
He discovered two planetoids, Pallas and Vesta, and five comets; he developed the method of calculating
parabolic orbits for comets which is still in use today. Olbers also actively supported the mathematician
Vol. I, page 150 and astronomer Friedrich Wilhelm Bessel in his career choice. The paradox is named after Olbers, though
motion in the universe 241

F I G U R E 109 Top: in a deep, or even inﬁnite forest, only trees are visible, and nothing behind them.
Bottom: at night, we can see the stars but also what is behind, namely the black sky. The universe is
thus of ﬁnite size. (© Aleks G, NASA/ESA)

paradox.
Today we know that two main effects explain the darkness of the night. First, since
the universe is finite in age, distant stars are shining for less time. We see them in a
younger stage or even during their formation, when they were darker. As a result, the
share of brightness of distant stars is smaller than that of nearby stars, so that the average

others had made similar points before, such as the Swiss astronomer Jean Philippe Loÿs de Cheseaux in
1744 and Johannes Kepler in 1610.
242 8 why can we see the stars?

temperature of the sky is reduced.* Today we know that even if all matter in the universe
were converted into radiation, the universe would still not be as bright as just calculated.
In other words, the power and lifetime of stars are much too low to produce the oven
Ref. 233 brightness just mentioned. Secondly, we can argue that the radiation of distant stars is
red-shifted and that the volume that the radiation must fill is increasing continuously, so
that the effective average temperature of the sky is also reduced.
Calculations are necessary to decide which reason for the darkness at night is the most
Ref. 234 important one. This issue has been studied in great detail by Paul Wesson; he explains
that the first effect, darkness due to a maximum finite star lifetime, is larger than the
second, darkness due to red-shift, by a factor of about three. However, both effects are
themselves due to the finite age of the universe. We may thus correctly state that the sky
is dark at night because the universe has a finite age.
Ref. 232 We note that the darkness of the sky arises only because the speed of light is finite.
Challenge 349 e Can you confirm this?
The darkness of the sky also tells us that the universe has a finite age that is large. In-
deed, the 2.7 K background radiation is that cold, despite having been emitted at 3000 K,

Motion Mountain – The Adventure of Physics
Ref. 235 because it is red-shifted, thanks to the Doppler effect. Under reasonable assumptions, the
temperature 𝑇 of this radiation changes with the scale factor 𝑎(𝑡) of the universe as

1
𝑇∼ . (259)
𝑎(𝑡)

In a young universe, we would thus not be able to see the stars, even if they existed.
From the brightness of the sky at night, measured to be about 3 ⋅ 10−13 times that of
an average star like the Sun, we can deduce something interesting: the density of stars in

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the universe must be much smaller than in our galaxy. The density of stars in the galaxy
can be deduced by counting the stars we see at night. But the average star density in
the galaxy would lead to much higher values for the night brightness if it were constant
Ref. 233 throughout the universe. We can thus deduce that the galaxy is much smaller than the
universe simply by measuring the brightness of the night sky and by counting the stars
Challenge 350 e in the sky. Can you make the explicit calculation?
In summary, the sky is black, or better, very dark at night because space-time and
matter are of finite, but old age. As a side issue, here is a quiz: is there an Olbers’ paradox
Challenge 351 ny also for gravitation?

The colour variations of the night sky
Not only is the night sky not black; the darkness of the night sky even depends on the
direction one is looking.
Since the Earth is moving when compared to the average stars, the dark colour of the
sky shows a Doppler shift. But even when this motion is compensated some colour vari-
ations remain. The variations are tiny, but they can be measured with special satellites.
The most precise results are those taken in 2013 by the European Planck satellite; they

* Can you explain that the sky is not black just because it is painted black or made of black chocolate? Or
more generally, that the sky is not made of and does not contain any dark and cold substance, as Olbers
Challenge 348 ny himself suggested, and as John Herschel refuted in 1848?
motion in the universe 243

Motion Mountain – The Adventure of Physics
-103 -102 -10 -1 0 1 10 102 103 104 105 106 107

Temperature fluctuations in μK

F I G U R E 110 A false colour image of the ﬂuctuations of the cosmic background radiation, after the
Doppler shift from our local motion and the signals from the Milky Way have been subtracted
(© Planck/ESA).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
are shown in Figure 110. These temperature variations are in the microkelvin range; they
show that the universe had already some inhomogeneities when the detected light was
emitted. Figure 110 thus gives an impression of the universe when it was barely 380 000
years ‘young’.
The data of Figure 110 is still being studied in great detail. It allows researchers to de-
duce the precise age of the universe – 13.8 Ga – its composition, and many other aspects.
These studies are still ongoing.

Is the universe open, closed or marginal?

“
– Doesn’t the vastness of the universe make you
feel small?
– I can feel small without any help from the

”
universe.
Anonymous

Sometimes the history of the universe is summed up in two words: bang!...crunch. But
will the universe indeed recollapse, or will it expand for ever? Or is it in an intermediate,
marginal situation? The parameters deciding its fate are the mass density and cosmolo-
gical constant.
The main news of the last decade of twentieth-century astrophysics are the experi-
244 8 why can we see the stars?

mental results allowing one to determine all these parameters. Several methods are being
used. The first method is obvious: determine the speed and distance of distant stars. For
large distances, this is difficult, since the stars are so faint. But it has now become possible
to search the sky for supernovae, the bright exploding stars, and to determine their dis-
tance from their brightness. This is presently being done with the help of computerized
Ref. 236 searches of the sky, using the largest available telescopes.
A second method is the measurement of the anisotropy of the cosmic microwave
background. From the observed power spectrum as a function of the angle, the curvature
of space-time can be deduced.
A third method is the determination of the mass density using the gravitational lens-
Page 252 ing effect for the light of distant quasars bent around galaxies or galaxy clusters.
A fourth method is the determination of the mass density using galaxy clusters. All
these measurements are expected to improve greatly in the years to come.
At present, these four completely independent sets of measurements provide the
Ref. 237 values
ΩM ≈ 0.3 , ΩΛ ≈ 0.7 , ΩK ≈ 0.0 (260)

Motion Mountain – The Adventure of Physics
where the errors are of the order of 0.1 or less. The values imply that

⊳ The universe is spatially flat, its expansion is accelerating and there will be
no big crunch.

However, no definite statement on the topology is possible. We will return to this last
Page 254 issue shortly.
In particular, the data show that the density of matter, including all dark matter, is

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
only about one third of the critical value.* Over two thirds are given by the cosmological
term. For the cosmological constant Λ the present measurements yield

3𝐻02
Λ = ΩΛ ≈ 10−52 /m2 . (261)
𝑐2
This value has important implications for quantum theory, since it corresponds to a va-
cuum energy density

Λ𝑐4 10−46 (GeV)4
𝜌Λ 𝑐2 = ≈ 0.5 nJ/m3 ≈ . (262)
8π𝐺 (ℏ𝑐)3

But the cosmological term also implies a negative vacuum pressure 𝑝Λ = −𝜌Λ 𝑐2 . In-
serting this result into the relation for the potential of universal gravity deduced from
Page 199 relativity
Δ𝜑 = 4π𝐺(𝜌 + 3𝑝/𝑐2 ) (263)

* The difference between the total matter density and the separately measurable baryonic matter density,
only about one sixth of the former value, is also not explained yet. It might even be that the universe contains
matter of a type unknown so far. We can say that the universe is not WYSIWYG; there is invisible, or dark
matter. This issue, the dark matter problem, is one of the important unsolved questions of cosmology.
motion in the universe 245

Ref. 238 we get
Δ𝜑 = 4π𝐺(𝜌M − 2𝜌Λ ) . (264)

Challenge 352 ny Thus the gravitational acceleration around a mass 𝑀 is

𝐺𝑀 Λ 2 𝐺𝑀
𝑎= 2
− 𝑐 𝑟 = 2 − ΩΛ 𝐻02 𝑟 , (265)
𝑟 3 𝑟
which shows that a positive vacuum energy indeed leads to a repulsive gravitational effect.
Inserting the mentioned value (261) for the cosmological constant Λ we find that the
repulsive effect is negligibly small even for the distance between the Earth and the Sun.
In fact, the order of magnitude of the repulsive effect is so much smaller than that of
attraction that one cannot hope for a direct experimental confirmation of this deviation
Challenge 353 ny from universal gravity at all. Probably astrophysical determinations will remain the only
possible ones. In particular, a positive gravitational constant manifests itself through a
positive component in the expansion rate.

Motion Mountain – The Adventure of Physics
But the situation is puzzling. The origin of the cosmological constant is not explained
by general relativity. This mystery will be solved only with the help of quantum theory. In
fact, the cosmological constant is the first and so far the only local and quantum aspect
of nature detected by astrophysical means.

Why is the universe transparent?
Could the universe be filled with water, which is transparent, as maintained by some
Ref. 239 popular books in order to explain rain? No. Even if the universe were filled with air, the
total mass would never have allowed the universe to reach the present size; it would have

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 354 ny recollapsed much earlier and we would not exist.
The universe is thus transparent because it is mostly empty. But why is it so empty?
First of all, in the times when the size of the universe was small, all antimatter annihilated
with the corresponding amount of matter. Only a tiny fraction of matter, which originally
was slightly more abundant than antimatter, was left over. This 10−9 fraction is the matter
Vol. V, page 255 we see now. As a consequence, there are 109 as many photons in the universe as electrons
or quarks.
In addition, 380 000 years after antimatter annihilation, all available nuclei and elec-
trons recombined, forming atoms, and their aggregates, like stars and people. No free
charges interacting with photons were lurking around any more, so that from that period
onwards light could travel through space as it does today, being affected only when it hits
a star or a dust particle or some other atom. The observation of this cosmic background
radiation shows that light can travel for over 13 000 million years without problems or
disturbance. Indeed, if we recall that the average density of the universe is 10−26 kg/m3
and that most of the matter is lumped by gravity in galaxies, we can imagine what an ex-
cellent vacuum lies in between. As a result, light can travel along large distances without
noticeable hindrance.
But why is the vacuum transparent? That is a deeper question. Vacuum is transpar-
ent because it contains no electric charges and no horizons: charges or horizons are in-
dispensable in order to absorb light. In fact, quantum theory shows that vacuum does
246 8 why can we see the stars?

Vol. V, page 122 contain so-called virtual charges. However, these virtual charges have no effects on the
transparency of vacuum.

The big bang and its consequences

“ ”
Μελέτη θανάτου. Learn to die.
Plato, Phaedo, 81a.

Above all, the hot big bang model, which is deduced from the colour of the stars and
Page 227 galaxies, states that about fourteen thousand million years ago the whole universe was
Vol. III, page 337 extremely small. This fact gave the big bang its name. The term was created (with a sar-
castic undertone) in 1950 by Fred Hoyle, who by the way never believed that it applies to
Ref. 240 nature. Nevertheless, the term caught on. Since the past smallness of the universe can-
not be checked directly, we need to look for other, verifiable consequences. The main
consequences are the following:
— All matter moves away from all other matter. This point was observed before the

Motion Mountain – The Adventure of Physics
model was proposed.
— The maximal age for any system in the universe is finite. Recently, it was found that
the maximal age is 13.8(1) Ga, around fourteen thousand million years.
— There is thermal background radiation. The observed temperature 𝑇𝛾 of about 2.7 K
was found independently of the big bang model; it agrees with deductions from the
maximal age value.
— The mass of the universe is made up of about 75 % hydrogen and 23 % helium. These
values agree with the expectations.
— For non-vanishing cosmological constant Λ, the expansion of the universe acceler-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ates. The acceleration has been observed, though its value cannot be predicted.
— For non-vanishing cosmological constant, universal gravity is slightly reduced. This
point has yet to be confirmed.
— There are background neutrinos with a temperature 𝑇𝜈 of about 2 K; the precise pre-
diction is 𝑇𝜈 /𝑇𝛾 ≈ (4/11)1/3 and that these neutrinos appeared about 0.3 s after the
big bang. This point has yet to be confirmed.
It must be stressed that these consequences confirm the hot big bang model, but that
historically, only the value of the background temperature was predicted from model.
The last two points, on the temperature of neutrinos and on the deviation from universal
gravity, are also true predictions, but they have not been confirmed yet. Technology will
probably not allow us to check these two predictions in the foreseeable future. On the
other hand, there is also no evidence against them.
Competing descriptions of the universe that avoid a hot early phase have not been too
Ref. 240 successful in matching observations. It could always be, however, that this might change
in the future.
In addition, mathematical arguments state that with matter distributions such as the
one observed in the universe, together with some rather weak general assumptions, there
is no way to avoid a period in the finite past in which the universe was extremely small
Ref. 241 and hot. Therefore it is worth having a closer look at the situation.
motion in the universe 247

Was the big bang a big bang?
First of all, was the big bang a kind of explosion? This description implies that some
material transforms internal energy into motion of its parts. However, there was no such
process in the early history of the universe. In fact, a better description is that space-
time is expanding, rather than matter moving apart. The mechanism and the origin of
the expansion is unknown at this point of our adventure. Because of the importance of
spatial expansion, the whole phenomenon cannot be called an explosion. And obviously
there neither was nor is any sound carrying medium in interstellar space, so that one
cannot speak of a ‘bang’ in any sense of the term.
Was the big bang big? About fourteen thousand million years ago, the visible universe
was rather small; much smaller than an atom. In summary, the big bang was neither big
nor a bang; but the rest is correct.

Was the big bang an event?

“
Quid faciebat deus, antequam faceret caelum et

Motion Mountain – The Adventure of Physics
”
terram? ...Non faciebat aliquid.*
Augustine of Hippo, Confessiones, XI, 12.

The big bang theory is a description of what happened in the whole of space-time. Despite
what is often written in careless newspaper articles, at every moment of the expansion
space has been of non-vanishing size: space was never a single point. People who pretend
it was are making ostensibly plausible, but false statements. The big bang theory is a
description of the expansion of space-time, not of its beginning. Following the motion of
matter back in time – even neglecting the issue of measurement errors – general relativity
can deduce the existence of an initial singularity only if point-like matter is assumed to

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
exist. However, this assumption is wrong. In addition, the effect of the non-linearities in
general relativity at situations of high energy densities is not even completely clarified
yet. Above all, the big bang occurred across the whole universe. (This is the reason that
researchers ponder ‘inflation’ to explain various aspects of the universe.) In short, the
big bang was no event.
Most importantly, quantum theory shows that the big bang was not a true singular-
ity, as no physical observable, neither density nor temperature, ever reaches an infinitely
Vol. VI, page 102 large (or infinitely small) value. Such values cannot exist in nature.** In any case, there is
a general agreement that arguments based on pure general relativity alone cannot make
correct statements about the big bang. Nevertheless, most statements in newspaper art-
icles are of this sort.

Was the big bang a beginning?

“
In the beginning there was nothing, which

”
exploded.
Terry Pratchett, Lords and Ladies.

* ‘What was god doing before he made heaven and earth? ...He didn’t do anything.’ Augustine of Hippo
(b. 354 Tagaste, d. 430 Hippo Regius) was an reactionary and influential theologian.
** Many physicists are still wary of making such strong statements on this point. The final part of our
Vol. VI, page 57 adventure gives the precise arguments leading to the conclusion.
248 8 why can we see the stars?

Asking what was before the big bang is like asking what is north of the North Pole. Just
as nothing is north of the North Pole, so nothing ‘was’ before the big bang. This analogy
could be misinterpreted to imply that the big bang took its start at a single point in time,
which of course is incorrect, as just explained. But the analogy is better than it looks: in
fact, there is no precise North Pole, since quantum theory shows that there is a funda-
mental indeterminacy as to its position. There is also a corresponding indeterminacy for
the big bang.
In fact, it does not take more than three lines to show with quantum theory that time
and space are not defined either at or near the big bang. We will give this simple argument
Vol. VI, page 65 in the first chapter of the final part of our adventure. The big bang therefore cannot be
called a ‘beginning’ of the universe. There never was a time when the scale factor 𝑎(𝑡) of
the universe was zero.
The conceptual mistake of stating that time and space exist from a ‘beginning’ on-
wards is frequently encountered. In fact, quantum theory shows that near the big bang,
events can neither be ordered nor even be defined. More bluntly, there is no beginning;

Motion Mountain – The Adventure of Physics
there has never been an initial event or singularity.
Obviously the concept of time is not defined ‘outside’ or ‘before’ the existence of
Ref. 242 the universe; this fact was already clear to thinkers over a thousand years ago. It is then
tempting to conclude that time must have started. But as we saw, that is a logical mistake
as well: first of all, there is no starting event, and secondly, time does not flow, as clarified
Vol. I, page 48 already in the beginning of our walk.
A similar mistake lies behind the idea that the universe had certain ‘initial condi-
Vol. I, page 237 tions.’ Initial conditions by definition make sense only for objects or fields, i.e., for entit-
ies which can be observed from the outside, i.e., for entities which have an environment.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The universe does not comply with this requirement; it thus cannot have initial condi-
tions. Nevertheless, many people still insist on thinking about this issue; interestingly,
Ref. 243 Stephen Hawking sold millions of copies of a book explaining that a description of the
universe without initial conditions is the most appealing, without mentioning anywhere
that there is no other possibility anyway.*
In summary, the big bang is not a beginning, nor does it imply one. We will uncover
Vol. VI, page 306 the correct way to think about it in the final part of our adventure.

Does the big bang imply creation?

“
[The general theory of relativity produces]

”
universal doubt about god and his creation.
A witch hunter

Creation, i.e., the appearance of something out of nothing, needs an existing concept of
Vol. III, page 330 space and time to make sense. The concept of ‘appearance’ makes no sense otherwise.
But whatever the description of the big bang, be it classical, as in this chapter, or quantum
mechanical, as in later ones, this condition is never fulfilled. Even in the present, clas-
sical description of the big bang, which gave rise to its name, there is no appearance of

* This statement will still provoke strong reactions among physicists; it will be discussed in more detail in
the section on quantum theory.
motion in the universe 249

F I G U R E 111 The transmittance of the atmosphere (NASA).

Motion Mountain – The Adventure of Physics
matter, nor of energy, nor of anything else. And this situation does not change in any
later, improved description, as time or space are never defined before the appearance of
matter.
In fact, all properties of a creation are missing: there is no ‘moment’ of creation, no
appearance from nothing, no possible choice of any ‘initial’ conditions out of some set
Vol. VI, page 148 of possibilities, and, as we will see in more detail in the last volume of this adventure, not
even any choice of particular physical ‘laws’ from any set of possibilities.
In summary, the big bang does not imply nor harbour a creation process. The big
bang was not an event, not a beginning and not a case of creation. It is impossible to

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 355 ny continue our adventure if we do not accept each of these three conclusions. To deny
them is to continue in the domain of beliefs and prejudices, thus effectively giving up on
the mountain ascent.

Why can we see the Sun?
First of all, the Sun is visible because air is transparent. It is not self-evident that air
is transparent; in fact it is transparent only to visible light and to a few selected other
frequencies. Infrared and ultraviolet radiation are mostly absorbed. The reasons lie in
the behaviour of the molecules the air consists of, namely mainly nitrogen, oxygen and a
few other transparent gases. Several moons and planets in the solar system have opaque
atmospheres: we are indeed lucky to be able to see the stars at all.
In fact, even air is not completely transparent; air molecules scatter light a little bit.
That is why the sky and distant mountains appear blue and sunsets red. However, our eyes
are not able to perceive this, and stars are invisible during daylight. At many wavelengths
far from the visible spectrum the atmosphere is even opaque, as Figure 111 shows. (It is
also opaque for all wavelengths shorter than 200 nm, up to gamma rays. On the long
wavelength range, it remains transparent up to wavelength of around 10 to 20 m, de-
pending on solar activity, when the extinction by the ionosphere sets in.)
Secondly, we can see the Sun because the Sun, like all hot bodies, emits light. We
Vol. III, page 239 describe the details of incandescence, as this effect is called, later on.
250 8 why can we see the stars?

F I G U R E 112 A hot red oven shows that at high
temperature, objects and their environment cannot be
distinguished from each other (© Wikimedia).

Motion Mountain – The Adventure of Physics
Thirdly, we can see the Sun because we and our environment and the Sun’s envir-
onment are colder than the Sun. In fact, incandescent bodies can be distinguished from
their background only if the background is colder. This is a consequence of the prop-
erties of incandescent light emission, usually called black-body radiation. The radiation
is material-independent, so that for an environment with the same temperature as the
body, nothing can be seen at all. Any oven, such as the shown in Figure 112 provides a
proof.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Finally, we can see the Sun because it is not a black hole. If it were, it would emit
(almost) no light.
Obviously, each of these conditions applies to stars as well. For example, we can only
see them because the night sky is black. But then, how to explain the multicoloured sky?

Why d o the colours of the stars differ?
Stars are visible because they emit visible light. We have encountered several important
effects which determine colours: the diverse temperatures among the stars, the Doppler
shift due to a relative speed with respect to the observer, and the gravitational red-shift.
Not all stars are good approximations to black bodies, so that the black-body radiation
Vol. III, page 148 law does not always accurately describe their colour. However, most stars are reasonable
approximations of black bodies. The temperature of a star depends mainly on its size,
Ref. 244 its mass, its composition and its age, as astrophysicists are happy to explain. Orion is a
good example of a coloured constellation: each star has a different colour. Long-exposure
Vol. I, page 87 photographs beautifully show this.
The basic colour determined by temperature is changed by two effects. The first, the
Challenge 356 ny Doppler red-shift 𝑧, depends on the speed 𝑣 between source and observer as

Δ𝜆 𝑓S 𝑐+𝑣
𝑧= = −1=√ −1. (266)
𝜆 𝑓O 𝑐−𝑣
motion in the universe 251

TA B L E 7 The colour of the stars.

C l as s Te mpe r - Example L o c at i o n Colour
at u r e
O 30 kK Mintaka δ Orionis blue-violet
O 31(10) kK Alnitak ζ Orionis blue-violet
B 22(6) kK Bellatrix γ Orionis blue
B 26 kK Saiph κ Orionis blue-white
B 12 kK Rigel β Orionis blue-white
B 25 kK Alnilam ε Orionis blue-white
B 17(5) kK Regulus α Leonis blue-white
A 9.9 kK Sirius α Canis Majoris blue-white
A 8.6 kK Megrez δ Ursae Majoris white
A 7.6(2) kK Altair α Aquilae yellow-white
F 7.4(7) kK Canopus α Carinae yellow-white

Motion Mountain – The Adventure of Physics
F 6.6 kK Procyon α Canis Minoris yellow-white
G 5.8 kK Sun ecliptic yellow
K 3.5(4) kK Aldebaran α Tauri orange
M 2.8(5) kK Betelgeuse α Orionis red
D <80 kK – – any

Note. White dwarfs, or class-D stars, are remnants of imploded stars, with a size of only a few tens of kilo-
metres. Not all are white; they can be yellow or red. They comprise 5 % of all stars. None is visible with the
naked eye. Temperature uncertainties in the last digit are given between parentheses.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The size of the stars is an independent variable and is sometimes added as roman numerals at the end of
the spectral type. (Sirius is an A1V star, Arcturus a K2III star.) Giants and supergiants exist in all classes
from O to M.
To accommodate brown dwarfs, two new star classes, L and T, have been proposed.

Such shifts play a significant role only for remote, and thus faint, stars visible through
the telescope. With the naked eye, Doppler shifts cannot be seen. But Doppler shifts can
make distant stars shine in the infrared instead of in the visible domain. Indeed, the
highest Doppler shifts observed for luminous objects are larger than 5.0, corresponding
Challenge 357 ny to a recessional speed of more than 94 % of the speed of light. In the universe, the red-
shift is related to the scale factor 𝑅(𝑡) by

𝑅(𝑡0 )
𝑧= −1. (267)
𝑅(𝑡emission)

Light at a red-shift of 5.0 was thus emitted when the universe was one sixth of its present
age.
The other colour-changing effect, the gravitational red-shift 𝑧g , depends on the matter
252 8 why can we see the stars?

density of the source and the light emission radius 𝑅; it is given by

Δ𝜆 𝑓S 1
𝑧g = = −1 = −1. (268)
𝜆 𝑓0 √1 − 2𝐺𝑀
𝑐2 𝑅

Challenge 358 e It is usually quite a bit smaller than the Doppler shift. Can you confirm this?
No other red-shift processes are known; moreover, such processes would contradict
Page 261 all the known properties of nature. But the colour issue leads to the next question.

Are there dark stars?
It could be that some stars are not seen because they are dark. This could be one ex-
planation for the large amount of dark matter seen in the recent measurements of the
background radiation. This issue is currently of great interest and hotly debated. It is
known that objects more massive than Jupiter but less massive than the Sun can exist in
states which emit hardly any light. Any star with a mass below 7.2 % of the mass of the

Motion Mountain – The Adventure of Physics
Sun cannot start fusion and is called a brown dwarf. It is unclear at present how many
such objects exist. Many of the so-called extrasolar ‘planets’ are probably brown dwarfs.
The issue is not yet settled.
Page 262 Another possibility for dark stars are black holes. These are discussed in detail below.

Are all stars different? – Gravitational lenses

“ ”
Per aspera ad astra.*

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Are we sure that at night, two stars are really different? The answer is no. Recently, it
was shown that two ‘stars’ were actually two images of the same object. This was found
by comparing the flicker of the two images. It was found that the flicker of one image
was exactly the same as the other, just shifted by 423 days. This result was found by the
Estonian astrophysicist Jaan Pelt and his research group while observing two images of
Ref. 245 quasars in the system Q0957+561.
The two images are the result of gravitational lensing, an effect illustrated in Figure 113.
Indeed, a large galaxy can be seen between the two images observed by Pelt, and much
nearer to the Earth than the star. This effect had been already considered by Einstein;
however he did not believe that it was observable. The real father of gravitational lensing
Ref. 246 is Fritz Zwicky, who predicted in 1937 that the effect would be quite common and easy to
observe, if lined-up galaxies instead of lined-up stars were considered, as indeed turned
out to be the case.
Interestingly, when the time delay is known, astronomers are able to determine the
Challenge 359 ny size of the universe from this observation. Can you imagine how?
If the two observed massive objects are lined up exactly behind each other, the more
distant one is seen as ring around the nearer one. Such rings have indeed been observed,
and the galaxy image around a central foreground galaxy at B1938+666, shown in Fig-
ure 114, is one of the most beautiful examples. In 2005, several cases of gravitational lens-
* ‘Through hardship to the stars.’ A famous Latin motto. Often incorrectly given as ‘per ardua ad astra’.
motion in the universe 253

Gravitational lensing Topological effect
first image first image

star star
Earth
galaxy
Earth

second image second image

F I G U R E 113 Two ways in which a single star can lead to several images.

Motion Mountain – The Adventure of Physics
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F I G U R E 114 The Zwicky–Einstein ring
B1938+666, seen in the radio spectrum (left) and
in the optical domain (right) (NASA).

ing by stars were also discovered. More interestingly, three events where one of the two
stars has a Earth-mass planet have also been observed. The coming years will surely lead
to many additional observations, helped by the sky observation programme in the south-
ern hemisphere that checks the brightness of about 100 million stars every night.
Generally speaking, images of nearby stars are truly unique, but for the distant stars
the problem is tricky. For single stars, the issue is not so important, seen overall. Reas-
suringly, only about 80 multiple star images have been identified so far. But when whole
galaxies are seen as several images at once (and several dozens are known so far) we
might start to get nervous. In the case of the galaxy cluster CL0024+1654, shown in Fig-
ure 115, seven thin, elongated, blue images of the same distant galaxy are seen around the
yellow, nearer, elliptical galaxies.
But multiple images can be created not only by gravitational lenses; the shape of the
universe could also play some tricks.
254 8 why can we see the stars?

F I G U R E 115 Multiple blue images of a galaxy
formed by the yellow cluster CL0024+1654
(NASA).

Motion Mountain – The Adventure of Physics
What is the shape of the universe?
A popular analogy for the expansion of the universe is the comparison to a rubber bal-
loon that increase in diameter by blowing air into it. The surface of the balloon is as-
sumed to correspond to the volume of the universe. The dots on the balloon correspond
to the galaxies; their distance continuously increases. The surface of the balloon is finite
and has no boundary. By analogy, this suggests that the volume of the universe has a fi-
nite volume, but no boundary. This analogy presupposes that the universe has the same

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
topology, the same ‘shape’ as that of a sphere with an additional dimension.
Ref. 247 But what is the experimental evidence for this analogy? Not much. Nothing definite
is known about the shape of the universe. It is extremely hard to determine it, simply
because of its sheer size. Experiments show that in the nearby region of the universe, say
within a few million light years, the topology is simply connected. But for large distances,
almost nothing is certain. Maybe research into gamma-ray bursts will tell us something
about the topology, as these bursts often originate from the dawn of time.* Maybe even
the study of fluctuations of the cosmic background radiation can tell us something. All
this research is still in its infancy.
Since little is known, we can ask about the range of possible answers. As just men-
tioned, in the concordance model of cosmology, there are three options. For 𝑘 = 0, com-
patible with experiments, the simplest topology of space is three-dimensional Euclidean
space ℝ3 . For 𝑘 = 1, space-time is usually assumed to be a product of linear time, with
the topology 𝑅 of the real line, and a sphere 𝑆3 for space. That is the simplest possible
shape, corresponding to a simply-connected universe. For 𝑘 = −1, the simplest option for
space is a hyperbolic manifold 𝐻3 .
Page 236 In addition, Figure 105 showed that depending on the value of the cosmological
constant, space could be finite and bounded, or infinite and unbounded. In most
Friedmann–Lemaître calculations, simple-connectedness is usually tacitly assumed,

* The story is told from the mathematical point of view by B ob Osserman, Poetry of the Universe, 1996.
motion in the universe 255

even though it is not at all required.
It could well be that space-time is multiply connected, like a higher-dimensional ver-
sion of a torus, as illustrated on the right-hand side of Figure 113. A torus still has 𝑘 = 0
everywhere, but a non-trivial global topology. For 𝑘 ≠ 0, space-time could also have even
more complex topologies.* If the topology is non-trivial, it could even be that the actual
number of galaxies is much smaller than the observed number. This situation would cor-
respond to a kaleidoscope, where a few beads produce a large number of images.
In fact, the range of possibilities is not limited to the simply and multiply connected
cases suggested by classical physics. If quantum effects are included, additional and much
Vol. VI, page 101 more complex options appear; they will be discussed in the last part of our walk.

What is behind the horizon?

“
If I arrived at the outermost edge of the heaven,
could I extend my hand or staff into what is
outside or not? It would be paradoxical not to

”
be able to extend it.

Motion Mountain – The Adventure of Physics
Archytas of Tarentum (428–347 bce)

“ ”
The universe is a big place; perhaps the biggest.
Kilgore Trout, Venus on the Half Shell.

The horizon of the night sky is a tricky entity. In fact, all cosmological models show that it
Ref. 249 moves rapidly away from us. A detailed investigation shows that for a matter-dominated
Challenge 360 ny universe the horizon moves away from us with a velocity

𝑣horizon = 3𝑐 . (269)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
A pretty result, isn’t it? Obviously, since the horizon does not transport any signal, this is
not a contradiction of relativity. Now, measurements of ΩK show that space is essentially
Page 243 flat. Thus we can ask: What is behind the horizon?
If the universe is open or marginal, the matter we see at night is predicted by naively
applied general relativity to be a – literally – infinitely small part of all matter existing.
Indeed, applying the field equations to an open or marginal universe implies that there
Challenge 361 s is an infinite amount of matter behind the horizon. Is such a statement testable?
In a closed universe, matter is still predicted to exist behind the horizon; however, in
Challenge 362 s this case it is only a finite amount. Is this statement testable?
In short, the concordance model of cosmology states that there is a lot of matter behind
the horizon. Like most cosmologists, we sweep the issue under the rug and take it up
only later in our walk. A precise description of the topic is provided by the hypothesis of
inflation.

* The Friedmann–Lemaître metric is also valid for any quotient of the just-mentioned simple topologies by
a group of isometries, leading to dihedral spaces and lens spaces in the case 𝑘 = 1, to tori in the case 𝑘 = 0,
Ref. 248 and to any hyperbolic manifold in the case 𝑘 = −1.
256 8 why can we see the stars?

Why are there stars all over the place? – Inflation
What were the initial conditions of matter? Matter was distributed in a constant density
over space expanding with great speed. How could this happen? The researcher who has
explored this question most thoroughly is Alan Guth. So far, we have based our studies
of the night sky, cosmology, on two observational principles: the isotropy and the ho-
mogeneity of the universe. In addition, the universe is (almost) flat. The conjecture of
inflation is an attempt to understand the origin of these observations.
Flatness at the present instant of time is strange: the flat state is an unstable solution of
the Friedmann equations. Since the universe is still flat after fourteen thousand million
years, it must have been even flatter near the big bang.
Ref. 250 Guth argued that the precise flatness, the homogeneity and the isotropy of the uni-
verse could follow if in the first second of its history, the universe had gone through a
short phase of exponential size increase, which he called inflation. This exponential size
increase, by a factor of about 1026 , would homogenize the universe. This extremely short
evolution would be driven by a still-unknown field, the inflaton field. Inflation also seems

Motion Mountain – The Adventure of Physics
to describe correctly the growth of inhomogeneities in the cosmic background radiation.
However, so far, inflation poses as many questions as it solves. Twenty years after his
initial proposal, Guth himself is sceptical on whether it is a conceptual step forward. The
final word on the issue has not been said yet.

Why are there so few stars? – The energy and entropy content
of the universe

“
Die Energie der Welt ist constant. Die Entropie

”
der Welt strebt einem Maximum zu.*
Rudolph Clausius

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The matter–energy density of the universe is near the critical one. Inflation, described in
the previous section, is the favourite explanation for this connection. This implies that
the actual number of stars is given by the behaviour of matter at extremely high temper-
atures, and by the energy density left over at lower temperature. The precise connection
is still the topic of intense research. But this issue also raises a question about the quota-
tion above. Was the creator of the term ‘entropy’, Rudolph Clausius, right when he made
this famous statement? Let us have a look at what general relativity has to say about all
this.
In general relativity, a total energy can indeed be defined, in contrast to localized en-
ergy, which cannot. The total energy of all matter and radiation is indeed a constant of
motion. It is given by the sum of the baryonic, luminous and neutrino parts:

𝑐2 𝑀0 𝑐2
𝐸 = 𝐸b + 𝐸𝛾 + 𝐸𝜈 ≈ + ... ≈ + ... . (270)
𝑇0 𝐺

This value is constant only when integrated over the whole universe, not when just the
inside of the horizon is taken.**

* ‘The energy of the universe is constant. Its entropy tends towards a maximum.’
** Except for the case when pressure can be neglected.
motion in the universe 257

Many people also add a gravitational energy term. If one tries to do so, one is obliged
to define it in such a way that it is exactly the negative of the previous term. This value
for the gravitational energy leads to the popular speculation that the total energy of the
universe might be zero. In other words, the number of stars could also be limited by this
relation.
However, the discussion of entropy puts a strong question mark behind all these seem-
ingly obvious statements. Many people have tried to give values for the entropy of the
Ref. 251 universe. Some have checked whether the relation

𝑘𝑐3 𝐴 𝑘𝐺
𝑆= = 4π𝑀2 , (271)
𝐺ℏ 4 ℏ𝑐

Challenge 363 ny which is correct for black holes, also applies to the universe. This assumes that all the
matter and all the radiation of the universe can be described by some average temperat-
ure. They argue that the entropy of the universe is surprisingly low, so that there must be
some ordering principle behind it. Others even speculate over where the entropy of the

Motion Mountain – The Adventure of Physics
universe comes from, and whether the horizon is the source for it.
But let us be careful. Clausius assumes, without the slightest doubt, that the universe is
a closed system, and thus deduces the statement quoted above. Let us check this assump-
tion. Entropy describes the maximum energy that can be extracted from a hot object.
After the discovery of the particle structure of matter, it became clear that entropy is also
given by the number of microstates that can make up a specific macrostate. But neither
definition makes any sense if applied to the universe as a whole. There is no way to ex-
tract energy from it, and no way to say how many microstates of the universe would look
like the macrostate.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
The basic reason is the impossibility of applying the concept of state to the universe.
Vol. I, page 27 We first defined the state as all those properties of a system which allow one to distin-
guish it from other systems with the same intrinsic properties, or which differ from one
observer to another. You might want to check for yourself that for the universe, such state
Challenge 364 s properties do not exist at all.
We can speak of the state of space-time and we can speak of the state of matter and
energy. But we cannot speak of the state of the universe, because the concept makes no
sense.If there is no state of the universe, there is no entropy for it. And neither is there
an energy value. This is in fact the only correct conclusion one can draw about the issue.

Why is mat ter lumped?
We are able to see the stars because the universe consists mainly of empty space, in other
words, because stars are small and far apart. But why is this the case? Cosmic expansion
was deduced and calculated using a homogeneous mass distribution. So why did matter
lump together?
It turns out that homogeneous mass distributions are unstable. If for any reason the
density fluctuates, regions of higher density will attract more matter than regions of lower
density. Gravitation will thus cause the denser regions to increase in density and the re-
gions of lower density to be depleted. Can you confirm the instability, simply by assuming
Challenge 365 ny a space filled with dust and 𝑎 = 𝐺𝑀/𝑟2 ? In summary, even a tiny quantum fluctuation
258 8 why can we see the stars?

in the mass density will lead, after a certain time, to lumped matter.
But how did the first inhomogeneities form? That is one of the big problems of mod-
ern physics and astrophysics, and there is no accepted answer yet. Several modern ex-
periments are measuring the variations of the cosmic background radiation spectrum
with angular position and with polarization; these results, which will be available in the
Ref. 252 coming years, might provide some information on the way to settle the issue.

Why are stars so small compared with the universe?
Given that the matter density is around the critical one, the size of stars, which contain
most of the matter, is a result of the interaction of the elementary particles composing
Page 282 them. Below we will show that general relativity (alone) cannot explain any size appear-
ing in nature. The discussion of this issue is a theme of quantum theory.

Are stars and galaxies moving apart or is the universe
expanding?

Motion Mountain – The Adventure of Physics
Can we distinguish between space expanding and galaxies moving apart? Yes, we can.
Challenge 366 ny Can you find an argument or devise an experiment to do so?
The expansion of the universe does not apply to the space on the Earth. The expan-
sion is calculated for a homogeneous and isotropic mass distribution. Matter is neither
homogeneous nor isotropic inside the galaxy; the approximation of the cosmological
principle is not valid down here. It has even been checked experimentally, by studying
Ref. 253 atomic spectra in various places in the solar system, that there is no Hubble expansion
taking place around us.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Is there more than one universe?
The existence of ‘several’ universes might be an option when we study the question
whether we see all the stars. But you can check that neither definition of ‘universe’ given
above, be it ‘all matter-energy’ or ‘all matter–energy and all space-time’, allows us to
Challenge 367 e speak of several universes.
There is no way to define a plural for universe: either the universe is everything, and
then it is unique, or it is not everything, and then it is not the universe. We will discover
Vol. IV, page 166 that also quantum theory does not change this conclusion, despite recurring reports to
the contrary.
Whoever speaks of many universes is talking gibberish.

Why are the stars fixed? – Arms, stars and Mach ’ s principle

“
Si les astres étaient immobiles, le temps et

”
l’espace n’existeraient plus.*
Maurice Maeterlink.

The two arms possessed by humans have played an important role in discussions about
motion, and especially in the development of relativity. Looking at the stars at night, we

* ‘If the stars were immobile, time and space would not exist any more.’ Maurice Maeterlink (1862–1949)
is a famous Belgian dramatist.
motion in the universe 259

can make a simple observation, if we keep our arms relaxed. Standing still, our arms hang
down. Then we turn rapidly. Our arms lift up. In fact they do so whenever we see the stars
turning. Some people have spent a large part of their lives studying this phenomenon.
Why?
Ref. 254 Stars and arms prove that motion is relative, not absolute.* This observation leads to
two possible formulations of what Einstein called Mach’s principle.
— Inertial frames are determined by the rest of the matter in the universe.
This idea is indeed realized in general relativity. No question about it.
— Inertia is due to the interaction with the rest of the universe.
This formulation is more controversial. Many interpret it as meaning that the mass of an
object depends on the distribution of mass in the rest of the universe. That would mean
that one needs to investigate whether mass is anisotropic when a large body is nearby.
Of course, this question has been studied experimentally; one simply needs to measure
whether a particle has the same mass values when accelerated in different directions.

Motion Mountain – The Adventure of Physics
Ref. 255 Unsurprisingly, to a high degree of precision, no such anisotropy has been found. Many
therefore conclude that Mach’s principle is wrong. Others conclude with some pain in
Ref. 256 their stomach that the whole topic is not yet settled.
But in fact it is easy to see that Mach cannot have meant a mass variation at all: one
then would also have to conclude that mass is distance-dependent, even in Galilean phys-
ics. But this is known to be false; nobody in his right mind has ever had any doubts about
Challenge 368 e it.
The whole debate is due to a misunderstanding of what is meant by ‘inertia’: one
can interpret it as inertial mass or as inertial motion (like the moving arms under the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
stars). There is no evidence that Mach believed either in anisotropic mass or in distance-
dependent mass; the whole discussion is an example people taking pride in not making
a mistake which is incorrectly imputed to another, supposedly more stupid, person.**
Obviously, inertial effects do depend on the distribution of mass in the rest of the
universe. Mach’s principle is correct. Mach made some blunders in his life (he is infam-
ous for opposing the idea of atoms until he died, against experimental evidence) but his
principle is not one of them. Unfortunately it is to be expected that the myth about the
Ref. 256 incorrectness of Mach’s principle will persist, like that of the derision of Columbus.
In fact, Mach’s principle is valuable. As an example, take our galaxy. Experiments
show that it is flattened and rotating. The Sun turns around its centre in about 250 million
years. Indeed, if the Sun did not turn around the galaxy’s centre, we would fall into it in
Page 211 about 20 million years. As mentioned above, from the shape of our galaxy we can draw
the powerful conclusion that there must be a lot of other matter, i.e., a lot of other stars
and galaxies in the universe.

* The original reasoning by Newton and many others used a bucket and the surface of the water in it; but
the arguments are the same.
** A famous example is often learned at school. It is regularly suggested that Columbus was derided be-
cause he thought the Earth to be spherical. But he was not derided at all for this reason; there were only
disagreements on the size of the Earth, and in fact it turned out that his critics were right, and that he was
wrong in his own, much too small, estimate of the radius.
260 8 why can we see the stars?

At rest in the universe
There is no preferred frame in special relativity, no absolute space. Is the same true in
the actual universe? No; there is a preferred frame. Indeed, in the standard big-bang
cosmology, the average galaxy is at rest. Even though we talk about the big bang, any
average galaxy can rightly maintain that it is at rest. Each one is in free fall. An even
better realization of this privileged frame of reference is provided by the background
radiation.
In other words, the night sky is black because we move with almost no speed through
background radiation. If the Earth had a large velocity relative to the background radi-
ation, the sky would be bright even at night, thanks to the Doppler effect for the back-
ground radiation. In other words, the night sky is dark in all directions because of our
slow motion against the background radiation.
This ‘slow’ motion has a speed of 368 km/s. (This is the value of the motion of the Sun;
there are variations due to addition of the motion of the Earth.) The speed value is large
in comparison to everyday life, but small compared to the speed of light. More detailed

Motion Mountain – The Adventure of Physics
studies do not change this conclusion. Even the motion of the Milky Way and that of the
local group against the cosmic background radiation is of the order of 600 km/s; that is
still much slower than the speed of light. The reasons why the galaxy and the solar system
move with these ‘low’ speeds across the universe have already been studied in our walk.
Challenge 369 e Can you give a summary?
By the way, is the term ‘universe’ correct? Does the universe rotate, as its name im-
plies? If by universe we mean the whole of experience, the question does not make sense,
because rotation is only defined for bodies, i.e., for parts of the universe. However, if by
Ref. 257 universe we only mean ‘all matter’, the answer can be determined by experiments. It
turns out that the rotation is extremely small, if there is any: measurements of the cos-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
mic background radiation show that in the lifetime of the universe, its matter cannot
have rotated by more than a hundredth of a millionth of a turn! In short, with a dose of
humour we can say that ‘universe’ is a misnomer.

Does light at tract light?
Another reason why we can see stars is that their light reaches us. But why are travelling
light rays not disturbed by each other’s gravitation? We know that light is energy and that
any energy attracts other energy through gravitation. In particular, light is electromag-
netic energy, and experiments have shown that all electromagnetic energy is subject to
gravitation. Could two light beams that are advancing with a small angle between them
converge, because of mutual gravitational attraction? That could have measurable and
possibly interesting effects on the light observed from distant stars.
The simplest way to explore the issue is to study the following question: Do parallel
light beams remain parallel? Interestingly, a precise calculation shows that mutual grav-
Ref. 258 itation does not alter the path of two parallel light beams, even though it does alter the
path of antiparallel light beams, i.e., parallel beams travelling in opposite directions. The
reason is that for parallel beams moving at light speed, the gravitomagnetic component
Challenge 370 ny exactly cancels the gravitoelectric component.
Since light does not attract light moving along, light is not disturbed by its own gravity
during the millions of years that it takes to reach us from distant stars. Light does not
motion in the universe 261

attract or disturb light moving alongside. So far, all known quantum-mechanical effects
also confirm this conclusion.

Does light decay?
In the section on quantum theory we will encounter experiments showing that light is
made of particles. It is plausible that these photons might decay into some other particle,
as yet unknown, or into lower-frequency photons. If that actually happened, we would
not be able to see distant stars.
Challenge 371 e But any decay would also mean that light would change its direction (why?) and thus
produce blurred images for remote objects. However, no blurring is observed. In addi-
tion, the Soviet physicist Matvey Bronshtein demonstrated in the 1930s that any light
Ref. 259 decay process would have a larger rate for smaller frequencies. When people checked
the shift of radio waves, in particular the famous 21 cm line, and compared it with the
shift of light from the same source, no difference was found for any of the galaxies tested.
People even checked that Sommerfeld’s fine-structure constant, which determines the

Motion Mountain – The Adventure of Physics
Ref. 260 colour of objects, does not change over time. Despite an erroneous claim in recent years,
no change could be detected over thousands of millions of years.
Of course, instead of decaying, light could also be hit by some hitherto unknown
Challenge 372 ny entity. But this possibility is excluded by the same arguments. These investigations also
show that there is no additional red-shift mechanism in nature apart from the Doppler
Page 252 and gravitational red-shifts.
The visibility of the stars at night has indeed shed light on numerous properties of
nature. We now continue our adventure with a more general issue, nearer to our quest
for the fundamentals of motion.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Summary on cosmolo gy
Asking what precisely we see at night leads to several awe-inspiring insights. First, the
universe is huge – but of finite size. Secondly, the universe is extremely old – but of finite
age. Thirdly, the universe is expanding.
If you ever have the chance to look through a big telescope, do so! It is wonderful.
Chapter 9

BL AC K HOL E S – FA L L I NG F OR EV E R

“ ”
Qui iacet in terra non habet unde cadat.**
Alanus de Insulis

Why explore black holes?

T
he most extreme gravitational phenomena in nature are black holes. They realize

Motion Mountain – The Adventure of Physics
he limit of length-to-mass ratios in nature. In other words, they produce
he highest force value possible in nature at their surface, the so-called hori-
zon. Black holes also produce the highest space-time curvature values for a given mass
value. In other terms, black holes are the most extreme general relativistic systems that
are found in nature. Due to their extreme properties, the study of black holes is also a
major stepping stone towards unification and the final description of motion.
Ref. 143 Black hole is shorthand for ‘gravitationally completely collapsed object’. Predicted
over two centuries ago, it was unclear for a long time whether or not they exist. Around
the year 2000, the available experimental data have now led most experts to conclude

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
that there is a black hole at the centre of almost all galaxies, including our own (see Fig-
Ref. 261 ure 116). Black holes are also suspected at the heart of quasars, of active galactic nuclei and
of gamma-ray bursters. In short, it seems that the evolution of galaxies is strongly tied to
the evolution of black holes. In addition, about a dozen smaller black holes have been
identified elsewhere in our galaxy. For these reasons, black holes, the most impressive,
Ref. 262 the most powerful and the most relativistic systems in nature, are a fascinating subject
of study.

Mass concentration and horizons
The escape velocity is the speed needed to launch an projectile in such a way that it never
falls back down. The escape velocity depends on the mass and the size of the planet from
which the launch takes place: the denser the planet is, the higher is the escape velocity.
What happens when a planet or star has an escape velocity that is larger than the speed of
light 𝑐? Such objects were first imagined by the British geologist John Michell in 1784, and
Ref. 263 independently by the French mathematician Pierre Laplace in 1795, long before general
relativity was developed. Michell and Laplace realized something fundamental: even if
an object with such a high escape velocity were a hot star, to a distant observer it would
appear to be completely black, as illustrated in Figure 117. The object would not allow
** ‘He who lies on the ground cannot fall down from it.’ The author’s original name is Alain de Lille (c. 1128
–1203).
black holes – falling forever 263

Motion Mountain – The Adventure of Physics
F I G U R E 116 A time-lapse ﬁlm, taken over a period of 16 years, of the orbits of the stars near the centre
of our Galaxy. The invisible central object is so massive and small that it is almost surely a black hole
(QuickTime ﬁlm © ESO).

any light to leave it; in addition, it would block all light coming from behind it. In 1967,
Ref. 143 John Wheeler* made the now standard term black hole, due to Anne Ewing, popular in
physics.
Challenge 373 e It only takes a few lines to show that light cannot escape from a body of mass 𝑀

2𝐺𝑀
𝑅S = (272)
𝑐2
called the Schwarzschild radius. The formula is valid both in universal gravity and in
general relativity, provided that in general relativity we take the radius as meaning the
circumference divided by 2π. Such a body realizes the limit value for length-to-mass
ratios in nature. For this and other reasons to be given shortly, we will call 𝑅S also the
size of the black hole of mass 𝑀. (But note that it is only half the diameter.) In principle,
it is possible to imagine an object with a smaller length-to-mass ratio; however, we will
discover that there is no way to observe an object smaller than the Schwarzschild radius,
just as an object moving faster than the speed of light cannot be observed. However, we
can observe black holes – the limit case – just as we can observe entities moving at the
speed of light.
When a test mass is made to shrink and to approach the critical radius 𝑅S , two
things happen. First, the local proper acceleration for (imaginary) point masses increases
* John Archibald Wheeler (1911–2008), US-American physicist, important expert on general relativity and
author of several excellent textbooks, among them the beautiful John A. Wheeler, A Journey into Grav-
ity and Spacetime, Scientific American Library & Freeman, 1990, in which he explains general relativity with
passion and in detail, but without any mathematics.
264 9 black holes – falling forever

Motion Mountain – The Adventure of Physics
F I G U R E 117 A simpliﬁed simulated image of how a black hole of ten solar masses, with Schwarzschild

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
radius of 30 km, seen from a constant distance of 600 km, will distort an image of the Milky Way in the
background. Note the Zwicky–Einstein ring formed at around twice the black hole radius and the thin
bright rim (image © Ute Kraus at www.tempolimit-lichtgeschwindigkeit.de).

without bound. For realistic objects of finite size, the black hole realizes the highest force
possible in nature. Something that falls into a black hole cannot be pulled back out. A
black hole thus swallows all matter that falls into it. It acts like a cosmic vacuum cleaner.
At the surface of a black hole, the red-shift factor for a distant observer also increases
without bound. The ratio between the two quantities is called the surface gravity of a
Challenge 374 ny black hole. It is given by
𝐺𝑀 𝑐4 𝑐2
𝑔surf = 2 = = . (273)
𝑅S 4𝐺𝑀 2𝑅S

A black hole thus does not allow any light to leave it.
A surface that realizes the force limit and an infinite red-shift makes it is impossible
to send light, matter, energy or signals of any kind to the outside world. A black hole
is thus surrounded by a horizon. We know that a horizon is a limit surface. In fact, a
horizon is a limit in two ways. First, a horizon is a limit to communication: nothing can
communicate across it. Secondly, a horizon is a surface of maximum force and power.
These properties are sufficient to answer all questions about the effects of horizons. For
black holes – falling forever 265

event horizon

black
hole

F I G U R E 118 The light cones in the
equatorial plane around a non-rotating
black hole, seen from above the plane.

Motion Mountain – The Adventure of Physics
Challenge 375 s example: What happens when a light beam is sent upwards from the horizon? And from
slightly above the horizon? Figure 118 provides some hints.
Black holes, regarded as astronomical objects, are thus different from planets. During
the formation of planets, matter lumps together; as soon as it cannot be compressed any
further, an equilibrium is reached, which determines the radius of the planet. That is the
same mechanism as when a stone is thrown towards the Earth: it stops falling when it
hits the ground. A ‘ground’ is formed whenever matter hits other matter. In the case of a

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
black hole, there is no ground; everything continues falling. That is why, in Russian, black
holes used to be called collapsars.
This continuous falling of a black hole takes place when the concentration of matter
is so high that it overcomes all those interactions which make matter impenetrable in
Ref. 264 daily life. In 1939, Robert Oppenheimer* and Hartland Snyder showed theoretically that
a black hole forms whenever a star of sufficient mass stops burning. When a star of suffi-
cient mass stops burning, the interactions that form the ‘floor’ disappear, and everything
continues falling without end.
A black hole is matter in permanent free fall. Nevertheless, its radius for an outside
observer remains constant! But that is not all. Furthermore, because of this permanent
free fall, black holes are the only state of matter in thermodynamic equilibrium! In a
sense, floors and all other every-day states of matter are metastable: these forms are not
as stable as black holes.

* Robert Oppenheimer (1904–1967), important US-American physicist. He can be called the father of the-
oretical physics in the USA. He worked on quantum theory and atomic physics. He then headed the team
that developed the nuclear bomb during the Second World War. He was also the most prominent (inno-
cent) victim of one of the greatest witch-hunts ever organized in his home country. See also the www.nap.
edu/readingroom/books/biomems/joppenheimer.html website.
266 9 black holes – falling forever

Black hole horizons as limit surfaces
The characterizing property of a black hole is thus its horizon. The first time we en-
Page 96 countered horizons was in special relativity, in the section on accelerated observers. The
horizons due to gravitation are similar in all their properties; the section on the max-
imum force and power gave a first impression. The only difference we have found is due
to the neglect of gravitation in special relativity. As a result, horizons in nature cannot
be planar, in contrast to what is suggested by the observations of the imagined point-like
observers assumed to exist in special relativity.
Both the maximum force principle and the field equations imply that the space-time
around a rotationally symmetric (thus non-rotating) and electrically neutral mass is de-
Page 146 scribed by
2𝐺𝑀 d𝑟2
d𝑖2 = (1 − ) d𝑡 2
− − 𝑟2 d𝜑2 /𝑐2 . (274)
𝑟𝑐2 1− 2 2𝐺𝑀
𝑟𝑐

This is the so-called Schwarzschild metric. As mentioned above, 𝑟 is the circumference

Motion Mountain – The Adventure of Physics
divided by 2π; 𝑡 is the time measured at infinity.
Let us now assume that the mass is strongly localized. We then find that no outside
observer will ever receive any signal emitted from a radius value 𝑟 = 2𝐺𝑀/𝑐2 or smaller.
We have a horizon at that distance, and the situation describes a black hole. Indeed, as the
proper time 𝑖 of an observer at radius 𝑟 is related to the time 𝑡 of an observer at infinity
through
2𝐺𝑀
d𝑖 = √1 − d𝑡 , (275)
𝑟𝑐2

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
we find that an observer at the horizon would have vanishing proper time. In other
words, at the horizon the red-shift is infinite. (More precisely, the surface of infinite
red-shift and the horizon coincide only for non-rotating black holes. For rotating black
holes, the two surfaces are distinct.) Everything happening at the horizon goes on in-
finitely slowly, as observed by a distant observer. In other words, for a distant observer
observing what is going on at the horizon itself, nothing at all ever happens.
In the same way that observers cannot reach the speed of light, observers cannot reach
a horizon. For a second observer, it can only happen that the first is moving almost as
fast as light; in the same way, for a second observer, it can only happen that the first
has almost reached the horizon. In addition, a traveller cannot feel how much he is near
the speed of light for another, and experiences light speed as unattainable; in the same
way, a traveller (into a large black hole) cannot feel how much he is near a horizon and
experiences the horizon as unattainable.
We cannot say what happens inside the horizon.* We can take this view to the extreme
and argue that the black hole metric is a type of vacuum metric. In this view, mass is a
quantity that is ‘built’ from vacuum.

* Of course, mathematicians do not care about physical arguments. Therefore, Martin Kruskal and George
Szekeres have defined coordinates for the inside of the black hole. However, these and similar coordinate
systems are unrealistic academic curiosities, as they contradict quantum theory. Coordinate systems for the
inside of a black hole horizon have the same status as coordinate systems behind the cosmological horizon:
they are belief systems that are not experimentally verifiable.
black holes – falling forever 267

black hole

impact
parameter

Motion Mountain – The Adventure of Physics
F I G U R E 119 Motions of massive objects around a non-rotating black hole – for different impact
parameters and initial velocities.

In general relativity, horizons of any kind are predicted to be black. Since light cannot
escape from them, classical horizons are completely dark surfaces. In fact, horizons are
the darkest entities imaginable: nothing in nature is darker. Nonetheless, we will discover

Orbits around black holes
Ref. 259 Since black holes curve space-time strongly, a body moving near a black hole behaves in
more complicated ways than predicted by universal gravity. In universal gravity, paths
are either ellipses, parabolas, or hyperbolas; all these are plane curves. It turns out that
paths lie in a plane only near non-rotating black holes.*
Around non-rotating black holes, also called Schwarzschild black holes, circular paths
Challenge 377 ny are impossible for radii less than 3𝑅S /2 (can you show why?) and are unstable to per-
turbations from there up to a radius of 3𝑅S . Only at larger radii are circular orbits stable.
Around black holes, there are no elliptic paths; the corresponding rosetta path is shown
in Figure 119. Such a path shows the famous periastron shift in all its glory.
Note that the potential around a black hole is not appreciably different from 1/𝑟 for
Challenge 378 e distances above about fifteen Schwarzschild radii. For a black hole of the mass of the

* For such paths, Kepler’s rule connecting the average distance and the time of orbit

𝐺𝑀𝑡3
= 𝑟3 (276)
(2π)2

Challenge 376 ny still holds, provided the proper time and the radius measured by a distant observer are used.
268 9 black holes – falling forever

limit orbit

black black
hole hole

the photon sphere the photon sphere

F I G U R E 120 Motions of light passing near a non-rotating black hole.

Motion Mountain – The Adventure of Physics
Sun, that would be 42 km from its centre; therefore, we would not be able to note any
difference for the path of the Earth around the Sun.
We have mentioned several times in our adventure that gravitation is characterized
by its tidal effects. Black holes show extreme properties in this respect. If a cloud of dust
falls into a black hole, the size of the cloud increases as it falls, until the cloud envelops
the whole horizon. In fact, the result is valid for any extended body. This property of
black holes will be of importance later on, when we will discuss the size of elementary
particles.
For falling bodies coming from infinity, the situation near black holes is even more in-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
teresting. Of course there are no hyperbolic paths, only trajectories similar to hyperbolas
for bodies passing far enough away. But for small, but not too small impact parameters,
a body will make a number of turns around the black hole, before leaving again. The
number of turns increases beyond all bounds with decreasing impact parameter, until a
value is reached at which the body is captured into an orbit at a radius of 2𝑅, as shown in
Figure 119. In other words, this orbit captures incoming bodies if they approach it below
a certain critical angle. For comparison, remember that in universal gravity, capture is
never possible. At still smaller impact parameters, the black hole swallows the incoming
mass. In both cases, capture and deflection, a body can make several turns around the
black hole, whereas in universal gravity it is impossible to make more than half a turn
around a body.
The most absurd-looking orbits, though, are those corresponding to the parabolic case
Challenge 379 ny of universal gravity. (These are of purely academic interest, as they occur with probability
zero.) In summary, relativity changes the motions due to gravity quite drastically.
Around rotating black holes, the orbits of point masses are even more complex than
those shown in Figure 119; for bound motion, for example, the ellipses do not stay in
one plane – thanks to the Thirring–Lense effect – leading to extremely involved orbits in
three dimensions filling the space around the black hole.
For light passing a black hole, the paths are equally interesting, as shown in Figure 120.
There are no qualitative differences with the case of rapid particles. For a non-rotating
black hole, the path obviously lies in a single plane. Of course, if light passes sufficiently
black holes – falling forever 269

nearby, it can be strongly bent, as well as captured. Again, light can also make one or
several turns around the black hole before leaving or being captured. The limit between
the two cases is the path in which light moves in a circle around a black hole, at 3𝑅/2.
If we were located on that orbit, we would see the back of our head by looking forward!
Challenge 380 ny However, this orbit is unstable. The surface containing all orbits inside the circular one
is called the photon sphere. The photon sphere thus divides paths leading to capture from
those leading to infinity. Note that there is no stable orbit for light around a black hole.
Challenge 381 ny Are there any rosetta paths for light around a black hole?
For light around a rotating black hole, paths are much more complex. Already in the
equatorial plane there are two possible circular light paths: a smaller one in the direction
Challenge 382 ny of the rotation, and a larger one in the opposite direction.
For charged black holes, the orbits for falling charged particles are even more com-
plex. The electrical field lines need to be taken into account. Several fascinating effects
appear which have no correspondence in usual electromagnetism, such as effects similar
to electrical versions of the Meissner effect. The behaviour of such orbits is still an active
area of research in general relativity.

Motion Mountain – The Adventure of Physics
Black holes have no hair
How is a black hole characterized? It turns out that all properties of black holes follow
from a few basic quantities characterizing them, namely their mass 𝑀, their angular mo-
mentum 𝐽, and their electric charge 𝑄.* All other properties – such as size, shape, colour,
magnetic field – are uniquely determined by these.** It is as though, to use Wheeler’s
colourful analogy, one could deduce every characteristic of a woman from her size, her
waist and her height. Physicists also say that black holes ‘have no hair,’ meaning that
(classical) black holes have no other degrees of freedom. This expression was also intro-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 267 duced by Wheeler.*** This fact was proved by Israel, Carter, Robinson and Mazur; they
showed that for a given mass, angular momentum and charge, there is only one possible
Ref. 268 black hole. (However, the uniqueness theorem is not valid any more if the black hole
carries nuclear quantum numbers, such as weak or strong charges.)
In other words, a black hole is independent of how it has formed, and of the materials
used when forming it. Black holes all have the same composition, or better, they have no
composition at all.
The mass 𝑀 of a black hole is not restricted by general relativity. It may be as small
as that of a microscopic particle and as large as many million solar masses. But for their
angular momentum 𝐽 and electric charge 𝑄, the situation is different. A rotating black

* The existence of three basic characteristics is reminiscent of particles. We will find out more about the
Vol. VI, page 148 relation between black holes and particles in the final part of our adventure.
** Mainly for marketing reasons, non-rotating and electrically neutral black holes are often called Schwarz-
Ref. 265 schild black holes; uncharged and rotating ones are often called Kerr black holes, after Roy Kerr, who
discovered the corresponding solution of Einstein’s field equations in 1963. Electrically charged but non-
rotating black holes are often called Reissner–Nordström black holes, after the German physicist Hans Re-
issner and the Finnish physicist Gunnar Nordström. The general case, charged and rotating, is sometimes
Ref. 266 named after Kerr and Newman.
Ref. 143 *** Wheeler claims that he was inspired by the difficulty of distinguishing between bald men; however, it is
not a secret that Feynman, Ruffini and others had a clear anatomical image in mind when they stated that
‘black holes, in contrast to their surroundings, have no hair.’
270 9 black holes – falling forever

rotation axis

event horizon

ergosphere

static limit

F I G U R E 121 The ergosphere of a rotating black hole.

hole has a maximum possible angular momentum and a maximum possible electric (and
magnetic) charge.* The limit on the angular momentum appears because its perimeter

Motion Mountain – The Adventure of Physics
Challenge 383 ny may not move faster than light. The electric charge is also limited. The two limits are not
independent: they are related by

𝐽 2 𝐺𝑄2 𝐺𝑀 2
( ) + ⩽ ( ) . (277)
𝑐𝑀 4π𝜀0 𝑐4 𝑐2

This follows from the limit on length-to-mass ratios at the basis of general relativity.
Challenge 384 ny Rotating black holes realizing the limit (277) are called extremal black holes. The limit
(277) implies that the horizon radius of a general black hole is given by

For example, for a black hole with the mass and half the angular momentum of the Sun,
namely 2 ⋅ 1030 kg and 0.45 ⋅ 1042 kg m2 /s, the charge limit is about 1.4 ⋅ 1020 C.
How does one distinguish rotating from non-rotating black holes? First of all by the
shape. Non-rotating black holes must be spherical (any non-sphericity is radiated away
Ref. 269 as gravitational waves) and rotating black holes have a slightly flattened shape, uniquely
determined by their angular momentum. Because of their rotation, their surface of in-
finite gravity or infinite red-shift, called the static limit, is different from their (outer)
horizon, as illustrated in Figure 121. The region in between is called the ergosphere; this
is a misnomer as it is not a sphere. (It is so called because, as we will see shortly, it can be
used to extract energy from the black hole.) The motion of bodies within the ergosphere
can be quite complex. It suffices to mention that rotating black holes drag any in-falling
body into an orbit around them; this is in contrast to non-rotating black holes, which
swallow in-falling bodies. In other words, rotating black holes are not really ‘holes’ at
all, but rather vortices.

Vol. III, page 55 * More about the conjectured magnetic charge later on. In black holes, it enters like an additional type of
charge into all expressions in which electric charge appears.
black holes – falling forever 271

The distinction between rotating and non-rotating black holes also appears in the ho-
rizon surface area. The (horizon) surface area 𝐴 of a non-rotating and uncharged black
Challenge 385 e hole is obviously related to its mass 𝑀 by

16π𝐺2 2
𝐴= 𝑀 . (279)
𝑐4
The relation between surface area and mass for a rotating and charged black hole is more
complex: it is given by

8π𝐺2 2 𝐽2 𝑐2 𝑄2
𝐴= 𝑀 (1 + √ 1 − − ) (280)
𝑐4 𝑀4 𝐺2 4π𝜀0 𝐺𝑀2

where 𝐽 is the angular momentum and 𝑄 the charge. In fact, the relation

8π𝐺

Motion Mountain – The Adventure of Physics
𝐴= 𝑀𝑟h (281)
𝑐2
is valid for all black holes. Obviously, in the case of an electrically charged black hole, the
rotation also produces a magnetic field around it. This is in contrast with non-rotating
black holes, which cannot have a magnetic field.

Black holes as energy sources
Can one extract energy from a black hole? Roger Penrose has discovered that this is
possible for rotating black holes. A rocket orbiting a rotating black hole in its ergosphere

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 270
could switch its engines on and would then get hurled into outer space at tremendous
velocity, much greater than what the engines could have produced by themselves. In fact,
the same effect is used by rockets on the Earth, and is the reason why all satellites orbit
the Earth in the same direction; it would require much more fuel to make them turn the
other way.*
The energy gained by the rocket would be lost by the black hole, which would thus
slow down and lose some mass; on the other hand, there is a mass increases due to the
exhaust gases falling into the black hole. This increase always is larger than, or at best
equal to, the loss due to rotation slowdown. The best one can do is to turn the engines on
exactly at the horizon; then the horizon area of the black hole stays constant, and only
its rotation is slowed down.**
As a result, for a neutral black hole rotating with its maximum possible angular mo-
mentum, 1 − 1/√2 = 29.3 % of its total energy can be extracted through the Penrose
Challenge 387 ny process. For black holes rotating more slowly, the percentage is obviously smaller.

* And it would be much more dangerous, since any small object would hit such an against-the-stream
Challenge 386 ny satellite at about 15.8 km/s, thus transforming the object into a dangerous projectile. In fact, any power
wanting to destroy satellites of the enemy would simply have to load a satellite with nuts or bolts, send it
into space the wrong way, and distribute the bolts into a cloud. It would make satellites impossible for many
decades to come.
** It is also possible to extract energy from rotational black holes through gravitational radiation.
272 9 black holes – falling forever

For charged black holes, such irreversible energy extraction processes are also pos-
Challenge 388 ny sible. Can you think of a way? Using expression (277), we find that up to 50 % of the
Challenge 389 ny mass of a non-rotating black hole can be due to its charge. In fact, in the quantum part
of our adventure we will encounter an energy extraction process which nature seems to
Vol. V, page 153 use quite frequently.
The Penrose process allows one to determine how angular momentum and charge
Ref. 271 increase the mass of a black hole. The result is the famous mass–energy relation

2 2
𝐸2
2 𝑄2 𝐽2 𝑐2 𝑄2 𝐽2 1
𝑀 = 4 = (𝑚irr + ) + = (𝑚irr + ) + (282)
𝑐 16π𝜀0 𝐺𝑚irr 4𝑚2irr 𝐺2 8π𝜀0 𝜌irr 2 2
𝜌irr 𝑐

which shows how the electrostatic and the rotational energy enter the mass of a black
hole. In the expression, 𝑚irr is the irreducible mass defined as
2
𝐴(𝑀, 𝑄 = 0, 𝐽 = 0) 𝑐4 𝑐2
𝑚2irr = = (𝜌 ) (283)

Motion Mountain – The Adventure of Physics
irr
16π 𝐺2 2𝐺

and 𝜌irr is the irreducible radius.
Detailed investigations show that there is no process which decreases the horizon area,
and thus the irreducible mass or radius, of the black hole. People have checked this in
all ways possible and imaginable. For example, when two black holes merge, the total
area increases. One calls processes which keep the area and energy of the black hole
constant reversible, and all others irreversible. In fact, the area of black holes behaves like
the entropy of a closed system: it never decreases. That the area in fact is an entropy was
first stated in 1970 by Jacob Bekenstein. He deduced that only when an entropy is ascribed

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Ref. 272
to a black hole, is it possible to understand where the entropy of all the material falling
into it is collected.
The black hole entropy is a function only of the mass, the angular momentum and
the charge of the black hole. You might want to confirm Bekenstein’s deduction that the
Challenge 390 ny entropy 𝑆 is proportional to the horizon area. Later it was found, using quantum theory,
that
𝐴 𝑘𝑐3 𝐴 𝑘
𝑆= = 2 . (284)
4 ℏ𝐺 4 𝑙Pl

This famous relation cannot be deduced without quantum theory, as the absolute value
of entropy, as for any other observable, is never fixed by classical physics alone. We will
Vol. V, page 154 discuss this expression later on in our adventure.
If black holes have an entropy, they also must have a temperature. If they have a tem-
perature, they must shine. Black holes thus cannot be black! This was proven by Stephen
Hawking in 1974 with extremely involved calculations. However, it could have been de-
duced in the 1930s, with a simple Gedanken experiment which we will present later on.
Vol. V, page 147 You might want to think about the issue, asking and investigating what strange con-
sequences would appear if black holes had no entropy. Black hole radiation is a further,
though tiny (quantum) mechanism for energy extraction, and is applicable even to non-
rotating, uncharged black holes. The interesting connections between black holes, ther-
black holes – falling forever 273

TA B L E 8 Types of black holes.

Black hole Mass Charge Angular E xpe ri m e n ta l
type momentum evidence
Supermassive black 105 to 1011 𝑚⊙ unknown unknown orbits of nearby stars,
holes light emission from
accretion
Intermediate black 50 to 105 𝑚⊙ unknown unknown X-ray emission of
holes accreting matter
Stellar black holes 1 to 50 𝑚⊙ unknown unknown X-ray emission from
double star companion
Primordial black below 1 𝑚⊙ unknown unknown undetected so far;
holes research ongoing
Micro black holes below 1 g n.a. n.a. none; appear only in
science fiction and in
the mind of cranks

Motion Mountain – The Adventure of Physics
Vol. V, page 147 modynamics, and quantum theory will be presented in the upcoming parts of our ad-
Challenge 391 ny venture. Can you imagine other mechanisms that make black holes shine?

Formation of and search for black holes
How might black holes form? At present, at least three possible mechanisms have been
distinguished; the question is still a hot subject of research. First of all, black holes could
Ref. 273 have formed during the early stages of the universe. These primordial black holes might

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
grow through accretion, i.e., through the swallowing of nearby matter and radiation, or
Vol. V, page 150 disappear through one of the mechanisms to be studied later on.
Of the observed black holes, the so-called supermassive black holes are found at the
centre of every galaxy studied so far. They have typical masses in the range from 106 to
109 solar masses and contain about 0.5 % of the mass of a galaxy. For example, the black
Ref. 261 hole at the centre of the Milky Way has about 2.6 million solar masses, while the central
black hole of the galaxy M87 has 6400 million solar masses. Supermassive black holes
seem to exist at the centre of almost all galaxies, and seem to be related to the formation
of galaxies themselves. Supermassive black holes are supposed to have formed through
the collapse of large dust clouds, and to have grown through subsequent accretion of
matter. The latest ideas imply that these black holes accrete a lot of matter in their early
stage; the matter falling in emits lots of radiation, which would explain the brightness
of quasars. Later on, the rate of accretion slows, and the less spectacular Seyfert galaxies
form. It may even be that the supermassive black hole at the centre of the galaxy triggers
the formation of stars. Still later, these supermassive black holes become almost dormant,
or quiescent, like the one at the centre of the Milky Way.
Ref. 274 On the other hand, black holes can form when old massive stars collapse. It is estim-
ated that when stars with at least three solar masses burn out their fuel, part of the matter
remaining will collapse into a black hole. Such stellar black holes have a mass between
one and a hundred solar masses; they can also continue growing through subsequent
accretion. This situation provided the first ever candidate for a black hole, Cygnus X-1,
274 9 black holes – falling forever

Ref. 261 which was discovered in 1971. Over a dozen stellar black holes of between 4 and 20 solar
masses are known to be scattered around our own galaxy; all have been discovered after
1971.
Recent measurements suggest also the existence of intermediate black holes, with typ-
ical masses around a thousand solar masses; the mechanisms and conditions for their
formation are still unknown. The first candidates were found in the year 2000. Astro-
nomers are also studying how large numbers of black holes in star clusters behave,
and how often they collide. Under certain circumstances, the two black holes merge.
Whatever the outcome, black hole collisions emit strong gravitational waves. In fact, this
Page 181 signal is being looked for at the gravitational wave detectors that are in operation around
the globe.
The search for black holes is a popular sport among astrophysicists. Conceptually, the
simplest way to search for them is to look for strong gravitational fields. But only double
stars allow one to measure gravitational fields directly, and the strongest ever measured
Ref. 275 is 30 % of the theoretical maximum value. Another obvious way is to look for strong
gravitational lenses, and try to get a mass-to-size ratio pointing to a black hole; however,

Motion Mountain – The Adventure of Physics
no black holes was found in this way yet. Still another method is to look at the dynamics
of stars near the centre of galaxies. Measuring their motion, one can deduce the mass
of the body they orbit. The most favoured method to search for black holes is to look
for extremely intense X-ray emission from point sources, using space-based satellites or
balloon-based detectors. If the distance to the object is known, its absolute brightness
can be deduced; if it is above a certain limit, it must be a black hole, since normal matter
cannot produce an unlimited amount of light. This method is being perfected with the
aim of directly observing of energy disappearing into a horizon. This disappearance may
Ref. 276 in fact have been observed recently.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Finally, there is the suspicion that small black holes might be found in the halos of
galaxies, and make up a substantial fraction of the so-called dark matter.
In summary, the list of discoveries about black holes is expected to expand dramatic-
ally in the coming years.

Singularities
Solving the equations of general relativity for various initial conditions, one finds that
a cloud of dust usually collapses to a singularity, i.e., to a point of infinite density. The
same conclusion appears when one follows the evolution of the universe backwards in
time. In fact, Roger Penrose and Stephen Hawking have proved several mathematical
theorems on the necessity of singularities for many classical matter distributions. These
theorems assume only the continuity of space-time and a few rather weak conditions on
Ref. 277 the matter in it. The theorems state that in expanding systems such as the universe itself,
or in collapsing systems such as black holes in formation, events with infinite matter
density should exist somewhere in the past, or in the future, respectively. This result is
usually summarized by saying that there is a mathematical proof that the universe started
in a singularity.
In fact, the derivation of the initial singularities makes a hidden, but strong assump-
tion about matter: that dust particles have no proper size, i.e., that they are point-like. In
other words, it is assumed that dust particles are singularities. Only with this assump-
black holes – falling forever 275

tion can one deduce the existence of initial or final singularities. However, we have seen
that the maximum force principle can be reformulated as a minimum size principle for
matter. The argument that there must have been an initial singularity of the universe is
thus flawed! The experimental situation is clear: there is overwhelming evidence for an
early state of the universe that was extremely hot and dense; but there is no evidence for
infinite temperature or density.
Mathematically inclined researchers distinguish two types of singularities: those with
a horizon – also called dressed singularities – and those without a horizon, the so-called
naked singularities. Naked singularities are especially strange: for example, a toothbrush
could fall into a naked singularity and disappear without leaving any trace. Since the
field equations are time invariant, we could thus expect that every now and then, na-
ked singularities emit toothbrushes. (Can you explain why dressed singularities are less
Challenge 392 ny dangerous?)
To avoid the spontaneous appearance of toothbrushes, over the years many people
have tried to discover some theoretical principles forbidding the existence of naked sin-
gularities. It turns out that there are two such principles. The first is the maximum force

Motion Mountain – The Adventure of Physics
or maximum power principle we encountered above. The maximum force implies that
no infinite force values appear in nature; in other words, there are no naked singularities
Ref. 278 in nature. This statement is often called cosmic censorship. Obviously, if general relativity
were not the correct description of nature, naked singularities could still appear. Cosmic
censorship is thus still discussed in research articles. The experimental search for naked
singularities has not yielded any success; in fact, there is not even a candidate observa-
tion for the – less abstruse – dressed singularities. But the theoretical case for ‘dressed’
singularities is also weak. Since there is no way to interact with anything behind a ho-
rizon, it is futile to discuss what happens there. There is no way to prove that behind a

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
horizon a singularity exists. Dressed singularities are articles of faith, not of physics.
In fact, there is another principle preventing singularities, namely quantum theory.
Whenever we encounter a prediction of an infinite value, we have extended our descrip-
tion of nature to a domain for which it was not conceived. To speak about singularities,
one must assume the applicability of pure general relativity to very small distances and
Vol. VI, page 102 very high energies. As will become clear in the last volume, nature does not allow this:
the combination of general relativity and quantum theory shows that it makes no sense
to talk about ‘singularities’, nor about what happens ‘inside’ a black hole horizon. The
Vol. VI, page 65 reason is that arbitrary small time and space values do not exist in nature.

Curiosities and fun challenges ab ou t black holes

“ ”
Tiens, les trous noirs. C’est troublant.*
Anonymous

Black holes have many counter-intuitive properties. We will first have a look at the clas-
Vol. V, page 156 sical effects, leaving the quantum effects for later on.
∗∗
Following universal gravity, light could climb upwards from the surface of a black hole
* No translation possible.
276 9 black holes – falling forever

observer dense
star

F I G U R E 122 Motion of some light rays from a
dense body to an observer.

and then fall back down. In general relativity, a black hole does not allow light to climb
Challenge 393 ny up at all; it can only fall. Can you confirm this?
∗∗
What happens to a person falling into a black hole? An outside observer gives a clear
answer: the falling person never arrives there since she needs an infinite time to reach the

Motion Mountain – The Adventure of Physics
Challenge 394 ny horizon. Can you confirm this result? The falling person, however, reaches the horizon
Challenge 395 ny in a finite amount of her own time. Can you calculate it?
This result is surprising, as it means that for an outside observer in a universe with
finite age, black holes cannot have formed yet! At best, we can only observe systems that
are busy forming black holes. In a sense, it might be correct to say that black holes do not
exist. Black holes could have existed right from the start in the fabric of space-time. On
the other hand, we will find out later why this is impossible. In short, it is important to
keep in mind that the idea of black hole is a limit concept but that usually, limit concepts
(like baths or temperature) are useful descriptions of nature. Independently of this last

𝐿 4𝐺
⩾ 2 . (285)
𝑀 𝑐
No exception has ever been observed.
∗∗
Interestingly, the size of a person falling into a black hole is experienced in vastly different
ways by the falling person and a person staying outside. If the black hole is large, the in-
falling observer feels almost nothing, as the tidal effects are small. The outside observer
makes a startling observation: he sees the falling person spread all over the horizon of
the black hole. In-falling, extended bodies cover the whole horizon. Can you explain this
Challenge 396 ny fact, for example by using the limit on length-to-mass ratios?
This strange result will be of importance later on in our exploration, and lead to im-
portant results about the size of point particles.
∗∗
An observer near a (non-rotating) black hole, or in fact near any object smaller than 7/4
times its gravitational radius, can even see the complete back side of the object, as shown
Challenge 397 ny in Figure 122. Can you imagine what the image looks like? Note that in addition to the
black holes – falling forever 277

paths shown in Figure 122, light can also turn several times around the black hole before
reaching the observer! Therefore, such an observer sees an infinite number of images of
the black hole. The resulting formula for the angular size of the innermost image was
Page 157 given above.
In fact, the effect of gravity means that it is possible to observe more than half the
surface of any spherical object. In everyday life, however, the effect is small: for example,
light bending allows us to see about 50.0002 % of the surface of the Sun.
∗∗
A mass point inside the smallest circular path of light around a black hole, at 3𝑅/2, can-
not stay in a circle, because in that region, something strange happens. A body which
circles another in everyday life always feels a tendency to be pushed outwards; this cent-
rifugal effect is due to the inertia of the body. But at values below 3𝑅/2, a circulating
body is pushed inwards by its inertia. There are several ways to explain this paradoxical
Ref. 279 effect. The simplest is to note that near a black hole, the weight increases faster than the
Challenge 398 ny centrifugal force, as you may want to check yourself. Only a rocket with engines switched

Motion Mountain – The Adventure of Physics
on and pushing towards the sky can orbit a black hole at 3𝑅/2.
∗∗
By the way, how can gravity, or an electrical field, come out of a black hole, if no signal
Challenge 399 s and no energy can leave it?
∗∗
Do white holes exist, i.e., time-inverted black holes, in which everything flows out of,
Challenge 400 ny instead of into, some bounded region?

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Challenge 401 ny Show that a cosmological constant Λ leads to the following metric for a black hole:

d𝑠2 2𝐺𝑀 Λ 2 d𝑟2 𝑟2 2
d𝜏2 = 2
= (1 − − 𝑟 ) d𝑡2 − Λ𝑐2 2
− d𝜑 . (286)
𝑐 𝑟𝑐2 3 𝑐2 − 2𝐺𝑀
− 𝑟 𝑐2
𝑟 3

Note that this metric does not turn into the Minkowski metric for large values of 𝑟.
However, in the case that Λ is small, the metric is almost flat for values of 𝑟 that satisfy
1/√Λ ≫ 𝑟 ≫ 2𝐺𝑀/𝑐2 .
As a result, the inverse square law is also modified:

𝐺𝑀 Λ𝑐2
𝑎=− + 𝑟. (287)
𝑟2 3
With the known values of the cosmological constant, the second term is negligible inside
the solar system.
∗∗
In quantum theory, the gyromagnetic ratio is an important quantity for any rotating
278 9 black holes – falling forever

Challenge 402 ny charged system. What is the gyromagnetic ratio for rotating black holes?
∗∗
A large black hole is, as the name implies, black. Still, it can be seen. If we were to travel
towards it in a spaceship, we would note that the black hole is surrounded by a bright
rim, like a thin halo, as shown in Figure 117. The ring at the radial distance of the photon
sphere is due to those photons which come from other luminous objects, then circle the
hole, and finally, after one or several turns, end up in our eye. Can you confirm this
Challenge 403 s result?
∗∗
Challenge 404 ny Do moving black holes Lorentz-contract? Black holes do shine a little bit. It is true that
the images they form are complex, as light can turn around them a few times before
reaching the observer. In addition, the observer has to be far away, so that the effects of
curvature are small. All these effects can be taken into account; nevertheless, the question
remains subtle. The reason is that the concept of Lorentz contraction makes no sense in

Motion Mountain – The Adventure of Physics
general relativity, as the comparison with the uncontracted situation is difficult to define
precisely.
∗∗
Are black holes made of space or of matter? Both answers are correct! Can you confirm
Challenge 405 s this?
∗∗
Power is energy change over time. General relativity limits power values to 𝑃 ⩽ 𝑐5 /4𝐺.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
In other words, no engine in nature can provide more than 0.92 ⋅ 1052 W or 1.2 ⋅ 1049
Challenge 406 e horsepower. Can you confirm that black holes support this limit?
∗∗
Black holes produce problems in the microscopic domain, where quantum theory holds,
Ref. 280 as was pointed out by Jürgen Ehlers. Quantum theory is built on point particles, and
point particles move on time-like world lines. But following general relativity, point
particles have a singularity inside their black hole horizon; and singularities always move
on space-like world lines. Microscopic black holes, in contrast to macroscopic black holes,
thus contradict quantum theory.

Summary on black holes
A black hole is matter in permanent free fall. Equivalently, a black hole is a strongly
curved type of space. Since black holes are defined through their horizon, they can be
seen either as limiting cases of matter systems or as limiting cases of curved empty space.
Black holes realize the maximum force. For a given mass value, black holes also real-
ize maximum density, maximum blackness and maximum entropy. Black holes deflect,
capture and emit matter and light in peculiar ways.
black holes – falling forever 279

A quiz – is the universe a black hole?
Could it be that we live inside a black hole? Both the universe and black holes have ho-
rizons. Interestingly, the horizon distance 𝑟0 of the universe is about

𝑟0 ≈ 3𝑐𝑡0 ≈ 4 ⋅ 1026 m (288)

and its matter content is about
4π 3 2𝐺𝑚0
𝑚0 ≈ 𝜌𝑟 whence = 72π𝐺𝜌0 𝑐𝑡30 = 6 ⋅ 1026 m (289)
3 o 0 𝑐2
for a density of 3 ⋅ 10−27 kg/m3 . Thus we have

2𝐺𝑚0
𝑟0 ≈ , (290)
𝑐2

Motion Mountain – The Adventure of Physics
which is similar to the black hole relation 𝑟S = 2𝐺𝑚/𝑐2 . Is this a coincidence? No, it is not:
all systems with high curvature more or less obey this relation. But are we nevertheless
Challenge 407 s falling into a large black hole? You can answer that question by yourself.

D OE S SPAC E DI F F E R F ROM T I M E ?

“ ”
Tempori parce.**
Seneca

T
ime is our master, says a frequently heard statement. Nobody says that of space.
ime and space are obviously different in everyday life. But what is

Motion Mountain – The Adventure of Physics
he difference between them in general relativity? Do we need them at all? These
questions are worth an exploration.
General relativity states that we live in a (pseudo-Riemannian) space-time of variable
curvature. The curvature is an observable and is related to the distribution and motion
of matter and energy. The precise relation is described by the field equations. However,
there is a fundamental problem.
The equations of general relativity are invariant under numerous transformations
which mix the coordinates 𝑥0 , 𝑥1 , 𝑥2 and 𝑥3 . For example, the viewpoint transforma-
tion

is allowed in general relativity, and leaves the field equations invariant. You might want
to search for other examples of transformations that follow from diffeomorphism invari-
Challenge 408 e ance.
Viewpoint transformations that mix space and time imply a consequence that is
clearly in sharp contrast with everyday life: diffeomorphism invariance makes it im-
possible to distinguish space from time inside general relativity. More explicitly, the co-
ordinate 𝑥0 cannot simply be identified with the physical time 𝑡, as we implicitly did up
to now. This identification is only possible in special relativity. In special relativity the
invariance under Lorentz (or Poincaré) transformations of space and time singles out
energy, linear momentum and angular momentum as the fundamental observables. In
general relativity, there is no (non-trivial) metric isometry group; consequently, there are
no basic physical observables singled out by their characteristic of being conserved. But

** ‘Care about time.’ Lucius Annaeus Seneca (c. 4 bce–65), Epistolae 14, 94, 28.
does space differ from time? 281

invariant quantities are necessary for communication! In fact, we can talk to each other
only because we live in an approximately flat space-time: if the angles of a triangle did
not add up to π (two right angles), there would be no invariant quantities and we would
not be able to communicate.
How have we managed to sweep this problem under the rug so far? We have done so
in several ways. The simplest way was to always require that in some part of the situation
under consideration space-time was our usual flat Minkowski space-time, where 𝑥0 can
be identified with 𝑡. We can fulfil this requirement either at infinity, as we did around
spherical masses, or in zeroth approximation, as we did for gravitational radiation and
for all other perturbation calculations. In this way, we eliminate the free mixing of co-
ordinates and the otherwise missing invariant quantities appear as expected. This prag-
matic approach is the usual way out of the problem. In fact, it is used in some otherwise
Ref. 235 excellent texts on general relativity that preclude any deeper questioning of the issue.
A common variation of this trick is to let the distinction between space and time
‘sneak’ into the calculations by the introduction of matter and its properties, or by the
introduction of radiation, or by the introduction of measurements. The material prop-

Motion Mountain – The Adventure of Physics
erties of matter, for example their thermodynamic state equations, always distinguish
between space and time. Radiation does the same, by its propagation. Obviously this is
true also for those special combinations of matter and radiation called clocks and metre
bars. Both matter and radiation distinguish between space and time simply by their pres-
ence.
In fact, if we look closely, the method of introducing matter to distinguish space and
time is the same as the method of introducing Minkowski space-time in some limit: all
properties of matter are defined using flat space-time descriptions.*
Another variation of the pragmatic approach is the use of the cosmological time co-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ordinate. An isotropic and homogeneous universe does have a preferred time coordinate,
Page 231 namely the one time coordinate that is used in all the tables on the past and the future
Vol. III, page 347 of the universe. This method is in fact a combination of the previous two.
But we are on a special quest here. We want to understand motion in principle, not
only to calculate it in practice. We want a fundamental answer, not a pragmatic one. And
for this we need to know how the positions 𝑥𝑖 and time 𝑡 are connected, and how we
can define invariant quantities. The question also prepares us for the task of combining
gravity with quantum theory, which is the aim of the final part of our adventure.
A fundamental solution to the problem requires a description of clocks together with
the system under consideration, and a deduction of how the reading 𝑡 of a clock relates to
the behaviour of the system in space-time. But we know that any description of a system
requires measurements: for example, in order to determine the initial conditions. And
initial conditions require space and time. We thus enter a vicious circle: that is precisely
what we wanted to avoid in the first place.
A suspicion arises. Is there in fact a fundamental difference between space and time?
Let us take a tour of various ways to investigate this question.

* We note something astonishing here: the inclusion of some condition at small distances (the description
Challenge 409 ny of matter) has the same effect as the inclusion of some condition at infinity (the asymptotic Minkowski
Vol. VI, page 113 space). Is this just coincidence? We will come back to this issue in the last part of our adventure.
282 10 does space differ from time?

C an space and time be measured?
In order to distinguish between space and time in general relativity, we must be able to
Vol. I, page 439 measure them. But already in the section on universal gravity we have mentioned the
impossibility of measuring lengths, times and masses with gravitational effects alone.
Does this situation change in general relativity? Lengths and times are connected by the
speed of light, and in addition lengths and masses are connected by the gravitational
constant. Despite this additional connection, it takes only a moment to convince oneself
that the problem persists.
In fact, we need electrodynamics and the granularity of matter to perform measure-
ments. In other words, we need the elementary charge 𝑒 in order to form length scales.
Ref. 281 The simplest one is
𝑒 √𝐺
𝑙em scale = 2
≈ 1.4 ⋅ 10−36 m . (292)
√4π𝜀0 𝑐

Vol. III, page 26 Here, 𝜀0 is the permittivity of free space. Alternatively, we can argue that quantum physics

Motion Mountain – The Adventure of Physics
provides a length scale, since we can use the quantum of action ℏ to define the length
scale
ℏ𝐺
𝑙qt scale = √ 3 ≈ 1.6 ⋅ 10−35 m , (293)
𝑐

which is called the Planck length or Planck’s natural length unit. However, this does not
change the argument, because we need electrodynamics to measure the value of ℏ.
The equivalence of the two arguments is shown by rewriting the elementary charge 𝑒
as a combination of nature’s fundamental constants:

Here, 𝛼 ≈ 1/137.06 is the fine-structure constant that characterizes the strength of elec-
tromagnetism. In terms of 𝛼, expression (292) becomes

𝛼ℏ𝐺
𝑙em scale = √ = √𝛼 𝑙qt scale . (295)
𝑐3
Summing up:

⊳ Every length measurement is based on the electromagnetic coupling con-
stant 𝛼 and on the Planck length.

Challenge 410 e Of course, the same is true for every time and every mass measurement. There is thus no
way to define or measure lengths, times and masses using gravitation or general relativity
only.*

Ref. 282 * In the past, John Wheeler used to state that his geometrodynamic clock, a device which measures time by
bouncing light back and forth between two parallel mirrors, was a counter-example. However, that is not
Challenge 411 s correct. Can you confirm this?
does space differ from time? 283

Given this sobering result, we can take the opposite point of view. We ask whether in
general relativity space and time are required at all.

Are space and time necessary?
Ref. 283 Robert Geroch answers this question in a beautiful five-page article. He explains how to
formulate the general theory of relativity without the use of space and time, by taking as
starting point the physical observables only.
Geroch starts with the set of all observables. Among them there is one, called 𝑣, which
stands out. It is the only observable which allows one to say that for any two observables
𝑎1 , 𝑎2 there is a third one 𝑎3 , for which

(𝑎3 − 𝑣) = (𝑎1 − 𝑣) + (𝑎2 − 𝑣) . (296)

Such an observable is called the vacuum. Geroch shows how to use such an observable to
construct derivatives of observables. Then he builds the so-called Einstein algebra, which

Motion Mountain – The Adventure of Physics
comprises the whole of general relativity.
Usually in general relativity, we describe motion in three steps: we deduce space-time
from matter observables, we calculate the evolution of space-time, and then we deduce
the motion of matter that follows from space-time evolution. Geroch’s description shows
that the middle step, and thus the use of space and time, is unnecessary.
Indirectly, the principle of maximum force makes the same statement. General relativ-
ity can be derived from the existence of limit values for force or power. Space and time
are only tools needed to translate this principle into consequences for real-life observers.
In short, it is possible to formulate general relativity without the use of space and
time. Since both are unnecessary, it seems unlikely that there should be a fundamental

Do closed time-like curves exist?
Is it possible that the time coordinate behaves, at least in some regions, like a torus? When
we walk, we can return to the point of departure. Is it possible, to come back in time to
where we have started? The question has been studied in great detail.
Ref. 241 The standard reference on closed time-like curves is the text by Hawking and Ellis;
they list the required properties of space-time, explaining which are mutually compat-
ible or exclusive. They find, for example, that space-times which are smooth, globally
hyperbolic, oriented and time-oriented do not contain any such curves. It is usually as-
sumed that the observed universe has these properties, so that the actual observation of
closed time-like curves is unlikely. Indeed, no candidate has ever been suggested – even
though it would be a scientific sensation. Later on, we will find that also searches for such
Vol. V, page 159 curves at the microscopic scale have also failed to find any example in nature.
In summary, there are no closed time-like curves in nature. The impossibility of such
curves seems to point to a difference between space and time. But in fact, this difference
is only apparent. These investigations are based on the behaviour of matter. Thus all argu-
ments assume a specific distinction between space and time right from the start. In short,
this line of enquiry cannot help us to decide whether space and time differ. Therefore we
look at the issue in another way.
284 10 does space differ from time?

hole

deformed
hole
𝑦 Mass

𝑥
F I G U R E 123 A ‘hole’ in
space in a schematic view.

Motion Mountain – The Adventure of Physics
Is general relativity lo cal? – The hole argument
When Albert Einstein developed general relativity, he had quite some trouble with diffeo-
morphism invariance. Most startling is his famous hole argument, better called the hole
paradox. Take the situation shown in Figure 123, in which a mass deforms the space-time
around it. Einstein imagined a small region of the vacuum, the hole, which is shown as
a small ellipse. What happens if we somehow change the curvature inside the hole while
Ref. 284 leaving the situation outside it unchanged, as shown in the inset of the picture?
On the one hand, the new situation is obviously physically different from the original

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
one, as the curvature inside the hole is different. This difference thus implies that the
curvature outside a region does not determine the curvature inside it. That is extremely
unsatisfactory. Worse, if we generalize this operation to the time domain, we seem to get
the biggest nightmare possible in physics: determinism is lost.
On the other hand, general relativity is diffeomorphism invariant. The deformation
shown in the figure is a diffeomorphism; so the new situation must be physically equi-
valent to the original situation.
Which argument is correct? Einstein first favoured the first point of view, and there-
fore dropped the whole idea of diffeomorphism invariance for about a year. Only later did
he understand that the second assessment is correct, and that the first argument makes a
fundamental mistake: it assumes an independent existence of the coordinate axes 𝑥 and
𝑦, as shown in the figure. But during the deformation of the hole, the coordinates 𝑥 and
𝑦 automatically change as well, so that there is no physical difference between the two
situations.
The moral of the story is that there is no difference between space-time and the gravit-
ational field. Space-time is a quality of the field, as Einstein put it, and not an entity with
a separate existence, as suggested by the graph. Coordinates have no physical meaning;
only distances (intervals) in space and time have one. In particular, diffeomorphism in-
variance proves that there is no flow of time. Time, like space, is only a relational entity:
time and space are relative; they are not absolute.
The relativity of space and time has practical consequences. For example, it turns out
does space differ from time? 285

that many problems in general relativity are equivalent to the Schwarzschild situation,
even though they appear completely different at first sight. As a result, researchers have
‘discovered’ the Schwarzschild solution (of course with different coordinate systems)
over twenty times, often thinking that they had found a new, unknown solution. We

Is the E arth hollow?

“
Any pair of shoes proves that we live on the
inside of a sphere. Their soles are worn out at

”
the ends, and hardly at all in between.
Anonymous

The hollow Earth hypothesis, i.e., the bizarre conjecture that we live on the inside of a
sphere, was popular in esoteric circles around the year 1900, and still remains so among
Vol. I, page 60 certain eccentrics today, especially in Britain, Germany and the US. They maintain, as
illustrated in Figure 124, that the solid Earth encloses the sky, together with the Moon, the
Sun and the stars. Most of us are fooled by education into another description, because
we are brought up to believe that light travels in straight lines. Get rid of this wrong belief,
they say, and the hollow Earth appears in all its glory.
Interestingly, the reasoning is partially correct. There is no way to disprove this sort of
Ref. 285 description of the universe. In fact, as the great physicist Roman Sexl used to explain, the
diffeomorphism invariance of general relativity even proclaims the equivalence between
the two views. The fun starts when either of the two camps wants to tell the other that only
its own description can be correct. You might check that any such argument is wrong;
it is fun to slip into the shoes of such an eccentric and to defend the hollow Earth hy-
286 10 does space differ from time?

Challenge 412 e pothesis against your friends. It is easy to explain the appearance of day and night, of
the horizon, and of the satellite images of the Earth. It is easy to explain what happened
during the flight to the Moon. You can drive many bad physicists crazy in this way! The
usual description and the hollow Earth description are exactly equivalent. Can you con-
firm that even quantum theory, with its introduction of length scales into nature, does
Challenge 413 s not change this situation?
In summary, diffeomorphism invariance is not an easy symmetry to swallow. But it is
best to get used to it now, as the rest of our adventure will throw up even more surprises.
Indeed, in the final part of our walk we will discover that there is an even larger sym-
metry of nature that is similar to the change in viewpoint from the hollow Earth view
to the standard view. This symmetry, space-time duality, is valid not only for distances
measured from the centre of the Earth, but for distances measured from any point in
Vol. VI, page 113 nature.

A summary: are space, time and mass independent?

Motion Mountain – The Adventure of Physics
We can conclude from this short discussion that there is no fundamental distinction
between space and time in general relativity. The only possible distinctions are the prag-
matic ones that make use of matter, radiation, or space-time at infinity.
Vol. I, page 438 In the beginning of our adventure we found that we needed matter to define space
and time. Now we have found that we even need matter to distinguish between space
and time. Similarly, in the beginning of our adventure we found that space and time are
Vol. I, page 203 required to define matter; now we have found that we even need flat space-time to define
matter. In these fundamental issues, general relativity has thus brought no improvement
over the results of Galilean physics.
In summary, general relativity does not provide a way out of the circular reasoning we

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
discovered in Galilean physics. Indeed, general relativity makes the issue even less clear
than before. Matter and radiation remain essential to define and distinguish space and
time, and space and time remain essential to define and distinguish matter and radiation.
Continuing our mountain ascent is the only way out.
In the next parts of our adventure, quantum physics will confirm that matter is needed
Vol. IV, page 167 to distinguish between space and time. No distinction between space and time without
matter is possible in principle. Then, in the last part of our adventure, we will discover
that mass and space are on an equal footing in nature. Because either is defined with
Vol. VI, page 80 the other, we will deduce that particles and vacuum are made of the same substance. It
will turn out that distinctions between space and time are possible only at low, everyday
energies; but no such distinction exists in principle.
C h a p t e r 11

G E N E R A L R E L AT I V I T Y I N A
N U T SH E L L – A SUM M A RY F OR T H E
L AYM A N

“ ”
Sapientia felicitas.**
Antiquity

G
eneral relativity is the final, correct description of macroscopic motion.
eneral relativity describes, first of all, all macroscopic motion due to

Motion Mountain – The Adventure of Physics
ravity, and in particular, describes how the observations of motion of any two
observers are related to each other. Above all, general relativity describes the most rapid,
the most powerful, the most violent and the most distant motions. For this reason, gen-
eral relativity describes the motion of matter and of empty space, including the motion
of horizons and the evolution of what is usually called the border of the universe.
The description of macroscopic motion with general relativity is final and correct. Cal-
culations and predictions from general relativity match all observations where the match
is possible. The match is not possible for dark matter; this issue is not settled yet.
General relativity is based on two principles deduced from observations:

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
— All observers agree that there is a ‘perfect’ speed in nature, namely a common
maximum energy speed relative to (nearby) matter. The invariant speed value 𝑐 =
299 792 458 m/s is realized by massless radiation, such as light or radio signals.
— All observers agree that there is a ‘perfect’ force in nature, a common maximum force
that can be realized relative to (nearby) matter. The invariant force value 𝐹 = 𝑐4 /4𝐺 =
3.0258(4) ⋅ 1043 N is realized on event horizons.
These two observations contain the full theory of relativity. In particular, from these two
observations we deduce:
— Space-time consists of events in 3+1 continuous dimensions, with a variable curvature.
The curvature can be deduced from distance measurements among events, for ex-
ample from tidal effects. Measured times, lengths and curvatures vary from observer
to observer in a predictable way. In short, we live in a pseudo-Riemannian space-time.
— Space-time and space are curved near mass and energy. The curvature at a point is
determined by the energy–momentum density at that point, and described by the
field equations. When matter and energy move, the space curvature moves along with
them. A built-in delay in this movement renders faster-than-light transport of energy
impossible. The proportionality constant between energy and curvature is so small

** ‘Wisdom is happiness.’ This old saying once was the motto of Oxford University.
288 11 general relativity in a nutshell

that the curvature is not observed in everyday life; only its indirect manifestation,
namely universal gravity, is observed.
— All macroscopic motion – that of matter, of radiation and of vacuum – is described
by the field equations of general relativity.
— Space is elastic: it prefers being flat. Being elastic, it can oscillate independently of
matter; one then speaks of gravitational radiation or of gravity waves.
— Freely falling matter moves along geodesics, i.e., along paths of maximal length in
curved space-time. In space this means that light bends when it passes near large
masses by twice the amount predicted by universal gravity.
— In order to describe gravitation we need curved space-time, i.e., general relativ-
ity, at the latest whenever distances are of the order of the Schwarzschild radius
𝑟S = 2𝐺𝑚/𝑐2 . When distances are much larger than this value, the relativistic de-
scription with gravity and gravitomagnetism (frame-dragging) is sufficient. When
distances are even larger and speeds much slower than those of light, the description
by universal gravity, namely 𝑎 = 𝐺𝑚/𝑟2 , together with flat Minkowski space-time,
will do as a good approximation.

Motion Mountain – The Adventure of Physics
— Space and time are not distinguished globally, but only locally. Matter and radiation
are required to make the distinction.
In addition, all the matter and energy we observe in the sky lead us to the following
conclusions:
— The universe has a finite size, given roughly by 𝑟max ≈ 1/√Λ ≈ 1026 m. The
Page 199 cosmological constant Λ also has the effect of an energy density. One speaks of dark
energy.
— The universe has a finite age; this is the reason for the darkness of the sky at night. A

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
horizon limits the measurable space-time intervals to about fourteen thousand mil-
lion years.
— On the cosmological scale, everything moves away from everything else: the universe
is expanding. The details of the underlying expansion of space, as well as the night-sky
horizon, are described by the field equations of general relativity.
In short, experiments show that all motion of energy, including matter and radiation, is
limited in speed and in momentum flow, or force. A maximum force implies that space
Vol. I, page 29 curves and that the curvature can move. The well-known basic properties of everyday
motion remain valid: also relativistic motion that includes gravity is continuous, con-
serves energy–momentum and angular momentum, is relative, is reversible, is mirror-
invariant (except for the weak interaction, where a generalized way to predict mirror-
Vol. V, page 245 inverse motion holds). Above all, like everyday motion, also every example of relativistic
motion that includes gravity is lazy: all motion minimizes action.
In summary, the principles of maximum force and of maximum speed hold for every
motion in nature. They are universal truths. The theory of general relativity that follows
from the two principles describes all macroscopic motion that is observed in the universe,
including the most rapid, the most powerful and the most distant motions known – be
it motion of matter, radiation, vacuum or horizons.
a summary for the layman 289

The accuracy of the description
Was general relativity worth the effort? The discussion of its accuracy is most conveni-
Ref. 286 ently split into two sets of experiments. The first set consists of measurements of how
matter moves. Do objects really follow geodesics? As summarized in Table 9, all experi-
ments agree with the theory to within measurement errors, i.e., at least within 1 part in
Ref. 287 1012 . In short, the way matter falls is indeed well described by general relativity.
The second set of measurements concerns the dynamics of space-time itself. Does
space-time move following the field equations of general relativity? In other words, is
space-time really bent by matter in the way the theory predicts? Many experiments have
been performed, near to and far from Earth, in both weak and strong gravitational fields.
Ref. 286, Ref. 287 All agree with the predictions to within measurement errors. However, the best meas-
urements so far have only about 3 significant digits. Note that even though numerous
experiments have been performed, there are only few types of tests, as Table 9 shows. The
Challenge 414 ny discovery of a new type of experiment almost guarantees fame and riches. Most sought
after, of course, is the direct detection of gravitational waves.

Motion Mountain – The Adventure of Physics
Another comment on Table 9 is in order. After many decades in which all measured
effects were only of the order 𝑣2 /𝑐2 , several so-called strong field effects in pulsars allowed
Ref. 286 us to reach the order 𝑣4 /𝑐4 . Soon a few effects of this order should also be detected even
inside the solar system, using high-precision satellite experiments. The present crown of
all measurements, the gravity wave emission delay, is the only 𝑣5 /𝑐5 effect measured so
Page 180 far.
The difficulty of achieving high precision for space-time curvature measurements is
the reason why mass is measured with balances, always (indirectly) using the prototype
kilogram in Paris, instead of defining some standard curvature and fixing the value of 𝐺.
Indeed, no useful terrestrial curvature experiment has ever been carried out. A break-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
through in this domain would make the news. The terrestrial curvature methods cur-
rently available would not even allow one to define a kilogram of oranges or peaches
with a precision high enough to distinguish it from the double amount!
A different way to check general relativity is to search for alternative descriptions of
gravitation. Quite a number of alternative theories of gravity have been formulated and
Ref. 287, Ref. 288 studied, but so far, only general relativity is in agreement with all experiments.
In summary, as Thibault Damour likes to explain, general relativity is at least
99.999 999 999 9 % correct concerning the motion of matter and energy, and at least
Ref. 286 99.9 % correct about the way matter and energy curve and move space-time. No excep-
tions, no anti-gravity and no unclear experimental data are known. All motion on Earth
and in the skies is described by general relativity. The most violent, the most rapid and
the most distant movements known behave as expected. Albert Einstein’s achievement
has no flaws.
We note that general relativity has not been tested for microscopic motion. In this con-
text, microscopic motion is any motion for which the action value is near the quantum of
action ℏ, namely 10−34 Js. The exploration of microscopic motion in strong gravitational
fields is the topic of the last part of our adventure.
290 11 general relativity in a nutshell

TA B L E 9 Types of tests of general relativity.

Measured effect Con- Type Refer-
firma- ence
tion
Equivalence principle 10−12 motion of matter Ref. 156,
Ref. 286,
Ref. 289
1/𝑟2 dependence (dimensionality of space-time) 10−10 motion of matter Ref. 290
Time independence of 𝐺 10−19 /s motion of matter Ref. 286
Red-shift (light and microwaves on Sun, Earth, 10−4 space-time curvature Ref. 135,
Sirius) Ref. 133,
Ref. 286
−3
Perihelion shift (four planets, Icarus, pulsars) 10 space-time curvature Ref. 286
Light deflection (light, radio waves around Sun, 10−3 space-time curvature Ref. 286
stars, galaxies)

Motion Mountain – The Adventure of Physics
Time delay (radio signals near Sun, near pulsars)10−3 space-time curvature Ref. 286,
Ref. 175
Gravitomagnetism (Earth, pulsar) 10−1 space-time curvature Ref. 167,
Ref. 168
−1
Geodesic effect (Moon, pulsars) 10 space-time curvature Ref. 164,
Ref. 286
Gravity wave emission delay (pulsars) 10−3 space-time curvature Ref. 286

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
R esearch in general relativity and cosmolo gy
Research in general relativity is still intense, though declining; it is declining most
strongly in Switzerland and Germany, the countries where Albert Einstein developed the
Ref. 291 theory. Research in cosmology and astrophysics, however, is at a high point at present.
Here is a short overview.
∗∗
The most interesting experimental studies of general relativity are the tests using double
pulsars, the search for gravitational waves, and the precision measurements using satel-
lites. Among others a special satellite will capture all possible pulsars of the galaxy. All
these experiments expand the experimental tests into domains that have not been ac-
cessible before. So far, all tests completely confirm general relativity – as expected.
∗∗
The investigation of cosmic collisions and many-body problems, especially those in-
volving neutron stars and black holes, helps astrophysicists to improve their understand-
Ref. 269 ing of the rich behaviour they observe in their telescopes.
∗∗
The study of chaos in the field equations is of fundamental interest in the study of the
a summary for the layman 291

early universe, and may be related to the problem of galaxy formation, one of the biggest
Ref. 292 open problems in physics.
∗∗
Gathering data about galaxy formation is the main aim of several satellite systems and
purpose-build telescopes. One focus is the search for localized cosmic microwave back-
Ref. 293 ground anisotropies due to protogalaxies.
∗∗
The precise determination of the cosmological parameters, such as the matter density,
Ref. 237 the curvature and the vacuum density, is a central effort of modern astrophysics. The
exploration of vacuum density – also called cosmological constant or dark energy – and
the clarification of the nature of dark matter occupy a large fraction of astrophysicists.
∗∗
Astronomers and astrophysicists regularly discover new phenomena in the skies. The

Motion Mountain – The Adventure of Physics
various types of gamma-ray bursts, X-ray bursts and optical bursts are still not com-
Ref. 294 pletely understood. Gamma-ray bursts, for example, can be as bright as 1017 sun-like
stars combined; however, they last only a few seconds. More details on this research topic
Vol. V, page 153 are given later on.
∗∗
A computer database of all known exact solutions of the field equations is being built.
Among other things, researchers are checking whether they really are all different from
Ref. 295 each other.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
∗∗
Ref. 296 Solutions of the field equations with non-trivial topology, such as wormholes and
particle-like solutions, constitue a fascinating field of enquiry. However, such solutions
Vol. V, page 159 are made impossible by quantum effects.
∗∗
Other formulations of general relativity, describing space-time with quantities other than
the metric, are continuously being developed, in the hope of clarifying the relationship
between gravity and the quantum world. The so-called Ashtekar variables are such a
Ref. 297 modern description.
∗∗
The study of the early universe and its relation of elementary particle properties, with
conjectures such as inflation, a short period of accelerated expansion during the first few
Ref. 298, Ref. 299 seconds after the big bang, is still an important topic of investigation.
∗∗
The unification of quantum physics, particle physics and general relativity is an import-
ant research field and will occupy researchers for many years to come. The aim is to find
Vol. VI, page 17 a complete description of motion. This is the topic of the final part of this adventure.
292 11 general relativity in a nutshell

∗∗
Finally, the teaching of general relativity, which for many decades has been hidden be-
hind Greek indices, differential forms and other antididactic approaches, will bene-
fit greatly from future improvements that focus more on the physics and less on the
Ref. 300 formalism.

C ould general relativit y be different?

“
It’s a good thing we have gravity, or else when
birds died they’d just stay right up there.

”
Hunters would be all confused.
Steven Wright

The constant of gravitation provides a limit for the density and the acceleration of objects,
as well as for the power of engines. We based all our deductions on its invariance. Is it
possible that the constant of gravitation 𝐺 changes from place to place or that it changes
with time? The question is tricky. At first sight, the answer is a loud: ‘Yes, of course! Just

Motion Mountain – The Adventure of Physics
see what happens when the value of 𝐺 is changed in formulae.’ However, this answer is
Page 104 wrong. It is as wrong as it was wrong for the speed of light 𝑐.
Since the constant of gravitation enters into our definition of gravity and acceleration,
it thus enters, even if we do not notice it, into the construction of all rulers, all meas-
urement standards and all measuring instruments. Therefore there is no way to detect
whether its value actually varies.

⊳ A change in the maximum force and thus in the gravitational constant 𝐺
cannot be measured.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Yes, the invariance of the limit force and of 𝐺 is counter-intuitive. Experiments are able to
detect the existence of a maximum force. Nevertheless, no imaginable experiment could
detect a variation of the maximum force value, neither over space nor over time. Just try!
Challenge 415 e Every measurement of force is, whether we like it or not, a comparison with the limit
force. There is no way, in principle, to falsify the invariance of a measurement standard.
This is even more astonishing because measurements of this type are regularly reported.
Page 290 In a sense, Table 9 is a list of such experiments! But the result of any such experiment
is easy to predict: no change will ever be found and no deviation from general relativity
will ever be found.
Are other changes possible? Could the number of space dimensions be different from
3? This issue is quite involved. For example, three is the smallest number of dimensions
for which a vanishing Ricci tensor is compatible with non-vanishing curvature. On the
other hand, more than three dimensions would give deviations from the inverse square
‘law’ of gravitation. There are no data pointing in this direction. All experiments confirm
that space has exactly three dimensions.
Could the equations of general relativity be different? During the past century, theor-
eticians have explored many alternative equations. However, almost none of the altern-
atives proposed so far seem to fit experimental data – nor the existence of a maximum
force. Only two candidates are still regularly discussed.
First, the inclusion of torsion in the field equations is one attempt to include particle
a summary for the layman 293

Ref. 302 spin in general relativity. The inclusion of torsion in general relativity does not re-
quire new fundamental constants; indeed, the absence of torsion was assumed in the
Ref. 301 Raychaudhuri equation. The use of the extended Raychaudhuri equation, which includes
torsion, might allow one to deduce the full Einstein–Cartan theory from the maximum
force principle. However, all arguments so far suggest that torsion is an unnecessary com-
plication.
Secondly, one issue remains unexplained: the question of the existence of dark matter.
The rotation speed of visible matter far from the centre of galaxies might be due to the
existence of dark matter or to some deviation from the inverse square dependence of
universal gravity. The latter option would imply a modification in the field equations for
Ref. 303 astronomically large distances. The dark matter option assumes that we have difficulties
observing something, the modified dynamics option assumes that we missed something
in the equations. Also certain experiments about light deflection seem to point to some
invisible kind of matter spread around galaxies. Is this a new form of matter? At present,
most researchers tend to assume the existence of dark matter, and further assume that
is some unknown type of matter. But since the nature of dark matter is not understood,

Motion Mountain – The Adventure of Physics
and since it has never been detected in the laboratory, the issue is not settled.
In summary, given the principle of maximum force, it seems extremely unlikely if not
impossible that nature is not described by general relativity.

“
It was, of course, a lie what you read about my
religious convictions, a lie which is being
systematically repeated. I do not believe in a
personal God and I have never denied this but
have expressed it clearly. If something is in me
which can be called religious then it is the
unbounded admiration for the structure of the

The limitations of general relativity
Despite its success and its fascination, the description of motion presented so far is un-
Challenge 416 e satisfactory; maybe you already have some gut feeling about certain unresolved issues.
First of all, even though the speed of light is the starting point of the whole theory,
we still do not know what light actually is. Understanding what light is will be our next
topic.
Secondly, we have seen that everything that has mass falls along geodesics. But a
mountain does not fall. Somehow the matter below prevents it from falling. How? And
where does mass come from anyway? What is matter? General relativity does not provide
any answer; in fact, it does not describe matter at all. Einstein used to say that the left-
hand side of the field equations, describing the curvature of space-time, was granite,
while the right-hand side, describing matter, was sand. Indeed, at this point we still do
not know what matter and mass are. (And we know even less what dark matter is.) As
already remarked, to change the sand into rock we first need quantum physics and then,
in a further step, its unification with relativity. This is the programme for the rest of our
adventure.
We have also seen that matter is necessary to clearly distinguish between space and
294 11 general relativity in a nutshell

time, and in particular, to understand the working of clocks, metre bars and balances.
But one question remains: why are there units of mass, length and time in nature at
all? Understanding why measurements are possible at all will be another of the topics of
quantum physics.
We also know too little about the vacuum. We need to understand the magnitude of
the cosmological constant, its time dependence and the number of space-time dimen-
sions. Only then can we answer the simple question: Why is the sky so far away? General
relativity does not help here. We will find out that the observed smallness of the cosmo-
logical constant contradicts the simplest version of quantum theory; this is one of the
reasons why we still have quite some height to scale before we reach the top of Motion
Vol. VI, page 58 Mountain.
Finally, we swept another important issue under the rug. General relativity forbids the
existence of point objects, and thus of point particles. But the idea of point particles is
one reason that we introduced space points in the first place. What is the final fate of the
idea of space point? What does this imply for the properties of horizons, for black holes
and the night sky? Also these issues remain open at this stage.

Motion Mountain – The Adventure of Physics
In short, to describe motion well, we need a more precise description of light, of matter
and of the vacuum. In other words, we need to know more about everything! Otherwise
Vol. V, page 67 we cannot hope to answer questions about mountains, clocks and stars. In particular, we
need to know more about light, matter and vacuum at small scales. We need to under-
stand the microscopic aspects of the world.
At small scales, the curvature of space is negligible. We therefore take a step back-
wards, to situations without gravity, and explore the microscopic motion of light, matter
Page 8 and vacuum. This domain is called quantum physics. Figure 1, shown in the preface, gives
an impression of the topics that await us. And despite the simplification to flat space-time,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
the adventure is beautiful and intense: we will explore the motion at the basis of life.
Appendix A

U N I T S , M E A SU R E M E N T S A N D
C ON STA N T S

M
easurements are comparisons with standards. Standards are based on units.
any different systems of units have been used throughout the world.
ost of these standards confer power to the organization in charge of them.
Such power can be misused; this is the case today, for example in the computer in-
dustry, and was so in the distant past. The solution is the same in both cases: organize

Motion Mountain – The Adventure of Physics
an independent and global standard. For measurement units, this happened in the
eighteenth century: in order to avoid misuse by authoritarian institutions, to eliminate
problems with differing, changing and irreproducible standards, and – this is not a joke
– to simplify tax collection and to make it more just, a group of scientists, politicians
and economists agreed on a set of units. It is called the Système International d’Unités,
abbreviated SI, and is defined by an international treaty, the ‘Convention du Mètre’.
The units are maintained by an international organization, the ‘Conférence Générale
des Poids et Mesures’, and its daughter organizations, the ‘Commission Internationale
des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM). All

SI units
All SI units are built from seven base units. Their simplest definitions, translated from
French into English, are the following ones, together with the dates of their formulation
and a few comments:
‘The second is the duration of 9 192 631 770 periods of the radiation corresponding
to the transition between the two hyperfine levels of the ground state of the caesium 133
atom.’ (1967) The 2019 definition is equivalent, but much less clear.*
‘The metre is the length of the path travelled by light in vacuum during a time inter-
val of 1/299 792 458 of a second.’ (1983) The 2019 definition is equivalent, but much less
clear.*
‘The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed
numerical value of the Planck constant h to be 6.626 070 15 ⋅ 10−34 when expressed in the
unit J ⋅ s, which is equal to kg ⋅ m2 ⋅ s−1 .’ (2019)*
‘The ampere, symbol A, is the SI unit of electric current. It is defined by taking the
fixed numerical value of the elementary charge e to be 1.602 176 634 ⋅ 10−19 when ex-
pressed in the unit C, which is equal to A ⋅ s.’ (2019)* This definition is equivalent to:
One ampere is 6.241 509 074... ⋅ 1018 elementary charges per second.
‘The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by
296 a units, measurements and constants

taking the fixed numerical value of the Boltzmann constant 𝑘 to be 1.380649 ⋅10−23 when
expressed in the unit J ⋅ K−1 .’ (2019)*
‘The mole, symbol mol, is the SI unit of amount of substance. One mole contains
exactly 6.02214076 ⋅ 1023 elementary entities.’ (2019)*
‘The candela is the luminous intensity, in a given direction, of a source that emits
monochromatic radiation of frequency 540 ⋅ 1012 hertz and has a radiant intensity in
that direction of (1/683) watt per steradian.’ (1979) The 2019 definition is equivalent, but
much less clear.*
We note that both time and length units are defined as certain properties of a standard
example of motion, namely light. In other words, also the Conférence Générale des Poids
et Mesures makes the point that the observation of motion is a prerequisite for the defin-
ition and construction of time and space. Motion is the fundament of every observation
and of all measurement. By the way, the use of light in the definitions had been proposed
already in 1827 by Jacques Babinet.**
From these basic units, all other units are defined by multiplication and division. Thus,

Motion Mountain – The Adventure of Physics
all SI units have the following properties:
SI units form a system with state-of-the-art precision: all units are defined with a pre-
cision that is higher than the precision of commonly used measurements. Moreover, the
precision of the definitions is regularly being improved. The present relative uncertainty
of the definition of the second is around 10−14 , for the metre about 10−10 , for the kilo-
gram about 10−9 , for the ampere 10−7 , for the mole less than 10−6 , for the kelvin 10−6 and
for the candela 10−3 .
SI units form an absolute system: all units are defined in such a way that they can
be reproduced in every suitably equipped laboratory, independently, and with high pre-

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cision. This avoids as much as possible any error or misuse by the standard-setting or-
ganization. In fact, the SI units are as now as near as possible to Planck’s natural units,
which are presented below. In practice, the SI is now an international standard defining
the numerical values of the seven constants Δ𝜈Cs , 𝑐, ℏ, 𝑒, 𝑘, 𝑁A and 𝐾cd . After over 200
years of discussions, the CGPM has little left to do.
SI units form a practical system: the base units are quantities of everyday magnitude.
Frequently used units have standard names and abbreviations. The complete list includes
the seven base units just given, the supplementary units, the derived units and the ad-
mitted units.
The supplementary SI units are two: the unit for (plane) angle, defined as the ratio
of arc length to radius, is the radian (rad). For solid angle, defined as the ratio of the
subtended area to the square of the radius, the unit is the steradian (sr).
The derived units with special names, in their official English spelling, i.e., without
capital letters and accents, are:

* The symbols of the seven units are s, m, kg, A, K, mol and cd. The full offical definitions are found at
Ref. 305 www.bipm.org. For more details about the levels of the caesium atom, consult a book on atomic physics.
The Celsius scale of temperature 𝜃 is defined as: 𝜃/°C = 𝑇/K − 273.15; note the small difference with the
number appearing in the definition of the kelvin. In the definition of the candela, the frequency of the light
corresponds to 555.5 nm, i.e., green colour, around the wavelength to which the eye is most sensitive.
** Jacques Babinet (1794–1874), French physicist who published important work in optics.
a units, measurements and constants 297

Name A bbre v iat i o n Name A b b r e v i at i o n

hertz Hz = 1/s newton N = kg m/s2
pascal Pa = N/m2 = kg/m s2 joule J = Nm = kg m2 /s2
watt W = kg m2 /s3 coulomb C = As
volt V = kg m2 /As3 farad F = As/V = A2 s4 /kg m2
ohm Ω = V/A = kg m2 /A2 s3 siemens S = 1/Ω
weber Wb = Vs = kg m2 /As2 tesla T = Wb/m2 = kg/As2 = kg/Cs
henry H = Vs/A = kg m2 /A2 s2 degree Celsius °C (see definition of kelvin)
lumen lm = cd sr lux lx = lm/m2 = cd sr/m2
becquerel Bq = 1/s gray Gy = J/kg = m2 /s2
sievert Sv = J/kg = m2 /s2 katal kat = mol/s

We note that in all definitions of units, the kilogram only appears to the powers of 1,
Challenge 417 s 0 and −1. Can you try to formulate the reason?

Motion Mountain – The Adventure of Physics
The admitted non-SI units are minute, hour, day (for time), degree 1° = π/180 rad,
minute 1 󸀠 = π/10 800 rad, second 1 󸀠󸀠 = π/648 000 rad (for angles), litre, and tonne. All
other units are to be avoided.
All SI units are made more practical by the introduction of standard names and ab-
breviations for the powers of ten, the so-called prefixes:*

Power Name Power Name Power Name Power Name
101 deca da 10−1 deci d 1018 Exa E 10−18 atto a
102 hecto h 10−2 centi c 1021 Zetta Z 10−21 zepto z

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103 kilo k 10−3 milli m 1024 Yotta Y 10−24 yocto y
106 Mega M 10−6 micro μ unofficial: Ref. 306
109 Giga G 10−9 nano n 1027 Xenta X 10−27 xenno x
1012 Tera T 10−12 pico p 1030 Wekta W 10−30 weko w
1015 Peta P 10−15 femto f 1033 Vendekta V 10−33 vendeko v
1036 Udekta U 10−36 udeko u

SI units form a complete system: they cover in a systematic way the full set of ob-
servables of physics. Moreover, they fix the units of measurement for all other sciences
as well.

* Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar
to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words ἑξάκις and
πεντάκις for ‘six times’ and ‘five times’, the unofficial ones to sound similar to the Greek words for nine,
ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten
‘fifteen’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from
nanus ‘dwarf’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from μικρός ‘small’,
deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from μέγας
‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’).
Translate: I was caught in such a traffic jam that I needed a microcentury for a picoparsec and that my
Challenge 418 e car’s fuel consumption was two tenths of a square millimetre.
298 a units, measurements and constants

SI units form a universal system: they can be used in trade, in industry, in commerce,
at home, in education and in research. They could even be used by extraterrestrial civil-
izations, if they existed.
SI units form a self-consistent system: the product or quotient of two SI units is also
an SI unit. This means that in principle, the same abbreviation, e.g. ‘SI’, could be used
for every unit.
The SI units are not the only possible set that could fulfil all these requirements, but they
are the only existing system that does so.*

The meaning of measurement
Every measurement is a comparison with a standard. Therefore, any measurement re-
Challenge 419 e quires matter to realize the standard (even for a speed standard), and radiation to achieve
the comparison. The concept of measurement thus assumes that matter and radiation ex-
ist and can be clearly separated from each other.
Every measurement is a comparison. Measuring thus implies that space and time ex-

Motion Mountain – The Adventure of Physics
ist, and that they differ from each other.
Every measurement produces a measurement result. Therefore, every measurement
implies the storage of the result. The process of measurement thus implies that the situ-
ation before and after the measurement can be distinguished. In other terms, every meas-
urement is an irreversible process.
Every measurement is a process. Thus every measurement takes a certain amount of
time and a certain amount of space.
All these properties of measurements are simple but important. Beware of anybody
who denies them.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Curiosities and fun challenges ab ou t units
The second does not correspond to 1/86 400th of the day any more, though it did in the
year 1900; the Earth now takes about 86 400.002 s for a rotation, so that the International
Earth Rotation Service must regularly introduce a leap second to ensure that the Sun is at
the highest point in the sky at 12 o’clock sharp.** The time so defined is called Universal
Time Coordinate. The speed of rotation of the Earth also changes irregularly from day to
day due to the weather; the average rotation speed even changes from winter to summer
because of the changes in the polar ice caps; and in addition that average decreases over
time, because of the friction produced by the tides. The rate of insertion of leap seconds
is therefore higher than once every 500 days, and not constant in time.

* Apart from international units, there are also provincial units. Most provincial units still in use are of
Roman origin. The mile comes from milia passum, which used to be one thousand (double) strides of about
1480 mm each; today a nautical mile, once defined as minute of arc on the Earth’s surface, is defined as
exactly 1852 m. The inch comes from uncia/onzia (a twelfth – now of a foot). The pound (from pondere ‘to
weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit
of counting in dozens instead of tens is Roman in origin. These and all other similarly funny units – like
the system in which all units start with ‘f’, and which uses furlong/fortnight as its unit of velocity – are now
officially defined as multiples of SI units.
** Their website at hpiers.obspm.fr gives more information on the details of these insertions, as does maia.
usno.navy.mil, one of the few useful military websites. See also www.bipm.fr, the site of the BIPM.
a units, measurements and constants 299

∗∗
Not using SI units can be expensive. In 1999, the space organisation NASA lost a satellite
on Mars because some software programmers had used provincial units instead of SI
units in part of the code. As a result of using feet instead of meters, the Mars Climate
Orbiter crashed into the planet, instead of orbiting it; the loss was around 100 million
euro.*
∗∗
The most precisely measured quantities in nature are the frequencies of certain milli-
second pulsars,** the frequency of certain narrow atomic transitions, and the Rydberg
constant of atomic hydrogen, which can all be measured as precisely as the second is
defined. The caesium transition that defines the second has a finite linewidth that limits
the achievable precision: the limit is about 14 digits.
∗∗
The least precisely measured of the fundamental constants of physics are the gravitational

Motion Mountain – The Adventure of Physics
constant 𝐺 and the strong coupling constant 𝛼s . Even less precisely known are the age of
Page 308 the universe and its density (see Table 14).
∗∗
Variations of quantities are often much easier to measure than their values. For example,
in gravitational wave detectors, the sensitivity achieved in 1992 was Δ𝑙/𝑙 = 3 ⋅ 10−19 for
Ref. 307 lengths of the order of 1 m. In other words, for a block of about a cubic metre of metal
it is possible to measure length changes about 3000 times smaller than a proton radius.
These set-ups are now being superseded by ring interferometers. Ring interferometers

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measuring frequency differences of 10−21 have already been built; and they are still being
Ref. 308 improved.
∗∗
The table of SI prefixes covers 72 orders of magnitude. How many additional prefixes will
be needed? Even an extended list will include only a small part of the infinite range of
possibilities. Will the Conférence Générale des Poids et Mesures have to go on forever,
Challenge 420 s defining an infinite number of SI prefixes? Why?
∗∗
The French philosopher Voltaire, after meeting Newton, publicized the now famous story
that the connection between the fall of objects and the motion of the Moon was dis-
covered by Newton when he saw an apple falling from a tree. More than a century later,
just before the French Revolution, a committee of scientists decided to take as the unit
of force precisely the force exerted by gravity on a standard apple, and to name it after
the English scientist. After extensive study, it was found that the mass of the standard

* This story revived an old but false urban legend claiming that only three countries in the world do not use
SI units: Liberia, the USA and Myanmar.
** An overview of this fascinating work is given by J. H. Taylor, Pulsar timing and relativistic gravity,
Philosophical Transactions of the Royal Society, London A 341, pp. 117–134, 1992.
300 a units, measurements and constants

apple was 101.9716 g; its weight was called 1 newton. Since then, visitors to the museum
in Sèvres near Paris have been able to admire the standard metre, the standard kilogram
and the standard apple.*

Precision and accuracy of measurements
Measurements are the basis of physics. Every measurement has an error. Errors are due
to lack of precision or to lack of accuracy. Precision means how well a result is reproduced
when the measurement is repeated; accuracy is the degree to which a measurement cor-
responds to the actual value.
Lack of precision is due to accidental or random errors; they are best measured by the
standard deviation, usually abbreviated 𝜎; it is defined through

2 1 𝑛
𝜎 = ∑(𝑥 − 𝑥)̄ 2 , (297)
𝑛 − 1 𝑖=1 𝑖

Motion Mountain – The Adventure of Physics
where 𝑥̄ is the average of the measurements 𝑥𝑖 . (Can you imagine why 𝑛 − 1 is used in
Challenge 421 s the formula instead of 𝑛?)
For most experiments, the distribution of measurement values tends towards a nor-
mal distribution, also called Gaussian distribution, whenever the number of measure-
ments is increased. The distribution, shown in Figure 304, is described by the expression

(𝑥−𝑥)̄ 2
𝑁(𝑥) ≈ e− 2𝜎2 . (298)

The square 𝜎2 of the standard deviation is also called the variance. For a Gaussian distri-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 422 e bution of measurement values, 2.35𝜎 is the full width at half maximum.
Lack of accuracy is due to systematic errors; usually these can only be estimated. This
estimate is often added to the random errors to produce a total experimental error, some-
Ref. 310 times also called total uncertainty. The relative error or uncertainty is the ratio between
the error and the measured value.
For example, a professional measurement will give a result such as 0.312(6) m. The
number between the parentheses is the standard deviation 𝜎, in units of the last digits.
As above, a Gaussian distribution for the measurement results is assumed. Therefore, a
Challenge 423 e value of 0.312(6) m implies that the actual value is expected to lie
— within 1𝜎 with 68.3 % probability, thus in this example within 0.312 ± 0.006 m;
— within 2𝜎 with 95.4 % probability, thus in this example within 0.312 ± 0.012 m;
— within 3𝜎 with 99.73 % probability, thus in this example within 0.312 ± 0.018 m;
— within 4𝜎 with 99.9937 % probability, thus in this example within 0.312 ± 0.024 m;
— within 5𝜎 with 99.999 943 % probability, thus in this example within 0.312 ± 0.030 m;
— within 6𝜎 with 99.999 999 80 % probability, thus within 0.312 ± 0.036 m;
— within 7𝜎 with 99.999 999 999 74 % probability, thus within 0.312 ± 0.041 m.

* To be clear, this is a joke; no standard apple exists. It is not a joke however, that owners of several apple
trees in Britain and in the US claim descent, by rerooting, from the original tree under which Newton had
Ref. 309 his insight. DNA tests have even been performed to decide if all these derive from the same tree. The result
was, unsurprisingly, that the tree at MIT, in contrast to the British ones, is a fake.
a units, measurements and constants 301

N
number of measurements

standard deviation

full width at half maximum
(FWHM)

limit curve for a large number
of measurements: the
Gaussian distribution

x x
average value measured values

Motion Mountain – The Adventure of Physics
F I G U R E 125 A precision experiment and its measurement distribution. The precision is high if the
width of the distribution is narrow; the accuracy is high if the centre of the distribution agrees with the
actual value.

Challenge 424 s (Do the latter numbers make sense?)
Note that standard deviations have one digit; you must be a world expert to use two,
and a fool to use more. If no standard deviation is given, a (1) is assumed. As a result,
among professionals, 1 km and 1000 m are not the same length!

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
What happens to the errors when two measured values 𝐴 and 𝐵 are added or subtrac-
ted? If the all measurements are independent – or uncorrelated – the standard deviation
of the sum and that of difference is given by 𝜎 = √𝜎𝐴2 + 𝜎𝐵2 . For both the product or ratio
of two measured and uncorrelated values 𝐶 and 𝐷, the result is 𝜌 = √𝜌𝐶2 + 𝜌𝐷2 , where the
𝜌 terms are the relative standard deviations.
Challenge 425 s Assume you measure that an object moves 1.0 m in 3.0 s: what is the measured speed
value?

Limits to precision
What are the limits to accuracy and precision? There is no way, even in principle, to
measure a length 𝑥 to a precision higher than about 61 digits, because in nature, the ratio
between the largest and the smallest measurable length is Δ𝑥/𝑥 > 𝑙Pl/𝑑horizon = 10−61 .
Challenge 426 e (Is this ratio valid also for force or for volume?) In the final volume of our text, studies
Vol. VI, page 94 of clocks and metre bars strengthen this theoretical limit.
But it is not difficult to deduce more stringent practical limits. No imaginable machine
can measure quantities with a higher precision than measuring the diameter of the Earth
within the smallest length ever measured, about 10−19 m; that is about 26 digits of preci-
sion. Using a more realistic limit of a 1000 m sized machine implies a limit of 22 digits.
If, as predicted above, time measurements really achieve 17 digits of precision, then they
are nearing the practical limit, because apart from size, there is an additional practical
302 a units, measurements and constants

restriction: cost. Indeed, an additional digit in measurement precision often means an
additional digit in equipment cost.

Physical constants
In physics, general observations are deduced from more fundamental ones. As a con-
sequence, many measurements can be deduced from more fundamental ones. The most
fundamental measurements are those of the physical constants.
The following tables give the world’s best values of the most important physical con-
stants and particle properties – in SI units and in a few other common units – as pub-
Ref. 311 lished in the standard references. The values are the world averages of the best measure-
ments made up to the present. As usual, experimental errors, including both random
and estimated systematic errors, are expressed by giving the standard deviation in the
last digits. In fact, behind each of the numbers in the following tables there is a long
Ref. 312 story which is worth telling, but for which there is not enough room here.
In principle, all quantitative properties of matter can be calculated with quantum the-

Motion Mountain – The Adventure of Physics
Vol. V, page 261 ory – more precisely, equations of the standard model of particle – and a set of basic
physical constants that are given in the next table. For example, the colour, density and
elastic properties of any material can be predicted, in principle, in this way.

TA B L E 11 Basic physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

Constants that define the SI measurement units
Vacuum speed of light 𝑐 𝑐 299 792 458 m/s 0

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
𝑐
Original Planck constant ℎ 6.626 070 15 ⋅ 10−34 Js 0
Reduced Planck constant, ℏ 1.054 571 817 ... ⋅ 10−34 Js 0
quantum of action
Positron charge 𝑐 𝑒 0.160 217 6634 aC 0
𝑐
Boltzmann constant 𝑘 1.380 649 ⋅ 10−23 J/K 0
Avogadro’s number 𝑁A 6.022 140 76 ⋅ 1023 1/mol 0
Constant that should define the SI measurement units
Gravitational constant 𝐺 6.674 30(15) ⋅ 10−11 Nm2 /kg2 2.2 ⋅ 10−5
Other fundamental constants
Number of space-time dimensions 3+1 0𝑏
2
Fine-structure constant 𝑑 or 𝛼 = 4π𝜀𝑒 ℏ𝑐 1/137.035 999 084(21) 1.5 ⋅ 10−10
0

e.m. coupling constant = 𝑔em (𝑚2e 𝑐2 ) = 0.007 297 352 5693(11) 1.5 ⋅ 10−10
Fermi coupling constant 𝑑 or 𝐺F /(ℏ𝑐)3 1.166 3787(6) ⋅ 10−5 GeV−2 5.1 ⋅ 10−7
weak coupling constant 𝛼w (𝑀Z ) = 𝑔w2 /4π 1/30.1(3) 1 ⋅ 10−2
Strong coupling constant 𝑑 𝛼s (𝑀Z ) = 𝑔s2 /4π 0.1179(10) 8.5 ⋅ 10−3
Weak mixing angle sin2 𝜃W (𝑀𝑆) 0.231 22(4) 1.7 ⋅ 10−4
sin2 𝜃W (on shell) 0.222 90(30) 1.3 ⋅ 10−3
= 1 − (𝑚W /𝑚Z )2
a units, measurements and constants 303

TA B L E 11 (Continued) Basic physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

0.97383(24) 0.2272(10) 0.00396(9)
CKM quark mixing matrix |𝑉| ( 0.2271(10) 0.97296(24) 0.04221(80) )
0.00814(64) 0.04161(78) 0.999100(34)
Jarlskog invariant 𝐽 3.08(18) ⋅ 10−5
0.82(2) 0.55(4) 0.150(7)
PMNS neutrino mixing m. |𝑃| (0.37(13) 0.57(11) 0.71(7) )
0.41(13) 0.59(10) 0.69(7)
Electron mass 𝑚e 9.109 383 7015(28) ⋅ 10−31 kg 3.0 ⋅ 10−10
5.485 799 090 65(16) ⋅ 10−4 u 2.9 ⋅ 10−11
0.510 998 950 00(15) MeV 3.0 ⋅ 10−10
−28
Muon mass 𝑚μ 1.883 531 627(42) ⋅ 10 kg 2.2 ⋅ 10−8
105.658 3755(23) MeV 2.2 ⋅ 10−8

Motion Mountain – The Adventure of Physics
Tau mass 𝑚𝜏 1.776 82(12) GeV/𝑐2 6.8 ⋅ 10−5
El. neutrino mass 𝑚𝜈e < 2 eV/𝑐2
Muon neutrino mass 𝑚𝜈𝜇 < 2 eV/𝑐2
Tau neutrino mass 𝑚𝜈𝜏 < 2 eV/𝑐2
Up quark mass 𝑢 21.6(+0.49/ − 0.26) MeV/𝑐2
Down quark mass 𝑑 4.67(+0.48/ − 0.17) MeV/𝑐2
Strange quark mass 𝑠 93(+11/ − 5) MeV/𝑐2
Charm quark mass 𝑐 1.27(2) GeV/𝑐2
4.18(3) GeV/𝑐2

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Bottom quark mass 𝑏
Top quark mass 𝑡 172.9(0.4) GeV/𝑐2
Photon mass γ < 2 ⋅ 10−54 kg
W boson mass 𝑊± 80.379(12) GeV/𝑐2
Z boson mass 𝑍0 91.1876(21) GeV/𝑐2
Higgs mass H 125.10(14) GeV/𝑐2
Gluon mass g1...8 c. 0 MeV/𝑐2

𝑎. Uncertainty: standard deviation of measurement errors.
𝑏. Measured from to 10−19 m to 1026 m.
𝑐. Defining constant.
𝑑. All coupling constants depend on the 4-momentum transfer, as explained in the section on
Page 131 renormalization. Fine-structure constant is the traditional name for the electromagnetic coup-
ling constant 𝑔em in the case of a 4-momentum transfer of 𝑄2 = 𝑚2e 𝑐2 , which is the smallest
one possible. At higher momentum transfers it has larger values, e.g., 𝑔em (𝑄2 = 𝑀W
2 2
𝑐 ) ≈ 1/128.
In contrast, the strong coupling constant has lover values at higher momentum transfers; e.g.,
𝛼s (34 GeV) = 0.14(2).

Why do all these basic constants have the values they have? For any basic constant with
a dimension, such as the quantum of action ℏ, the numerical value has only historical
meaning. It is 1.054 ⋅ 10−34 Js because of the SI definition of the joule and the second.
304 a units, measurements and constants

The question why the value of a dimensional constant is not larger or smaller therefore
always requires one to understand the origin of some dimensionless number giving the
ratio between the constant and the corresponding natural unit that is defined with 𝑐, 𝐺,
Vol. IV, page 208 𝑘, 𝑁A and ℏ. Details and values for the natural units are given in the dedicated section.
In other words, understanding the sizes of atoms, people, trees and stars, the duration
of molecular and atomic processes, or the mass of nuclei and mountains, implies under-
standing the ratios between these values and the corresponding natural units. The key to
understanding nature is thus the understanding of all measurement ratios, and thus of
all dimensionless constants. This quest, including the understanding of the fine-structure
constant 𝛼 itself, is completed only in the final volume of our adventure.
The basic constants yield the following useful high-precision observations.

TA B L E 12 Derived physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t.

Vacuum permeability 𝜇0 1.256 637 062 12(19) μH/m 1.5 ⋅ 10−10

Motion Mountain – The Adventure of Physics
Vacuum permittivity 𝜀0 = 1/𝜇0 𝑐2 8.854 187 8128(13) pF/m 1.5 ⋅ 10−10
Vacuum impedance 𝑍0 = √𝜇0 /𝜀0 376.730 313 668(57) Ω 1.5 ⋅ 10−10
Loschmidt’s number 𝑁L 2.686 780 111... ⋅ 1025 1/m3 0
at 273.15 K and 101 325 Pa
Faraday’s constant 𝐹 = 𝑁A 𝑒 96 485.332 12... C/mol 0
Universal gas constant 𝑅 = 𝑁A 𝑘 8.314 462 618... J/(mol K) 0
Molar volume of an ideal gas 𝑉 = 𝑅𝑇/𝑝 22.413 969 54... l/mol 0
at 273.15 K and 101 325 Pa
Rydberg constant 𝑎 𝑅∞ = 𝑚e 𝑐𝛼2 /2ℎ 10 973 731.568 160(21) m−1 1.9 ⋅ 10−12

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Conductance quantum 𝐺0 = 2𝑒2 /ℎ 77.480 917 29... μS 0
Magnetic flux quantum 𝜑0 = ℎ/2𝑒 2.067 833 848... fWb 0
Josephson frequency ratio 2𝑒/ℎ 483.597 8484... THz/V 0
Von Klitzing constant ℎ/𝑒2 = 𝜇0 𝑐/2𝛼 25 812.807 45... Ω 0
Bohr magneton 𝜇B = 𝑒ℏ/2𝑚e 9.274 010 0783(28) yJ/T 3.0 ⋅ 10−10
Classical electron radius 𝑟e = 𝑒2 /4π𝜀0 𝑚e 𝑐2 2.817 940 3262(13) f m 4.5 ⋅ 10−10
Compton wavelength 𝜆 C = ℎ/𝑚e 𝑐 2.426 310 238 67(73) pm 3.0 ⋅ 10−10
of the electron 𝜆c = ℏ/𝑚e 𝑐 = 𝑟e /𝛼 0.386 159 267 96(12) pm 3.0 ⋅ 10−10
Bohr radius 𝑎 𝑎∞ = 𝑟e /𝛼2 52.917 721 0903(80) pm 1.5 ⋅ 10−10
Quantum of circulation ℎ/2𝑚e 3.636 947 5516(11) cm2 /s 3.0 ⋅ 10−10
Specific positron charge 𝑒/𝑚e 175.882 001 076(55) GC/kg 3.0 ⋅ 10−10
Cyclotron frequency 𝑓c /𝐵 = 𝑒/2π𝑚e 27.992 489 872(9) GHz/T 3.0 ⋅ 10−10
of the electron
Electron magnetic moment 𝜇e −9.284 764 7043(28) yJ/T 3.0 ⋅ 10−10
𝜇e /𝜇B −1.001 159 652 181 28(18) 1.7 ⋅ 10−13
𝜇e /𝜇N −1 838.281 971 88(11) ⋅ 103 6.0 ⋅ 10−11
Electron g-factor 𝑔e −2.002 319 304 362 56(35) 1.7 ⋅ 10−13
Muon–electron mass ratio 𝑚μ /𝑚e 206.768 2830(46) 2.2 ⋅ 10−8
Muon magnetic moment 𝜇μ −4.490 448 30(10) ⋅ 10−26 J/T 2.2 ⋅ 10−8
a units, measurements and constants 305

TA B L E 12 (Continued) Derived physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t.

Muon g-factor 𝑔μ −2.002 331 8418(13) 6.3 ⋅ 10−10
Atomic mass unit 1 u = 𝑚12C /12 1.660 539 066 60(50) ⋅ 10 kg 3.0 ⋅ 10−10
−27

Proton mass 𝑚p 1.672 621 923 69(51) ⋅ 10−27 kg 3.1 ⋅ 10−10
1.007 276 466 621(53) u 5.3 ⋅ 10−11
938.272 088 16(29) MeV 3.1 ⋅ 10−10
Proton–electron mass ratio 𝑚p /𝑚e 1 836.152 673 43(11) 6.0 ⋅ 10−11
Specific proton charge 𝑒/𝑚p 9.578 833 1560(29) ⋅ 107 C/kg 3.1 ⋅ 10−10
Proton Compton wavelength 𝜆 C,p = ℎ/𝑚p 𝑐 1.321 409 855 39(40) f m 3.1 ⋅ 10−10
Nuclear magneton 𝜇N = 𝑒ℏ/2𝑚p 5.050 783 7461(15) ⋅ 10 J/T 3.1 ⋅ 10−10
−27

Proton magnetic moment 𝜇p 1.410 606 797 36(60) ⋅ 10−26 J/T 4.2 ⋅ 10−10
𝜇p /𝜇B 1.521 032 202 30(46) ⋅ 10−3 3.0 ⋅ 10−10
𝜇p /𝜇N 2.792 847 344 63(82) 2.9 ⋅ 10−10

Motion Mountain – The Adventure of Physics
Proton gyromagnetic ratio 𝛾p = 2𝜇𝑝 /ℎ 42.577 478 518(18) MHz/T 4.2 ⋅ 10−10
Proton g factor 𝑔p 5.585 694 6893(16) 2.9 ⋅ 10−10
Neutron mass 𝑚n 1.674 927 498 04(95) ⋅ 10−27 kg 5.7 ⋅ 10−10
1.008 664 915 95(43) u 4.8 ⋅ 10−10
939.565 420 52(54) MeV 5.7 ⋅ 10−10
Neutron–electron mass ratio 𝑚n /𝑚e 1 838.683 661 73(89) 4.8 ⋅ 10−10
Neutron–proton mass ratio 𝑚n /𝑚p 1.001 378 419 31(49) 4.9 ⋅ 10−10
Neutron Compton wavelength 𝜆 C,n = ℎ/𝑚n 𝑐 1.319 590 905 81(75) f m 5.7 ⋅ 10−10

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Neutron magnetic moment 𝜇n −0.966 236 51(23) ⋅ 10−26 J/T 2.4 ⋅ 10−7
𝜇n /𝜇B −1.041 875 63(25) ⋅ 10−3 2.4 ⋅ 10−7
𝜇n /𝜇N −1.913 042 73(45) 2.4 ⋅ 10−7
Stefan–Boltzmann constant 𝜎 = π2 𝑘4 /60ℏ3 𝑐2 56.703 744 19... nW/m K 2 4
0
Wien’s displacement constant 𝑏 = 𝜆 max 𝑇 2.897 771 955... mmK 0
58.789 257 57... GHz/K 0
Electron volt eV 0.160 217 6634... aJ 0
Bits to entropy conversion const. 𝑘 ln 2 1023 bit = 0.956 994... J/K 0
TNT energy content 3.7 to 4.0 MJ/kg 4 ⋅ 10−2

𝑎. For infinite mass of the nucleus.

Some useful properties of our local environment are given in the following table.

TA B L E 13 Astronomical constants.

Q ua nt it y Symbol Va l u e

Tropical year 1900 𝑎 𝑎 31 556 925.974 7 s
Tropical year 1994 𝑎 31 556 925.2 s
Mean sidereal day 𝑑 23ℎ 56󸀠 4.090 53󸀠󸀠
306 a units, measurements and constants

TA B L E 13 (Continued) Astronomical constants.

Q ua nt it y Symbol Va l u e

Average distance Earth–Sun 𝑏 149 597 870.691(30) km
Astronomical unit 𝑏 AU 149 597 870 691 m
Light year, based on Julian year 𝑏 al 9.460 730 472 5808 Pm
Parsec pc 30.856 775 806 Pm = 3.261 634 al
Earth’s mass 𝑀♁ 5.973(1) ⋅ 1024 kg
Geocentric gravitational constant 𝐺𝑀 3.986 004 418(8) ⋅ 1014 m3 /s2
Earth’s gravitational length 𝑙♁ = 2𝐺𝑀/𝑐2 8.870 056 078(16) mm
Earth’s equatorial radius 𝑐 𝑅♁eq 6378.1366(1) km
Earth’s polar radius 𝑐 𝑅♁p 6356.752(1) km
𝑐
Equator–pole distance 10 001.966 km (average)
Earth’s flattening 𝑐 𝑒♁ 1/298.25642(1)
Earth’s av. density 𝜌♁ 5.5 Mg/m3

Motion Mountain – The Adventure of Physics
Earth’s age 𝑇♁ 4.50(4) Ga = 142(2) Ps
Earth’s normal gravity 𝑔 9.806 65 m/s2
Earth’s standard atmospher. pressure 𝑝0 101 325 Pa
Moon’s radius 𝑅v 1738 km in direction of Earth
Moon’s radius 𝑅h 1737.4 km in other two directions
Moon’s mass 𝑀 7.35 ⋅ 1022 kg
Moon’s mean distance 𝑑 𝑑 384 401 km
Moon’s distance at perigee 𝑑 typically 363 Mm, historical minimum

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
359 861 km
Moon’s distance at apogee 𝑑 typically 404 Mm, historical maximum
406 720 km
Moon’s angular size 𝑒 average 0.5181° = 31.08 󸀠 , minimum
0.49°, maximum 0.55°
Moon’s average density 𝜌 3.3 Mg/m3
Moon’s surface gravity 𝑔 1.62 m/s2
Moon’s atmospheric pressure 𝑝 from 10−10 Pa (night) to 10−7 Pa (day)
Jupiter’s mass 𝑀 1.90 ⋅ 1027 kg
Jupiter’s radius, equatorial 𝑅 71.398 Mm
Jupiter’s radius, polar 𝑅 67.1(1) Mm
Jupiter’s average distance from Sun 𝐷 778 412 020 km
Jupiter’s surface gravity 𝑔 24.9 m/s2
Jupiter’s atmospheric pressure 𝑝 from 20 kPa to 200 kPa
Sun’s mass 𝑀⊙ 1.988 43(3) ⋅ 1030 kg
Sun’s gravitational length 2𝐺𝑀⊙ /𝑐2 2.953 250 08(5) km
Heliocentric gravitational constant 𝐺𝑀⊙ 132.712 440 018(8) ⋅ 1018 m3 /s2
Sun’s luminosity 𝐿⊙ 384.6 YW
Solar equatorial radius 𝑅⊙ 695.98(7) Mm
a units, measurements and constants 307

TA B L E 13 (Continued) Astronomical constants.

Q ua nt it y Symbol Va l u e

Sun’s angular size 0.53∘ average; minimum on fourth of July
(aphelion) 1888 󸀠󸀠 , maximum on fourth of
January (perihelion) 1952 󸀠󸀠
Sun’s average density 𝜌⊙ 1.4 Mg/m3
Sun’s average distance AU 149 597 870.691(30) km
Sun’s age 𝑇⊙ 4.6 Ga
Solar velocity 𝑣⊙g 220(20) km/s
around centre of galaxy
Solar velocity 𝑣⊙b 370.6(5) km/s
against cosmic background
Sun’s surface gravity 𝑔⊙ 274 m/s2
Sun’s lower photospheric pressure 𝑝⊙ 15 kPa

Motion Mountain – The Adventure of Physics
Distance to Milky Way’s centre 8.0(5) kpc = 26.1(1.6) kal
Milky Way’s age 13.6 Ga
Milky Way’s size c. 1021 m or 100 kal
Milky Way’s mass 1012 solar masses, c. 2 ⋅ 1042 kg
Most distant galaxy cluster known SXDF-XCLJ 9.6 ⋅ 109 al
0218-0510

𝑎. Defining constant, from vernal equinox to vernal equinox; it was once used to define the

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
second. (Remember: π seconds is about a nanocentury.) The value for 1990 is about 0.7 s less,
Challenge 427 s corresponding to a slowdown of roughly 0.2 ms/a. (Watch out: why?) There is even an empirical
Ref. 313 formula for the change of the length of the year over time.
𝑏. The truly amazing precision in the average distance Earth–Sun of only 30 m results from time
averages of signals sent from Viking orbiters and Mars landers taken over a period of over twenty
years. Note that the International Astronomical Union distinguishes the average distance Earth–
Sun from the astronomical unit itself; the latter is defined as a fixed and exact length. Also the
light year is a unit defined as an exact number by the IAU. For more details, see www.iau.org/
public/measuring.
𝑐. The shape of the Earth is described most precisely with the World Geodetic System. The last
edition dates from 1984. For an extensive presentation of its background and its details, see the
www.wgs84.com website. The International Geodesic Union refined the data in 2000. The radii
and the flattening given here are those for the ‘mean tide system’. They differ from those of the
‘zero tide system’ and other systems by about 0.7 m. The details constitute a science in itself.
𝑑. Measured centre to centre. To find the precise position of the Moon at a given date, see
the www.fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page www.
fourmilab.ch/solar/solar.html and the other pages on the same site.
𝑒. Angles are defined as follows: 1 degree = 1∘ = π/180 rad, 1 (first) minute = 1 󸀠 = 1°/60, 1 second
(minute) = 1 󸀠󸀠 = 1 󸀠 /60. The ancient units ‘third minute’ and ‘fourth minute’, each 1/60th of the
preceding, are not in use any more. (‘Minute’ originally means ‘very small’, as it still does in
modern English.)

Some properties of nature at large are listed in the following table. (If you want a chal-
308 a units, measurements and constants

Challenge 428 s lenge, can you determine whether any property of the universe itself is listed?)

TA B L E 14 Cosmological constants.

Q ua nt it y Symbol Va l u e

Cosmological constant Λ c. 1 ⋅ 10−52 m−2
Age of the universe 𝑎 𝑡0 4.333(53) ⋅ 1017 s = 13.8(0.1) ⋅ 109 a
(determined from space-time, via expansion, using general relativity)
Age of the universe 𝑎 𝑡0 over 3.5(4) ⋅ 1017 s = 11.5(1.5) ⋅ 109 a
(determined from matter, via galaxies and stars, using quantum theory)
Hubble parameter 𝑎 𝐻0 2.3(2) ⋅ 10−18 s−1 = 0.73(4) ⋅ 10−10 a−1
= ℎ0 ⋅ 100 km/s Mpc = ℎ0 ⋅ 1.0227 ⋅ 10−10 a−1
Reduced Hubble parameter 𝑎 ℎ0 0.71(4)
Deceleration parameter 𝑎 ̈ 0 /𝐻02 −0.66(10)
𝑞0 = −(𝑎/𝑎)
Universe’s horizon distance 𝑎 𝑑0 = 3𝑐𝑡0 40.0(6) ⋅ 1026 m = 13.0(2) Gpc

Motion Mountain – The Adventure of Physics
Universe’s topology trivial up to 1026 m
Number of space dimensions 3, for distances up to 1026 m
2
Critical density 𝜌c = 3𝐻0 /8π𝐺 ℎ20 ⋅ 1.878 82(24) ⋅ 10−26 kg/m3
of the universe = 0.95(12) ⋅ 10−26 kg/m3
𝑎
(Total) density parameter Ω0 = 𝜌0 /𝜌c 1.02(2)
𝑎
Baryon density parameter ΩB0 = 𝜌B0 /𝜌c 0.044(4)
Cold dark matter density parameter 𝑎 ΩCDM0 = 𝜌CDM0 /𝜌c 0.23(4)
Neutrino density parameter 𝑎 Ω𝜈0 = 𝜌𝜈0 /𝜌c 0.001 to 0.05
Dark energy density parameter 𝑎 ΩX0 = 𝜌X0 /𝜌c 0.73(4)

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Dark energy state parameter 𝑤 = 𝑝X /𝜌X −1.0(2)
Baryon mass 𝑚b 1.67 ⋅ 10−27 kg
Baryon number density 0.25(1) /m3
Luminous matter density 3.8(2) ⋅ 10−28 kg/m3
Stars in the universe 𝑛s 1022±1
Baryons in the universe 𝑛b 1081±1
𝑏
Microwave background temperature 𝑇0 2.725(1) K
Photons in the universe 𝑛𝛾 1089
Photon energy density 𝜌𝛾 = π2 𝑘4 /15𝑇04 4.6 ⋅ 10−31 kg/m3
Photon number density 410.89 /cm3 or 400 /cm3 (𝑇0 /2.7 K)3
Density perturbation amplitude √𝑆 5.6(1.5) ⋅ 10−6
Gravity wave amplitude √𝑇 < 0.71√𝑆
Mass fluctuations on 8 Mpc 𝜎8 0.84(4)
Scalar index 𝑛 0.93(3)
Running of scalar index d𝑛/d ln 𝑘 −0.03(2)
Planck length 𝑙Pl = √ℏ𝐺/𝑐3 1.62 ⋅ 10−35 m
Planck time 𝑡Pl = √ℏ𝐺/𝑐5 5.39 ⋅ 10−44 s
Planck mass 𝑚Pl = √ℏ𝑐/𝐺 21.8 μg
a units, measurements and constants 309

TA B L E 14 (Continued) Cosmological constants.

Q ua nt it y Symbol Va l u e

Instants in history 𝑎 𝑡0 /𝑡Pl 8.7(2.8) ⋅ 1060
Space-time points 𝑁0 = (𝑅0 /𝑙Pl )3 ⋅ 10244±1
inside the horizon 𝑎 (𝑡0 /𝑡Pl )
Mass inside horizon 𝑀 1054±1 kg

𝑎. The index 0 indicates present-day values.
𝑏. The radiation originated when the universe was 380 000 years old and had a temperature of
about 3000 K; the fluctuations Δ𝑇0 which led to galaxy formation are today about 16 ± 4 μK =
Page 231 6(2) ⋅ 10−6 𝑇0 .

Useful numbers

Motion Mountain – The Adventure of Physics
π 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105
e 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959
γ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923
Ref. 314
ln 2 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255
ln 10 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772
√10 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216

Challenge 1, page 10: Do not hesitate to be demanding and strict. The next edition of the text
will benefit from it.
Challenge 2, page 15: A cone or a circular hyperboloid also looks straight from ‘all’ directions,
provided the positioning of the eye is suitably chosen. Therefore, to check planarity, we need not
only to turn the object, but also to displace it. The best method to check planarity is to use inter-
ference between an arriving and a departing beam of coherent light* with a diameter that covers

Motion Mountain – The Adventure of Physics
Vol. III, page 180 the whole object. If the interference fringes in such an interferogram are straight, the surface is
planar.
Challenge 3, page 16: A finite fraction of infinity is still infinite. Infinity cannot be used as a unit.
Challenge 4, page 17: The time at which the Moon Io enters the shadow in the second meas-
urement occurs about 1000 s later than predicted from the first measurement. Since the Earth is
about 3 ⋅ 1011 m further away from Jupiter and Io, we get the usual value for the speed of light.
Challenge 5, page 18: The rain and wind diagrams in Figure 4 suggest to use the tangent in equa-
tion (2). This is the Galilean expression; however, for light, it would imply velocities above 𝑐, and
thus cannot be correct. The light diagrams suggest to use the sine. This is the correct relativ-
istic expression for the special case of a star precisely above the ecliptic. More general relativistic

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
expressions, for stars of general declination, are easily derived.
Challenge 6, page 18: To compensate for the aberration, the telescope has to be inclined along
the direction of motion of the Earth; to compensate for parallax, perpendicularly to the motion.
Challenge 7, page 19: The drawing shows it. Observer, Moon and Sun form a triangle. When the
Moon is half full, the angle at the Moon is a right angle. Thus the distance ratio can be determined,
though not easily, as the angle at the observer is very close to a right angle as well.
Challenge 8, page 19: There are Cat’s-eyes on the Moon deposited there during the Apollo and
Page 168 Lunokhod missions. They are used to reflect laser 35 ps light pulses sent there through telescopes.
The timing of the round trip then gives the distance to the Moon. Of course, absolute distance is
not know to high precision, but the variations are. The thickness of the atmosphere is the largest
source of error. See the www.csr.utexas.edu/mlrs and ilrs.gsfc.nasa.gov websites.
Challenge 9, page 19: Fizeau used a mirror about 8.6 km away. As the picture shows, he only
had to count the teeth of his cog-wheel and measure its rotation speed when the light goes in
one direction through one tooth and comes back to the next.
Challenge 10, page 20: The shutter time must be shorter than 𝑇 = 𝑙/𝑐, in other words, shorter
than 30 ps; it was a gas shutter, not a solid one. It was triggered by a red light pulse (shown in the
photograph) timed by the pulse to be photographed; for certain materials, such as the used gas,
strong light can lead to bleaching, so that they become transparent. For more details about the

* Generally speaking, two light beams – or any other two waves – are called coherent if they have constant
phase difference and frequency. Coherence enables and is required for interference.
challenge hints and solutions 311

shutter and its neat trigger technique, see the paper by the authors. For even faster shutters, see
also the discussion in volume VI, on page 120.
Challenge 11, page 21: Just take a photograph of a lightning while moving the camera horizont-
ally. You will see that a lightning is made of several discharges; the whole shows that lightning is
much slower than light.
If lightning moved only nearly as fast as light itself, the Doppler effect would change its colour
depending on the angle at which we look at it, compared to its direction of motion. A nearby
lightning would change colour from top to bottom.
Challenge 12, page 23: The fastest lamps were subatomic particles, such as muons, which decay
by emitting a photon, thus a tiny flash of light. However, also some stars emit fasts jets of matter,
which move with speeds comparable to that of light.
Challenge 13, page 24: The speed of neutrinos is the same as that of light to 9 decimal digits,
since neutrinos and light were observed to arrive together, within 12 seconds of each other, after
Ref. 44 a trip of 170 000 light years from a supernova explosion.
Challenge 14, page 25: Even the direction of the arriving light pulse is hard to measure before
it arrives. But maybe one could play on the surface of a black hole? Or under water? Or using a

Motion Mountain – The Adventure of Physics
mirror as tennis court? Enjoy the exploration.
Challenge 16, page 29: This is best discussed by showing that other possibilities make no sense.
Challenge 17, page 29: The spatial coordinate of the event at which the light is reflected is 𝑐(𝑘2 −
1)𝑇/2; the time coordinate is (𝑘2 + 1)𝑇/2. Their ratio must be 𝑣. Solving for 𝑘 gives the result.
Challenge 19, page 31: The motion of radio waves, infrared, ultraviolet and gamma rays is also
unstoppable. Another past suspect, the neutrino, has been found to have mass and to be thus in
principle stoppable. The motion of gravity is also unstoppable.
Challenge 21, page 35: 𝜆 R /𝜆 S = 𝛾.
Challenge 22, page 35: To change from bright red (650 nm) to green (550 nm), 𝑣 = 0.166𝑐 is

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
necessary.
Challenge 23, page 35: People measure the shift of spectral lines, such as the shift of the so-called
Lyman-𝛼 line of hydrogen, that is emitted (or absorbed) when a free electron is captured (or
Vol. IV, page 180 ejected) by a proton. It is one of the famous Fraunhofer lines.
Challenge 24, page 35: The speeds are given by

(𝑧 + 1)2 − 1
𝑣/𝑐 = (299)
(𝑧 + 1)2 + 1
which implies 𝑣(𝑧 = −0.1) = 31 Mm/s = 0.1𝑐 towards the observer and 𝑣(𝑧 = 5) = 284 Mm/s =
0.95𝑐 away from the observer.
A red-shift of 6 implies a speed of 0.96𝑐; such speeds appear because, as we will see in the
section of general relativity, far away objects recede from us. And high red-shifts are observed
only for objects which are extremely far from Earth, and the faster the further they are away. For
a red-shift of 6 that is a distance of several thousand million light years.
Challenge 25, page 36: No Doppler effect is seen for a distant observer at rest with respect to the
large mass. In other cases there obviously is a Doppler effect, but it is not due to the deflection.
Challenge 26, page 36: Sound speed is not invariant of the speed of observers. As a result, the
Doppler effect for sound even confirms – within measurement differences – that time is the same
for observers moving against each other.
Challenge 29, page 38: Inside colour television tubes (they used higher voltages, typically 30 kV,
than black and white ones did), electrons are described by 𝑣/𝑐 ≈ √2 ⋅ 30/511 or 𝑣 ≈ 0.3𝑐.
312 challenge hints and solutions

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
F I G U R E 126 The original lines published by Fraunhofer and the meridian instrument that he used
(© Fraunhofer Gesellschaft).

Challenge 30, page 38: If you can imagine this, publish it. Readers will be delighted to hear the
story.
Challenge 32, page 39: The connection between observer invariance and limit property seems to
Vol. VI, page 27 be generally valid in nature, as shown in chapter 2. However, a complete and airtight argument
is not yet at hand. If you have one, publish it!
Challenge 35, page 42: If the speed of light is the same for all observers, no observer can pretend
to be more at rest than another (as long as space-time is flat), because there is no observation from
electrodynamics, mechanics or another part of physics that allows such a statement.
Challenge 39, page 43: The human value is achieved in particle accelerators; the value in nature
is found in cosmic rays of the highest energies.
challenge hints and solutions 313

home home
time time
in years in years away
time
in years
away
away away twin
home twin home twin
twin twin
home
twin

turn- turn-
around around turn-
around

Motion Mountain – The Adventure of Physics
away
space
home space home space in light-
in light-years in light-years years

F I G U R E 127 The twin paradox: (left and centre) the clock timing for both twins with the signals sent

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
among the twins in the inertial frame of the home twin, and (right) the description by the away twin, in
a frame that, however, is not inertial.

Challenge 41, page 44: Redrawing Figure 11 on page 29 for the other observer makes the point.
Challenge 42, page 45: The set of events behaves like a manifold, because it behaves like a four-
dimensional space: it has infinitely many points around any given starting point, and distances
behave as we are used to, limits behave as we are used to. It differs by one added dimension, and
by the sign in the definition of distance; thus, properly speaking, it is a Riemannian manifold.
Challenge 43, page 46: Infinity is obvious, as is openness. Thus the topology equivalence can be
shown by imagining that the manifold is made of rubber and wrapped around a sphere.
Challenge 44, page 46: The light cone remains unchanged; thus causal connection as well.
Challenge 47, page 47: In such a case, the division of space-time around an inertial observer
into future, past and elsewhere would not hold any more, and the future could influence the past
(as seen from another observer).
Challenge 53, page 50: To understand the twin paradox, the best way is to draw a space-time
diagram showing how each twin sends a time signal at regular intervals, as seen on his own clock,
to his brother. Some examples are given in Figure 127. These time signals show how much he has
aged. You will see directly that, during the trip, one twin sends fewer signals than the other.
Challenge 54, page 51: The ratio predicted by naive reasoning is (1/2)(6.4/2.2) = 0.13.
Challenge 55, page 51: The time dilation factor for 𝑣 = 0.9952𝑐 is 10.2, giving a proper time of
0.62 μs; thus the ratio predicted by special relativity is (1/2)(0.62/2.2) = 0.82.
314 challenge hints and solutions

Challenge 57, page 51: Send a light signal from the first clock to the second clock and back.
Take the middle time between the departure and arrival, and then compare it with the time at
the reflection. Repeat this a few times. See also Figure 11.
Challenge 59, page 52: Not with present experimental methods.
Challenge 60, page 52: Hint: think about different directions of sight.
Challenge 62, page 53: Hint: be careful with the definition of ‘rigidity’.
Challenge 64, page 53: While the departing glider passes the gap, the light cannot stay on at any
speed, if the glider is shorter than the gap. This is strange at first sight, because the glider does
not light the lamp even at high speeds, even though in the frame of the glider there is contact
at both ends. The reason is that in this case there is not enough time to send the signal to the
battery that contact is made, so that the current cannot start flowing.
Assume that current flows with speed 𝑢, which is of the order of 𝑐. Then, as Dirk Van de
Moortel showed, the lamp will go off if the glider length 𝑙 and the gap length 𝑑 obey 𝑙/𝑑 < 𝛾(𝑢 +
𝑣)/𝑢. See also the cited reference.
For a glider approaching the gap and the lamp, the situation is different: a glider shorter than
the gap can keep the lamp on all the time, as pointed out by Madhu Rao.

Motion Mountain – The Adventure of Physics
Why are the debates often heated? Some people will (falsely) pretend that the problem is un-
physical; other will say that Maxwell’s equations are needed. Still others will say that the problem
is absurd, because for larger lengths of the glider, the on/off answer depends on the precise speed
value. However, this actually is the case in this situation.
Challenge 65, page 54: Yes, the rope breaks; in accelerated cars, distance changes, as shown later
on in the text.
Challenge 66, page 54: The submarine will sink. The fast submarine will even be heavier, as his
kinetic energy adds to his weight. The contraction effect would make it lighter, as the captain
says, but by a smaller amount. The total weight – counting upwards as positive – is given by
𝐹 = −𝑚𝑔(𝛾 − 1/𝛾).

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 67, page 54: A relativistic submarine would instantly melt due to friction with the
water. If not, it would fly off the planet because it would move faster than the escape velocity –
and would produce several other disasters.
Challenge 68, page 55: A relativistic pearl necklace at constant speed does get shorter, but as
usual, the shortening can only be measured, not photographed. At relativistic speeds, the meas-
ured sizes of the pearls are flattened ellipsoids. Spheres do not get transformed into spheres. The
observed necklace consists of overlapping elliposids.
Challenge 69, page 55: No: think about it!
Challenge 72, page 58: Yes, ageing in a valley is slowed compared to mountain tops. However,
the proper sensation of time is not changed. The reason for the appearance of grey hair is not
known; if the timing is genetic, the proper time at which it happens is the same in either location.
Challenge 73, page 58: There is no way to put an observer at the specified points. Proper velocity
can only be defined for observers, i.e., for entities which can carry a clock. That is not the case
for images.
Challenge 74, page 59: Just use plain geometry to show this.
Challenge 75, page 60: Most interestingly, the horizon can easily move faster than light, if you
move your head appropriately, as can the end of the rainbow.
Challenge 77, page 63: Light is necessary to determine distance and to synchronize clocks; thus
there is no way to measure the speed of light from one point to another alone. The reverse motion
needs to be included. However, some statements on the one-way speed of light can still be made
challenge hints and solutions 315

(see math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html). All experiments on the
one-way speed of light performed so far are consistent with an isotropic value that is equal to
the two-way velocity. However, no experiment is able to rule out a group of theories in which
the one-way speed of light is anisotropic and thus different from the two-way speed. All theories
from this group have the property that the round-trip speed of light is isotropic in any inertial
frame, but the one-way speed is isotropic only in a preferred ‘ether’ frame. In all of these theories,
in all inertial frames, the effects of slow clock transport exactly compensate the effects of the
anisotropic one-way speed of light. All these theories are experimentally indistinguishable from
special relativity. In practice, therefore, the one-way speed of light has been measured and is
constant. But a small option remains.
The subtleties of the one-way and two-way speed of light have been a point of discussion
for a long time. It has been often argued that a factor different than two, which would lead to a
distinction between the one-way speed of light and the two-way speed of light, cannot be ruled
out by experiment, as long as the two-way speed of light remains 𝑐 for all observers.
Ref. 18 Many experiments on the one-way velocity of light are explained and discussed by Zhang.
He says in his summary on page 171, that the one-way velocity of light is indeed independent
of the light source; however, no experiment really shows that it is equal to the two-way velocity.

Motion Mountain – The Adventure of Physics
Ref. 78 Moreover, almost all so-called ‘one-way’ experiments are in fact still hidden ‘two-way’ experi-
ments (see his page 150).
Ref. 79 In 2004, Hans Ohanian showed that the question can be settled by discussing how a non-
standard one-way speed of light would affect dynamics. He showed that a non-standard one-way
speed of light would introduce pseudoaccelerations and pseudoforces (similar to the Coriolis
acceleration and force); since these pseudoaccelerations and pseudoforces are not observed, the
one-way speed of light is the same as the two-way speed of light.
In short, the issues of the one-way speed of light do not need to worry us here.
Challenge 78, page 65: The expression does not work for a photon hitting a mirror, for example.
Challenge 79, page 65: Teleportation contradicts, in an inertial reference frame, the conserva-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
tion of the centre of mass. Quick teleportation would lead to strong acceleration of the sending
and the receiving environment.
Challenge 85, page 68: The lower collision in Figure 41 shows the result directly, from energy
conservation. For the upper collision the result also follows, if one starts from momentum con-
servation 𝛾𝑚𝑣 = Γ𝑀𝑉 and energy conservation (𝑔𝑎𝑚𝑚𝑎 + 1)𝑚 = Γ𝑀.
Challenge 97, page 74: Just turn the left side of Figure 45 a bit in anti-clockwise direction.
Challenge 98, page 75: In collisions between relativistic charges, part of the energy is radiated
away as light, so that the particles effectively lose energy.
Challenge 99, page 76: Probably not, as all relations among physical quantities are known now.
However, you might check for yourself; one might never know. It is worth to mention that the
maximum force in nature was discovered (in this text) after remaining hidden for over 80 years.
Challenge 101, page 79: Write down the four-vectors 𝑈󸀠 and 𝑈 and then extract 𝑣󸀠 as function
of 𝑣 and the relative coordinate speed 𝑉. Then rename the variables.
Challenge 102, page 79: No example of motion of a massive body has! Only the motion of light
waves has null phase 4-velocity and null group 4-velocity, as explained on page 86.
Challenge 107, page 82: For ultrarelativistic particles, like for massless particles, one has 𝐸 = 𝑝𝑐.
Challenge 108, page 82: Hint: evaluate 𝑃1 and 𝑃2 in the rest frame of one particle.
Challenge 110, page 83: Use the definition 𝐹 = d𝑝/d𝑡 and the relation 𝐾𝑈 = 0 = 𝐹𝑣 − d𝐸/d𝑡
valid for rest-mass preserving forces.
Challenge 112, page 84: The story is told on page 108.
316 challenge hints and solutions

Challenge 116, page 85: This problem is called the Ehrenfest paradox. There are many publica-
tions about it. Enjoy the exploration!
Challenge 117, page 85: Yes, one can see such an object: the searchlight effect and the Doppler
effect do not lead to invisibility. However, part of the object, namely the region rotating away
from the observer, may become very dark.
Challenge 119, page 86: If the rotating particle has a magnetic moment, one can send it through
an inhomogeneous magnetic field and observe whether the magnetic moment changes direction.
Challenge 121, page 86: No.
Challenge 122, page 86: For a discussion of relativistic angular momentum and a pretty effect,
see K. Y. Bliokh & F. Nori, Relativistic Hall Effect, Physical Review Letters 108, p. 120403,
2012, preprint at arxiv.org/abs/1112.5618.
Challenge 125, page 86: The relation for the frequency follows from the definition of the phase.
Challenge 128, page 88: Planck invited Einstein to Berlin and checked his answers with him.
The expression 𝐸 = ℏ𝜔 for the photon energy implies the invariance of ℏ.
Challenge 144, page 96: The energy contained in the fuel must be comparable to the rest mass
of the motorbike, multiplied by 𝑐2 . Since fuel contains much more mass than energy, that gives

Motion Mountain – The Adventure of Physics
a big problem.
Challenge 146, page 96: Constant acceleration and gravity are similar in their effects, as dis-
cussed in the section on general relativity.
Challenge 149, page 98: Yes, it is true.
Challenge 150, page 98: It is flat, like a plane.
Challenge 151, page 98: Despite the acceleration towards the centre of the carousel, no horizon
appears.
Challenge 153, page 99: Yes; however, the effect is minimal and depends on the position of the
Sun. In fact, what is white at one height is not white at another.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 155, page 100: Locally, light always moves with speed 𝑐.
Challenge 156, page 100: Away from Earth, 𝑔 decreases; it is effectively zero over most of the
distance.
Challenge 159, page 102: As shown in the cited reference, the limit follows from the condition
𝑙𝛾3 𝑎 ⩽ 𝑐2 .
Challenge 161, page 102: Yes.
Challenge 162, page 102: Yes. Take Δ𝑓 Δ𝑡 ⩾ 1 and substitute Δ𝑙 = 𝑐/Δ𝑓 and Δ𝑎 = 𝑐/Δ𝑡.
Challenge 164, page 105: Though there are many publications pretending to study the issue,
there are also enough physicists who notice the impossibility. Measuring a variation of the speed
of light is not much far from measuring the one way speed of light: it is not possible. However,
the debates on the topic are heated; the issue will take long to be put to rest.
Challenge 166, page 107: The inverse square law of gravity does not comply with the maximum
speed principle; it is not clear how it changes when one changes to a moving observer.
Challenge 167, page 112: If you hear about a claim to surpass the force or power limit, let me
know.
Challenge 168, page 112: Take a surface moving with the speed of light, or a surface defined
with a precision smaller than the Planck length.
Challenge 169, page 117: Also shadows do not remain parallel on curved surfaces. Forgetting
this leads to strange mistakes: many arguments allegedly ‘showing’ that men have never been on
the moon neglect this fact when they discuss the photographs taken there.
challenge hints and solutions 317

Challenge 170, page 120: If you find one, publish it and then send it to me.
Challenge 172, page 125: This is tricky. Simple application of the relativistic transformation rule
for 4-vectors can result in force values above the limit. But in every such case, a horizon has
appeared that prevents the observation of this higher value.
Challenge 173, page 125: If so, publish it; then send it to me.
Challenge 174, page 127: For example, it is possible to imagine a surface that has such an intric-
ate shape that it will pass all atoms of the universe at almost the speed of light. Such a surface is
not physical, as it is impossible to imagine observers on all its points that move in that way all at
the same time.
Challenge 176, page 127: Publish it – and then send it to me.
Challenge 177, page 128: New sources cannot appear from nowhere. Any ‘new’ power source
results from the transformation of other radiation found in the universe already before the ap-
pearance.
Challenge 178, page 128: Many do not believe the limits yet; so any proposed counter-example
or any additional paradox is worth a publication.

Motion Mountain – The Adventure of Physics
Challenge 181, page 133: If so, publish it; then send it to me.
Challenge 185, page 135: If so, publish it; then send it to me.
Challenge 187, page 137: They are accelerated upwards.
Challenge 188, page 137: In everyday life, (a) the surface of the Earth can be taken to be flat, (b)
the vertical curvature effects are negligible, and (c) the lateral length effects are negligible.
Challenge 192, page 138: For a powerful bus, the acceleration is 2 m/s2 ; in 100 m of acceleration,
this makes a relative frequency change of 2.2 ⋅ 10−15 .
Challenge 193, page 139: Yes, light absorption and emission are always lossless conversions of
energy into mass.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 196, page 140: For a beam of light, in both cases the situation is described by an en-
vironment in which masses ‘fall’ against the direction of motion. If the Earth and the train walls
were not visible – for example if they were hidden by mist – there would not be any way to de-
termine by experiment which situation is which. Or again, if an observer would be enclosed in a
box, he could not distinguish between constant acceleration or constant gravity. (Important: this
impossibility only applies if the observer has negligible size!)
Challenge 200, page 141: Length is time times the speed of light. If time changes with height, so
do lengths.
Challenge 202, page 141: Both fall towards the centre of the Earth. Orbiting particles are also in
free fall; their relative distance changes as well, as explained in the text.
Challenge 205, page 144: Such a graph would need four or even 5 dimensions.
Challenge 206, page 145: The experiments about change of time with height can also be used in
this case.
Challenge 208, page 146: The energy due to the rotation can be neglected compared with all
other energies in the problem.
Challenge 218, page 151: Different nucleons, different nuclei, different atoms and different mo-
lecules have different percentages of binding energies relative to the total mass.
Challenge 220, page 153: In free fall, the bottle and the water remain at rest with respect to each
other.
Challenge 221, page 153: Let the device fall. The elastic rubber then is strong enough to pull the
ball into the cup. See M. T. Westra, Einsteins verjaardagscadeau, Nederlands tijdschrift voor
318 challenge hints and solutions

natuurkunde 69, p. 109, April 2003. The original device also had a spring connected in series to
the rubber.
Challenge 222, page 153: Apart from the chairs and tables already mentioned, important anti-
gravity devices are suspenders, belts and plastic bags.
Challenge 224, page 154: The same amount.
Challenge 225, page 154: Yes, in gravity the higher twin ages more. The age difference changes
with height, and reaches zero for infinite height.
Challenge 226, page 154: The mass flow limit is 𝑐3 /4𝐺.
Challenge 227, page 154: No, the conveyer belt can be built into the train.
Challenge 228, page 154: They use a spring scale, and measure the oscillation time. From it they
deduce their mass. NASA’s bureaucracy calls it a BMMD, a body mass measuring device. A
Vol. I, page 108 photograph is found in the first volume.
Challenge 229, page 154: The apple hits the wall after about half an hour.
Challenge 232, page 155: Approaches with curved light paths, or with varying speed of light do
not describe horizons properly.

Motion Mountain – The Adventure of Physics
Challenge 233, page 155: With ℏ as smallest angular momentum one get about 100 Tm.
Challenge 234, page 155: No. The diffraction of the beams does not allow it. Also quantum the-
ory makes this impossible; bound states of massless particles, such as photons, are not stable.
Challenge 236, page 156: The orbital radius is 4.2 Earth radii; that makes c. 38 μs every day.
Challenge 237, page 157: To be honest, the experiments are not consistent. They assume that
some other property of nature is constant – such as atomic size – which in fact also depends on
𝐺. More on this issue on page 292.
Challenge 238, page 157: Of course other spatial dimensions could exist which can be detected
only with the help of measurement apparatuses. For example, hidden dimensions could appear

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
at energies not accessible in everyday life.
Challenge 239, page 157: On this tiny effect, see the text by Ohanian, Ref. 113, on page 147.
Challenge 262, page 172: Since there is no negative mass, gravitoelectric fields cannot be neut-
ralized. In contrast, electric fields can be neutralized around a metallic conductor with a Faraday
cage.
Challenge 265, page 174: To find the answer, thinking about the electromagnetic analogy helps.
Challenge 274, page 182: One needs to measure the timing of pulses which cross the Earth at
different gravitational wave detectors on Earth.
Challenge 248, page 162: They did so during a solar eclipse.
Challenge 275, page 183: No. For the same reasons that such a electrostatic field is not possible.
Challenge 280, page 186: No, a line cannot have intrinsic curvature. A torus is indeed intrinsic-
ally curved; it cannot be cut open to yield a flat sheet of paper.
Challenge 285, page 188: No, they cannot be made from a sheet of paper. The curvature is
nonzero everywhere.
Challenge 302, page 196: The trace of the Einstein tensor is the negative of the Ricci scalar; it is
thus the negative of the trace of the Ricci tensor.
Challenge 306, page 198: The concept of energy makes no sense for the universe, as the concept
is only defined for physical systems, and thus not for the universe itself. See also page 256.
Challenge 313, page 204: Indeed, in general relativity gravitational energy cannot be localized
in space, in contrast to what one expects and requires from an interaction.
challenge hints and solutions 319

Challenge 324, page 208: Errors in the south-pointing carriage are due to the geometric phase,
an effect that appears in any case of parallel transport in three dimensions. It is the same effect
that makes Foucault’s pendulum turn. Parallel transport is sometimes also called Fermi-Walker
Vol. III, page 139 transport. The geometric phase is explained in detail in the volume on optics.
Challenge 328, page 210: The European Space Agency is exploring the issue. Join them!
Challenge 331, page 220: There is a good chance that some weak form of Sun jets exist; but a
detection will not be easy. (The question whether the Milky Way has jets was part of this text
since 2006; they have been discovered in 2010.)
Challenge 333, page 224: If you believe that the two amounts differ, you are prisoner of a belief,
namely the belief that your ideas of classical physics and general relativity allow you to extrapolate
these ideas into domains where they are not valid, such as behind a horizon. At every horizon,
quantum effects are so strong that they invalidate such classical extrapolations.
Challenge 334, page 224: A few millimetres.
Challenge 335, page 226: If we assume a diameter of 150 μm and a density of 1000 kg/m3 for
the flour particles, then there are about 566 million particles in one kg of flour. A typical galaxy
contains 1011 stars; that corresponds to 177 kg of flour.

Motion Mountain – The Adventure of Physics
Challenge 336, page 226: Speed is measured with the Doppler effect, usually by looking at the
Lyman-alpha line. Distance is much more difficult to explain. Measuring distances is a science on
its own, depending on whether one measures distances of stars in the galaxy, to other galaxies,
or to quasars. Any book on astronomy or astrophysics will tell more.
Challenge 337, page 226: See the challenge on page 234.
Challenge 339, page 234: The rabbit observes that all other rabbits seem to move away from him.
Challenge 347, page 240: Stand in a forest in winter, and try to see the horizon. If the forest is
very deep, you hit tree trunks in all directions. If the forest is finite in depth, you have chance to
see the horizon.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Challenge 361, page 255: No. This is an example of how a seemingly exact description of nature
can lead to an unscientific statement, a belief, without any relation to reality.
Challenge 362, page 255: Again no. The statement is a pure belief.
Challenge 364, page 257: The universe does not allow observation from outside. It thus has no
state properties.
Challenge 375, page 265: At the horizon, light cannot climb upwards.
Challenge 399, page 277: This happens in the same way that the static electric field comes out
of a charge. In both cases, the transverse fields do not get out, but the longitudinal fields do.
Quantum theory provides the deeper reason. Real radiation particles, which are responsible for
free, transverse fields, cannot leave a black hole because of the escape velocity. However, virtual
particles can, as their speed is not bound by the speed of light. All static, longitudinal fields are
produced by virtual particles. In addition, there is a second reason. Classical field can come out
of a black hole because for an outside observer everything that constitutes the black hole is con-
tinuously falling, and no constituent has actually crossed the horizon. The field sources thus are
not yet out of reach.
Challenge 403, page 278: The description says it all. A visual impression can be found in the
room on black holes in the ‘Deutsches Museum’ in Munich.
Challenge 405, page 278: On the one hand, black holes can occur through collapse of matter.
On the other hand, black holes can be seen as a curved horizon.
Challenge 407, page 279: So far, it seems that all experimental consequences from the analogy
match observations; it thus seems that we can claim that the night sky is a black hole horizon.
320 challenge hints and solutions

Nevertheless, the question is not settled, and some prominent physicists do not like the analogy.
The issue is also related to the question whether nature shows a symmetry between extremely
large and extremely small length scales. This topic is expanded in the last leg of our adventure.
Challenge 411, page 282: Any device that uses mirrors requires electrodynamics; without elec-
trodynamics, mirrors are impossible.
Challenge 413, page 286: The hollow Earth theory is correct if usual distances are consistently
changed according to 𝑟he = 𝑅2Earth /𝑟. This implies a quantum of action that decreases towards the
centre of the hollow sphere. Then there is no way to prefer one description over the other, except
for reasons of simplicity.
Challenge 417, page 297: Mass is a measure of the amount of energy. The ‘square of mass’ makes
no sense.
Challenge 420, page 299: Probably the quantity with the biggest variation is mass, where a prefix
for 1 eV/c2 would be useful, as would be one for the total mass in the universe, which is about
1090 times larger.
Challenge 421, page 300: The formula with 𝑛 − 1 is a better fit. Why?
Challenge 424, page 301: No! They are much too precise to make sense. They are only given as

Motion Mountain – The Adventure of Physics
an illustration for the behaviour of the Gaussian distribution. Real measurement distributions
are not Gaussian to the precision implied in these numbers.
Challenge 425, page 301: About 0.33 m/s. It is not 0.333 m/s and it is not any longer strings of
threes!
Challenge 427, page 307: The slowdown goes quadratically with time, because every new slow-
down adds to the old one!
Challenge 428, page 308: No, only properties of parts of the universe are listed. The universe
Vol. VI, page 112 itself has no properties, as shown in detail in the last part of this adventure.
Challenge 429, page 349: This could be solved with a trick similar to those used in the irration-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ality of each of the two terms of the sum, but nobody has found one.
Challenge 430, page 349: There are still discoveries to be made in modern mathematics, espe-
cially in topology, number theory and algebraic geometry.
BI BL IO G R A PH Y

“
A man will turn over half a library to make one

”
book.
Samuel Johnson*

1 Aristotle, On sense and the sensible, section 1, part 1, 350 bce. Cited in Jean-
Paul Dumont, Les écoles présocratiques, Folio Essais, Gallimard, p. 157, 1991. Cited on

Motion Mountain – The Adventure of Physics
page 15.
2 The original Latin text of Descartes’ letter, dated 22 August 1634, can be read online on
gallica.bnf.fr/ark:/12148/bpt6k20740j/f419.image. Cited on page 16.
3 Anonyme, Demonstration touchant le mouvement de la lumière trouvé par M. Römer de
l’Academie Royale des Sciences, Journal des Scavans pp. 233–236, 1676. An English summary
is found in O. C. Rømer, A demonstration concerning the motion of light, Philosophical
Transactions of the Royal Society 136, pp. 893–894, 1677. You can read the two papers at
dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Roemer-1677/Roemer-1677.html. Cited on
page 16.
4 F. Tuinstra, Rømer and the finite speed of light, Physics Today 57, pp. 16–17, December

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
2004. Cited on page 17.
5 The history of the measurement of the speed of light can be found in chapter 19 of the text
by Francis A. Jenkins & Harvey E. White, Fundamentals of Optics, McGraw-Hill,
New York, 1957. Cited on page 17.
6 On the way to perform such measurements, see Sydney G. Brewer, Do-it-yourself As-
tronomy, Edinburgh University Press, 1988. Kepler himself never measured the distances of
planets to the Sun, but only ratios of planetary distances. The parallax of the Sun from two
points of the Earth is at most 8.79 󸀠󸀠 ; it was first measured in the eighteenth century. Cited
on page 19.
7 For a detailed discussion, see note 32 of A. Gualandi & F. B ònoli, The search for stellar
parallaxes and the discovery of the aberration of light: the observational proofs of the Earth’s
revolution, Eustachio Manfredi, and the ‘Bologna case’, Journal for the History of Astro-
nomy 20, pp. 155–172, 2009. Indeed, the term ‘aberration’ does not appear in Bradley’s
publication on the issue. Cited on page 18.
8 Aristarchus of Samos, On the sizes and the distances of the Sun and the Moon, c. 280
bce, in Michael J. Crowe, Theories of the World From Antiquity to the Copernican Re-
volution, Dover, 1990. Cited on page 19.
9 J. Frercks, Creativity and technology in experimentation: Fizeau’s terrestrial determin-
ation of the speed of light, Centaurus 42, pp. 249–287, 2000. See also the beautiful web-

* Samuel Johnson (1709–1784), famous English poet and intellectual.
322 bibliography

site on reconstrutions of historical science experiments at www.uni-oldenburg.de/histodid/
forschung/nachbauten. Cited on page 19.
10 The way to make pictures of light pulses with an ordinary photographic camera, without
any electronics, is described by M. A. Duguay & A. T. Mattick, Ultrahigh speed pho-
tography of picosecond light pulses and echoes, Applied Optics 10, pp. 2162–2170, 1971. The
picture on page 20 is taken from it. Cited on page 20.
11 You can learn the basics of special relativity with the help of the web; the simplest
and clearest introduction is part of the Karlsruhe physics course, downloadable at www.
physikdidaktik.uni-karlsruhe.de. You can also use the physics.syr.edu/research/relativity/
RELATIVITY.html web page as a starting point; the page mentions many of the English-
language relativity resources available on the web. Links in other languages can be found
with search engines. Cited on page 22.
12 See the classic paper G. J. Fishman, Gamma ray observations of the Crab pulsar – past,
present, future, Annals of the New York Academy of Sciences 655, pp. 309–318, 1992. Cited
on page 22.
13 Observations of gamma-ray bursts show that the speed of light does not depend on the lamp

Motion Mountain – The Adventure of Physics
speed to within one part in 1020 , as shown by the well-known paper by B. E. Schaefer,
Severe limits on variations of the speed of light with frequency, Physical Review Letters 82,
pp. 4964–4966, 1999, preprint at arxiv.org/abs/astro-ph/9810479. The result was confirmed
most impressively with a bright gamma-ray burst observed in 2009. In this gamma-ray
burst, after travelling for ten thousand million years, photons of frequencies that differed
by a factor 3 ⋅ 1010 – gamma rays and visible light – still arrived within less than a second
from each other. See A.A. Abdo & al., (Fermi GBM/LAT collaborations) Testing Einstein’s
special relativity with Fermi’s short hard gamma-ray burst GRB090510, preprint at arxiv.org/
abs/0908.1832. Cited on pages 22, 23, 322, and 324.
14 M. Füllekrug, Probing the speed of light with radio waves at extremely low frequencies,

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Physical Review Letters 93, p. 043901, 2004. Cited on page 23.
15 See, e.g., C. Will, Theory and Experiment in Gravitational Physics, revised edition, Cam-
bridge University Press, 1993. See also Ref. 27. Cited on pages 23 and 27.
16 The discussions using binary stars are W. de Sitter, A proof of the constancy of the
speed of light, Proceedings of the Section of the Sciences – Koninklijke Academie der
Wetenschappen 15, pp. 1297–1298, 1913, W. de Sitter, On the constancy of the speed of
light, Proceedings of the Section of the Sciences – Koninklijke Academie der Wetenschap-
pen 16, pp. 395–396, 1913, W. de Sitter, Ein astronomischer Beweis für die Konstanz der
Lichtgeschwindigkeit, Physikalische Zeitschrift 14, p. 429, 1913, W. de Sitter, Über die
Genauigkeit, innerhalb welcher die Unabhängigkeit der Lichtgeschwindigkeit von der Bewe-
gung der Quelle behauptet werden kann, Physikalische Zeitschrift 14, p. 1267, 1913, For a
more recent version, see K. Brecher, Is the speed of light independent of the velocity of the
source?, Physics Letters 39, pp. 1051–1054, Errata 1236, 1977. Cited on page 23.
17 The strong limits from observations of gamma-ray bursts were explored in Ref. 13.
Measuring the light speed from rapidly moving stars is another possible test; see the
previous reference. Some of these experiments are not completely watertight; a specula-
tion about electrodynamics, due to Ritz, maintains that the speed of light is 𝑐 only when
measured with respect to the source; the light from stars, however, passes through the at-
mosphere, and its speed might thus be reduced to 𝑐.
The famous experiment with light emitted from rapid pions at CERN is not subject to this
criticism. It is described in T. Alväger, J. M. Bailey, F. J. M. Farley, J. Kjellman
& I. Wallin, Test of the second postulate of relativity in the GeV region, Physics Letters 12,
bibliography 323

pp. 260–262, 1964. See also T. Alväger & al., Velocity of high-energy gamma rays, Arkiv
för Fysik 31, pp. 145–157, 1965.
Another precise experiment at extreme speeds is described by G. R. Kalbfleisch,
N. Baggett, E. C. Fowler & J. Alspector, Experimental comparison of neutrino,
anti-neutrino, and muon velocities, Physical Review Letters 43, pp. 1361–1364, 1979. Cited
on page 23.
18 An overview of experimental results is given in Yuan Zhong Zhang, Special Relativity
and its Experimental Foundations, World Scientific, 1998. Cited on pages 23, 32, 40, 51, 68,
315, and 326.
19 The beginning of the modern theory of relativity is the famous paper by Al-
bert Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik 17, pp. 891–
921, 1905. It still well worth reading, and every physicist should have done so. The same
can be said of the famous paper, probably written after he heard of Olinto De Pretto’s idea,
found in Albert Einstein, Ist die Trägheit eines Körpers von seinem Energieinhalt ab-
hängig?, Annalen der Physik 18, pp. 639–641, 1905. See also the review Albert Einstein,
Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen, Jahrbuch der
Radioaktivität und Elektronik 4, pp. 411–462, 1907. These papers are now available in

Motion Mountain – The Adventure of Physics
many languages. A later, unpublished review is available in facsimile and with an English
translation as Albert Einstein, Hanoch Gutfreund, ed., Einstein’s 1912 Manuscript
on the Theory of Relativity, George Braziller, 2004. Also interesting is the later paper Al-
bert Einstein, Elementary derivation of the equivalence of mass and energy, American
Mathematical Society Bulletin 41, pp. 223–230, 1935. All papers and letters of Einstein are
now available, both in their original language and in English translation, at einsteinpapers.
press.princeton.edu. Cited on pages 24, 27, and 76.
20 Jean van Bladel, Relativity and Engineering, Springer, 1984. Cited on page 24.
21 Albert Einstein, Mein Weltbild, edited by Carl Selig, Ullstein Verlag, 1998. Cited

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
on page 26.
22 There is even a book on the topic: Hans C. Ohanian, Einstein’s Mistakes: The Human
Failings of Genius, Norton, 2009. On 26 December 1915, Einstein wrote, in a letter to Paul
Ehrenfest: ‘‘Es ist bequem mit dem Einstein, jedes Jahr widerruft er, was er das vorige Jahr
geschrieben hat. ’’ A few weeks later, on 17 January 1916, Einstein wrote to Hendrik Lorentz:
‘‘Die Serie meiner Gravitationsarbeiten ist eine Kette von Irrwegen, die aber doch allmäh-
lich dem Ziele näher führten.’’ See for, example, the book by Henoch Gutfreund &
Jürgen Renn, editors, The Road to Relativity, Princeton University Press, 2015. Cited on
page 26.
23 Albrecht Fölsing, Albert Einstein – eine Biographie, Suhrkamp, p. 237, 1993. Cited on
pages 27 and 41.
24 Einstein’s beautiful introduction, almost without formulae, is Albert Einstein, Über
die spezielle und allgemeine Relativitätstheorie, Vieweg, 1917 and 1997. For a text with all re-
quired mathematics, see Albert Einstein, The Meaning of Relativity, Methuen, 1921 and
1956. The posthumous edition also contains Einstein’s last printed words on the theory, in
the appendix. See also the German text Albert Einstein, Grundzüge der Relativitätsthe-
orie, Springer, 1921 expanded 1954, and republished 2002, which also contains the relevant
mathematics. Cited on page 27.
25 Julian Schwinger, Einstein’s Legacy, Scientific American, 1986. Edwin F. Taylor
& John A. Wheeler, Spacetime Physics – Introduction to Special Relativity, second edi-
tion, Freeman, 1992. See also Nick M. J. Woodhouse, Special Relativity, Springer, 2003.
Cited on pages 27 and 87.
324 bibliography

26 Wolf gang Rindler, Relativity – Special, General and Cosmological, Oxford University
Press, 2001. This is a beautiful book by one of the masters of the field. Cited on pages 27
and 85.
27 R. J. Kennedy & E. M. Thorndike, Experimental establishment of the relativity of time,
Physical Review 42, pp. 400–418, 1932. See also H. E. Ives & G. R. Stilwell, An experi-
mental study of the rate of a moving atomic clock, Journal of the Optical Society of America
28, pp. 215–226, 1938, and 31, pp. 369–374, 1941. For a modern, high-precision versions,
see C. Braxmeier, H. Müller, O. Pradl, J. Mlynek, A. Peters & S. Schiller,
New tests of relativity using a cryogenic optical resonator, Physical Review Letters 88,
p. 010401, 2002. The newest result is in P. Antonini, M. Okhapkin, E. Göklü &
S. Schiller, Test of constancy of speed of light with rotating cryogenic optical resonat-
ors, Physical Review A 71, p. 050101, 2005, preprint at arxiv.org/abs/gr-qc/0504109, and
the subsequent arxiv.org/abs/physics/0510169. See also P. Antonini, M. Okhapkin,
E. Göklü & S. Schiller, Reply to “Comment on ‘Test of constancy of speed of light with
rotating cryogenic optical resonators’ ”, Physical Review A 72, p. 066102, 2005, preprint at
arxiv.org/abs/physics/0602115. Cited on pages 28 and 322.

Motion Mountain – The Adventure of Physics
28 The slowness of the speed of light inside the Sun is due to the frequent scattering of photons
by solar matter. The most serious estimate is by R. Mitalas & K. R. Sills, On the photon
diffusion time scale for the Sun, The Astrophysical Journal 401, pp. 759–760, 1992, They give
a photon escape time of 0.17 Ma, an average photon free mean path of 0.9 mm, an average
speed of 0.97 cm/s and a speed at the centre that is ten times smaller. Cited on page 28.
29 L. Vestergaard Hau, S. E. Harris, Z. Dutton & C. H. Behroozi, Light speed
reduction to 17 meters per second in an ultracold atomic gas, Nature 397, pp. 594–598, 1999.
See also C. Liu, Z. Dutton, C. H. Behroozi & L. Vestergaard Hau, Observa-
tion of coherent optical information storage in an atomic medium using halted light pulses,
Nature 409, pp. 490–493, 2001, and the comment E. A. Cornell, Stopping light in its

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
track, 409, pp. 461–462, 2001. However, despite the claim, the light pulses have not been
halted. Cited on page 28.
30 The method of explaining special relativity by drawing a few lines on paper is due to Her-
mann B ondi, Relativity and Common Sense: A New Approach to Einstein, Dover, New
York, 1980. See also Dierck-Ekkehard Liebscher, Relativitätstheorie mit Zirkel und
Lineal, Akademie-Verlag Berlin, 1991. Cited on page 28.
31 S. Reinhardt & al., Test of relativistic time dilation with fast optical clocks at different
velocities, Nature Physics 3, pp. 861–864, 2007. Cited on page 30.
32 Rod S. Lakes, Experimental limits on the photon mass and cosmic vector potential, Phys-
ical Review Letters 80, pp. 1826–1829, 1998. A maximum photon mass of 10−47 kg was also
deduced from gamma-ray bursts in the paper Ref. 13. Cited on page 32.
33 F. Tuinstra, De lotgevallen van het dopplereffect, Nederlands tijdschrift voor
natuurkunde 75, p. 296, August 2009. Cited on page 33.
34 R. W. McGowan & D. M. Giltner, New measurement of the relativistic Doppler shift in
neon, Physical Review Letters 70, pp. 251–254, 1993. Cited on page 35.
35 R. Lambourne, The Doppler effect in astronomy, Physics Education 32, pp. 34–40, 1997,
Cited on page 35.
36 D. Kiefer & al., Relativistic electron mirrors from nanoscale foils for coherent frequency
upshift to the extreme ultraviolet, Nature Communications 4, p. 1763, 2013. The exploration
of relativistic charge systems promises interesting results and applications in the coming
decade. Cited on page 35.
bibliography 325

37 See, for example, the paper by T. Wilken & al., A spectrograph for exoplanet observations
calibrated at the centimetre-per-second level, Nature 485, pp. 611–614, 2012. Cited on page
36.
38 The present record for clock synchronization seems to be 1 ps for two clocks distant 3 km
from each other. See A. Valencia, G. Scarcelli & Y. Shih, Distant clock synchron-
ization using entangled photon pairs, Applied Physics Letters 85, pp. 2655–2657, 2004, or
arxiv.org/abs/quant-ph/0407204. Cited on page 36.
39 J. Frenkel & T. Kontorowa, Über die Theorie der plastischen Verformung, Physikali-
sche Zeitschrift der Sowietunion 13, p. 1, 1938. F. C. Frank, On the equations of
motion of crystal dislocations, Proceedings of the Physical Society A 62, pp. 131–134,
1949. J. Eshelby, Uniformly moving dislocations, Proceedings of the Physical So-
ciety A 62, pp. 307–314, 1949. See also G. Leibfried & H. Dietze, Zeitschrift
für Physik 126, p. 790, 1949. A general introduction can be found in A. Seeger &
P. Schiller, Kinks in dislocation lines and their effects in internal friction in crystals,
Physical Acoustics 3A, W. P. Mason, ed., Academic Press, 1966. See also the textbooks by
Frank R. N. Nabarro, Theory of Crystal Dislocations, Oxford University Press, 1967,
or J. P. Hirth & J. Lothe, Theory of Dislocations, McGraw Hill, 1968. Cited on page 36.

Motion Mountain – The Adventure of Physics
40 This beautiful graph is taken from Z. G. T. Guiragossian, G. B. Rothbart,
M. R. Yearian, R. Gearhart & J. J. Murray, Relative velocity measurements of
electrons and gamma rays at 15 GeV, Physical Review Letters 34, pp. 335–338, 1975. Cited
on page 37.
41 A provocative attempt to explain the lack of women in physics in general is made in Mar-
garet Wertheim, Pythagoras’ Trousers – God, Physics and the Gender Wars, Fourth Es-
tate, 1997. Cited on page 37.
42 To find out more about the best-known crackpots, and their ideas, send an email to
majordomo@zikzak.net with the one-line body ‘subscribe psychoceramics’. Cited on page

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
37.
43 The accuracy of Galilean mechanics was discussed by Simon Newcomb already in 1882. For
details, see Steven Weinberg, Gravitation and Cosmology, Wiley, 1972. Cited on page
38.
44 The speed of neutrinos is the same as that of light to 9 decimal digits. This is explained
by Leo Stodolsky, The speed of light and the speed of neutrinos, Physics Letters B 201,
p. 353, 1988. An observation of a small mass for the neutrino has been published by the
Japanese Super-Kamiokande collaboration, in Y. Fukuda & al., Evidence for oscillation of
atmospheric neutrinos, Physical Review Letters 81, pp. 1562–1567, 1998. The newer results
published by the Canadian Sudbury Neutrino Observatory, as Q.R. Ahmad & al., Direct
evidence for neutrino flavor transformation from neutral-current interactions in the Sudbury
Neutrino Observatory, Physical Review Letters 89, p. 011301, 2002, also confirm that neut-
rinos have a mass in the 1 eV region. Cited on pages 39 and 311.
45 B. Rothenstein & G. Eckstein, Lorentz transformations directly from the speed of
light, American Journal of Physics 63, p. 1150, 1995. See also the comment by E. Kapuścik,
Comment on “Lorentz transformations directly from the speed of light” by B. Rothenstein and
G. Eckstein, American Journal of Physics 65, p. 1210, 1997. Cited on page 40.
46 See e.g. the 1922 lectures by Lorentz at Caltech, published as H. A. Lorentz, Problems of
Modern Physics, edited by H. Bateman, Ginn and Company, page 99, 1927. Cited on page
40.
47 Max B orn, Die Relativitätstheorie Einsteins, Springer, 2003, a new, commented edition of
the original text of 1920. Cited on page 40.
326 bibliography

48 A. A. Michelson & E. W. Morley, On the relative motion of the Earth and the lumini-
ferous ether, American Journal of Science (3rd series) 34, pp. 333–345, 1887. Michelson pub-
lished many other papers on the topic after this one. Cited on page 40.
49 The newest result is Ch. Eisele, A. Yu. Nevsky & S. Schiller, Laboratory test of the
isotropy of light propagation at the 10−17 level, Physics Review Letters 103, p. 090401, 2009.
See also the older experiment at S. Schiller, P. Antonini & M. Okhapkin, A pre-
cision test of the isotropy of the speed of light using rotating cryogenic cavities, arxiv.org/abs/
physics/0510169. See also the institute page at www.exphy.uni-duesseldorf.de/ResearchInst/
WelcomeFP.html. Cited on page 41.
50 H. A. Lorentz, De relative beweging van de aarde en dem aether, Amst. Versl. 1, p. 74,
1892, and also H. A. Lorentz, Electromagnetic phenomena in a system moving with any
velocity smaller than that of light, Amst. Proc. 6, p. 809, 1904, or Amst. Versl. 12, p. 986,
1904. Cited on page 44.
51 A general refutation of such proposals is discussed by S. R. Mainwaring &
G. E. Stedman, Accelerated clock principles, Physical Review A 47, pp. 3611–3619, 1993.
Experiments on muons at CERN in 1968 showed that accelerations of up to 1020 m/s2 have

Motion Mountain – The Adventure of Physics
no effect, as explained by D. H. Perkins, Introduction to High Energy Physics, Addison-
Wesley, 1972, or by J. Bailey & al., Il Nuovo Cimento 9A, p. 369, 1972. Cited on page
44.
52 W. Rindler, General relativity before special relativity: an unconventional overview of re-
lativity theory, American Journal of Physics 62, pp. 887–893, 1994. Cited on page 45.
53 Steven K. Blau, Would a topology change allow Ms. Bright to travel backward in time?,
American Journal of Physics 66, pp. 179–185, 1998. Cited on page 47.
54 On the ‘proper’ formulation of relativity, see for example D. Hestenes, Proper particle
mechanics, Journal of Mathematical Physics 15, pp. 1768–1777, 1974. See also his numerous

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
other papers, his book David Hestenes, Spacetime Algebra, Gordon and Breach, 1966,
and his webpage modelingnts.la.asu.edu. A related approach is W. E. Baylis, Relativity in
introductory physics, preprint at arxiv.org/abs/physics/0406158. Cited on page 49.
55 The simple experiment to take a precise clock on a plane, fly it around the world and
then compare it with an identical one left in place was first performed by J. C. Hafele
& R. E. Keating, Around-the-world atomic clocks: predicted relativistic time gains, Sci-
ence 177, pp. 166–167, and Around-the-world atomic clocks: observed relativistic time gains,
pp. 168–170, 14 July 1972. See also Ref. 18. Cited on pages 49 and 140.
56 A readable introduction to the change of time with observers, and to relativity in gen-
eral, is Roman U. Sexl & Herbert Kurt Schmidt, Raum-Zeit-Relativität, 2. Au-
flage, Vieweg & Sohn, Braunschweig, 1991. Cited on page 49.
57 Most famous is the result that moving muons stay younger, as shown for example by
D. H. Frisch & J. B. Smith, Measurement of the relativistic time dilation using 𝜇-mesons,
American Journal of Physics 31, pp. 342–355, 1963. For a full pedagogical treatment of the
twin paradox, see E. Sheldon, Relativistic twins or sextuplets?, European Journal of Phys-
ics 24, pp. 91–99, 2003. Cited on page 50.
58 Paul J. Nahin, Time Machines – Time Travel in Physics, Metaphysics and Science Fiction,
Springer Verlag and AIP Press, second edition, 1999. Cited on page 50.
59 The first muon experiment was B. Rossi & D. B. Hall, Variation of the rate of decay of
mesotrons with momentum, Physical Review 59, pp. 223–228, 1941. ‘Mesotron’ was the old
name for muon. Cited on page 51.
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60 J. Bailey & al., Final report on the CERN muon storage ring including the anomalous
magnetic moment and the electric dipole moment of the muon, and a direct test of relativistic
time dilation, Nuclear Physics B 150, pp. 1–75, 1979. Cited on page 51.
61 A. Harvey & E. Schucking, A small puzzle from 1905, Physics Today, pp. 34–36,
March 2005. Cited on page 51.
62 Search for ‘fuel’ and ‘relativistic rocket’ on the internet. Cited on page 51.
63 W. Rindler, Length contraction paradox, American Journal of Physics 29, pp. 365–366,
1961. For a variation without gravity, see R. Shaw, Length contraction paradox, American
Journal of Physics 30, p. 72, 1962. Cited on page 53.
64 H. van Lintel & C. Gruber, The rod and hole paradox re-examined, European Journal
of Physics 26, pp. 19–23, 2005. Cited on page 53.
65 See the clear discussion by C. Iyer & G. M. Prabhu, Differing observations on the land-
ing of the rod into the slot, American Journal of Physics 74, pp. 998–1001, 2006, preprint at
arxiv.org/abs/0809.1740. Cited on page 53.
66 This situation is discussed by G. P. Sastry, Is length contraction paradoxical?, American
Journal of Physics 55, 1987, pp. 943–946. This paper also contains an extensive literature

Motion Mountain – The Adventure of Physics
list covering variants of length contraction paradoxes. Cited on page 53.
67 S. P. B oughn, The case of the identically accelerated twins, American Journal of Physics
57, pp. 791–793, 1989. Cited on pages 54 and 55.
68 J. M. Supplee, Relativistic buoyancy, American Journal of Physics 57 1, pp. 75–77, January
1989. See also G. E. A. Matsas, Relativistic Arquimedes law for fast moving bodies and the
general-relativistic resolution of the ‘submarine paradox’, Physical Review D 68, p. 027701,
2003, or arxiv.org/abs/gr-qc/0305106. Cited on page 54.
69 The distinction was first published by J. Terrell, Invisibility of Lorentz contraction, Phys-
ical Review 116, pp. 1041–1045, 1959, and R. Penrose, The apparent shape of a relativist-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
ically moving sphere, Proceedings of the Cambridge Philosophical Society 55, pp. 137–139,
1959. Cited on page 55.
70 G. R. Rybicki, Speed limit on walking, American Journal of Physics 59, pp. 368–369, 1991.
Cited on page 58.
71 See, for example, R. J. Nemiroff, Q. Zhong & E. Lilleskov, Lights illuminate sur-
faces superluminally, preprint available at arXiv.org/abs/1506.02643; see also the related and
beautiful paper R. J. Nemiroff, Superluminal spot pair events in astronomical settings:
sweeping beams, preprint available at arXiv.org/abs/1412.7581. Cited on page 60.
72 The first examples of such astronomical observations were provided by A.R. Whitney &
al., Quasars revisited: rapid time variations observed via very-long-baseline interferometry,
Science 173, pp. 225–230, 1971, and by M.H. Cohen & al., The small-scale structure of ra-
dio galaxies and quasi-stellar sources at 3.8 centimetres, Astrophysical Journal 170, pp. 207–
217, 1971. See also T. J. Pearson, S. C. Unwin, M. H. Cohen, R. P. Linfield,
A. C. S. Readhead, G. A. Seielstad, R. S. Simon & R. C. Walker, Superluminal
expansion of quasar 3C 273, Nature 290, pp. 365–368, 1981. An overview is given in
J. A. Zensus & T. J. Pearson, editors, Superluminal radio sources, Cambridge Univer-
sity Press, 1987. Another measurement, using very long baseline interferometry with radio
waves on jets emitted from a binary star (thus not a quasar), was shown on the cover of
Nature: I. F. Mirabel & L. F. Rodríguez, A superluminal source in the galaxy, Nature
371, pp. 46–48, 1994. A more recent example was reported in Science News 152, p. 357, 6
December 1997.
Pedagogical explanations are given by D. C. Gabuzda, The use of quasars in teaching
328 bibliography

introductory special relativity, American Journal of Physics 55, pp. 214–215, 1987, and by
Ref. 119 on pages 89-92. Cited on page 60.
73 A somewhat old-fashioned review paper on tachyons mentioning the doubling issue, on
pages 52 and 53, is E. Recami, Classical tachyons and possible applications, Rivista del
Nuovo Cimento 9, pp. 1–178, 1986. It also discusses a number of paradoxes. By the way,
a simple animation claiming to show a flying tachyon can be found in Wikipedia. Cited on
page 61.
74 O. M. Bilaniuk & E. C. Sudarshan, Particles beyond the light barrier, Phys-
ics Today 22, pp. 43–51, 1969, and O. M. P. Bilaniuk, V. K. Deshpande &
E. C. G. Sudarshan, ‘Meta’ relativity, American Journal of Physics 30, pp. 718–723,
1962. See also the book E. Recami, editor, Tachyons, Monopoles and Related Topics,
North-Holland, 1978. Cited on page 61.
75 J. P. Costella, B. H. J. McKellar, A. A. Rawlinson & G. J. Stephenson, The
Thomas rotation, American Journal of Physics 69, pp. 837–847, 2001. Cited on page 62.
76 Planck wrote this in a letter in 1908. Cited on page 63.

Motion Mountain – The Adventure of Physics
77 See for example S. S. Costa & G. E. A. Matsas, Temperature and relativity, preprint
available at arxiv.org/abs/gr-qc/9505045. Cited on page 63.
78 One of the latest of these debatable experiments is T. P. Krisher, L. Maleki,
G. F. Lutes, L. E. Primas, R. T. Logan, J. D. Anderson & C. M. Will, Test of
the isotropy of the one-way speed of light using hydrogen-maser frequency standards, Phys-
ical Review D 42, pp. 731–734, 1990. Cited on pages 63 and 315.
79 H. C. Ohanian, The role of dynamics in the synchronization problem, American Journal
of Physics 72, pp. 141–148, 2004. Cited on pages 63 and 315.
80 R. C. Tolman & G. N. Lewis, The principle of relativity and non-Newtonian mechanics,
Philosophical Magazine 18, pp. 510–523, 1909, and R. C. Tolman, Non-Newtonian mech-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
anics: the mass of a moving body, Philosophical Magazine 23, pp. 375–380, 1912. Cited on
page 65.
81 S. Rainville, J. K. Thompson, E. G. Myers, J. M. Brown, M. S. Dewey,
E. G. Kessler, R. D. Deslattes, H. G. B örner, M. Jentschel, P. Mutti &
D. E. Pritchard, World year of physics: a direct test of 𝐸 = 𝑚𝑐2 , Nature 438, pp. 1096–
1097, 2005. Cited on page 71.
82 This information is due to a private communication by Frank DiFilippo; part of the story
is given in F. DiFilippo, V. Natarajan, K. R. B oyce & D. E. Pritchard, Accurate
atomic masses for fundamental metrology, Physical Review Letters 73, pp. 1481–1484, 1994.
These measurements were performed with Penning traps; a review of the possibilities they
offer is given by R. C. Thompson, Precision measurement aspects of ion traps, Measure-
ment Science and Technology 1, pp. 93–105, 1990. The most important experimenters in
the field of single particle levitation were awarded the Nobel Prize in 1989. One of the No-
bel Prize lectures can be found in W. Paul, Electromagnetic traps for neutral and charged
particles, Reviews of Modern Physics 62, pp. 531–540, 1990. Cited on page 71.
83 J. L. Synge, Relativity: The Special Theory, North-Holland, 1956, pp. 208–213. More about
antiparticles in special relativity can be found in J. P. Costella, B. H. J. McKellar &
A. A. Rawlinson, Classical antiparticles, American Journal of Physics 65, pp. 835–841,
1997. See also Ref. 103. Cited on page 72.
84 M. Cannoni, Lorentz invariant relative velocity and relativistic binary collisions, preprint
at arxiv.org/abs/1605.00569. Cited on page 74.
bibliography 329

85 A. Papapetrou, Drehimpuls- und Schwerpunktsatz in der relativistischen Mechanik,
Praktika Acad. Athenes 14, p. 540, 1939, and A. Papapetrou, Drehimpuls- und Schwer-
punktsatz in der Diracschen Theorie, Praktika Acad. Athenes 15, p. 404, 1940. See also
M. H. L. Pryce, The mass-centre in the restricted theory of relativity and its connexion with
the quantum theory of elementary particles, Proceedings of the Royal Society in London, A
195, pp. 62–81, 1948. Cited on page 74.
86 The references preceding Einstein’s 𝐸 = 𝑐2 𝑚 are: S. Tolver Preston, Physics of the
Ether, E. & F.N. Spon, 1875, J. H. Poincaré, La théorie de Lorentz et le principe de
réaction, Archives néerlandaises des sciences exactes et naturelles 5, pp. 252–278, 1900,
O. De Pretto, Ipotesi dell’etere nella vita dell’universo, Reale Istituto Veneto di Scienze,
Lettere ed Arti tomo LXIII, parte 2, pp. 439–500, Febbraio 1904, F. Hasenöhrl, Berichte
der Wiener Akademie 113, p. 1039, 1904, F. Hasenöhrl, Zur Theorie der Strahlung in be-
wegten Körpern, Annalen der Physik 15, pp. 344–370, 1904, F. Hasenöhrl, Zur Theorie
der Strahlung in bewegten Körpern – Berichtigung, Annalen der Physik 16, pp. 589–592,
1905. Hasenöhrl died in 1915, De Pretto in 1921. All these papers were published before the
famous paper by Albert Einstein, Ist die Trägheit eines Körpers von seinem Energiein-
halt abhängig?, Annalen der Physik 18, pp. 639–641, 1905. Cited on page 76.

Motion Mountain – The Adventure of Physics
87 Umberto Bartocci, Albert Einstein e Olinto De Pretto: la vera storia della formula più
famosa del mondo, Ultreja, 1998. Cited on page 76.
88 For a clear overview of the various sign conventions in general relativity, see the front cover
of the text by Misner, Thorne and Wheeler, Ref. 98. We use the gravitational sign conven-
tions of the text by Ohanian and Ruffini, Ref. 113. Cited on page 78.
89 A jewel among the texts on special relativity is the booklet by Ulrich E. Schröder,
Spezielle Relativitätstheorie, Verlag Harri Deutsch, 1981. Cited on pages 80 and 83.
90 An improved proposal for the definition of 4-jerk that also cites this book is J. G. Russo
& P. K. Townsend, Relativistic kinematics and stationary motions, Journal of Physics A:

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Mathematical and Theoretical 42, p. 445402, 2009, preprint at arxiv.org/abs/0902.4243.
Cited on page 80.
91 G. Stephenson & C. W. Kilmister, Special Relativity for Physicists, Longmans, Lon-
don, 1965. See also W. N. Matthews, Relativistic velocity and acceleration transforma-
tions from thought experiments, American Journal of Physics 73, pp. 45–51, 2005, and the
subsequent J. M. Lév y, A simple derivation of teh Lorentz transformation and of the ac-
companying velocity and acceleration changes, American Journal of Physics 75, pp. 615–618,
2007. Cited on page 80.
92 A readable article showing a photocopy of a letter by Einstein making this point is
Lev B. Okun, The concept of mass, Physics Today, pp. 31–36, June 1989. The topic is not
without controversy, as the letters by readers following that article show; they are found in
Physics Today, pp. 13–14 and pp. 115–117, May 1990. The topic is still a source of debates.
Cited on page 82.
93 Christian Møller, The Theory of Relativity, Clarendon Press, 1952, 1972. This standard
textbook has been translated in several languages. Cited on page 83.
94 The famous no-interaction theorem states that there is no way to find a Lagrangian that only
depends on particle variables, is Lorentz invariant and contains particle interactions. It was
shown by D. G. Currie, T. F. Jordan & E. C. G. Sudarshan, Relativistic invariance
and Hamiltonian theories of interacting particles, Review of Modern Physics 35, pp. 350–
375, 1963. Cited on page 83.
95 P. Ehrenfest, Gleichförmige Rotation starrer Körper und Relativitätstheorie, Physikalis-
che Zeitschrift 10, pp. 918–928, 1909. Ehrenfest (incorrectly) suggested that this meant
330 bibliography

that relativity cannot be correct. A good modern summary of the issue can be found in
M. L. Ruggiero, The relative space: space measurements on a rotating platform, arxiv.org/
abs/gr-qc/0309020. Cited on page 85.
96 R. J. Low, When moving clocks run fast, European Journal of Physics 16, pp. 228–229, 1995.
Cited on pages 91, 92, and 93.
97 E. A. Desloge & R. J. Philpott, Uniformly accelerated reference frames in special re-
lativity, American Journal of Physics 55, pp. 252–261, 1987. Cited on page 94.
98 The impossibility of defining rigid coordinate frames for non-uniformly accelerating ob-
servers is discussed by Charles Misner, Kip Thorne & John A. Wheeler, Grav-
itation, Freeman, p. 168, 1973. Cited on pages 94 and 329.
99 R. H. Good, Uniformly accelerated reference frame and twin paradox, American Journal
of Physics 50, pp. 232–238, 1982. Cited on pages 94, 95, and 99.
100 J. D. Hamilton, The uniformly accelerated reference frame, American Journal of Physics
46, pp. 83–89, 1978. Cited on page 95.
101 The best and cheapest mathematical formula collection remains the one by

Motion Mountain – The Adventure of Physics
K. Rottmann, Mathematische Formelsammlung, BI Hochschultaschenbücher, 1960.
Cited on page 95.
102 C. G. Adler & R. W. Brehme, Relativistic solutions to a falling body in a uniform grav-
itation field, American Journal of Physics 59, pp. 209–213, 1991. Cited on page 96.
103 See for example the excellent lecture notes by D. J. Raymond, A radically modern ap-
proach to freshman physics, on the www.physics.nmt.edu/~raymond/teaching.html website.
Cited on pages 96 and 328.
104 Edward A. Desloge, The gravitational red-shift in a uniform field, American Journal of
Physics 58, pp. 856–858, 1990. Cited on page 99.
105 L. Mishra, The relativistic acceleration addition theorem, Classical and Quantum Gravity

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
11, pp. L97–L102, 1994. Cited on page 100.
106 Edwin F. Taylor & A. P. French, Limitation on proper length in special relativity,
American Journal of Physics 51, pp. 889–893, 1983. Cited on page 102.
107 Clear statements against a varying speed of light are made by Michael Duff in several of his
publications, such as M. J. Duff, Comment on time-variation of fundamental constants,
arxiv.org/abs/hep-th/0208093. The opposite point of view, though incorrect, has been pro-
posed by John Moffat and by João Magueijo, but also by various other authors. Cited on
page 104.
108 The quote is form a letter of Gibbs to the American Academy of Arts and Sciences, in which
he thanks the Academy for their prize. The letter was read in a session of the Academy and
thus became part of the proceedings: J. W. Gibbs, Proceedings of the American Academy
of Arts and Sciences, 16, p. 420, 1881. Cited on page 108.
109 It seems that the first published statement of the maximum force as a fundamental prin-
ciple was around the year 2000, in this text, in the chapter on gravitation and relativity. The
author discovered the maximum force principle, not knowing the work of others, when
Vol. VI, page 57 searching for a way to derive the results of the last part of this adventure that would be so
simple that it would convince even a secondary-school student. In the year 2000, the author
told his friends in Berlin about his didactic approach.
The concept of a maximum force was first proposed, most probably, by Venzo de Sabbata
and C. Sivaram in 1993. Also this physics discovery was thus made much too late. In 1995,
Corrado Massa took up the idea. Independently, Ludwik Kostro in 1999, Christoph Schiller
bibliography 331

just before 2000 and Gary Gibbons in the years before 2002 arrived at the same concept.
Gary Gibbons was inspired by a book by Oliver Lodge; he explains that the maximum force
value follows from general relativity; he does not make a statement about the converse, nor
do the other authors. The statement of maximum force as a fundamental principle seems
original to Christoph Schiller.
The temporal order of the first papers on maximum force seems to start with a first
mention in E. A. Rauscher, The Minkowski metric for a multidimensional geometry, Lett.
Nuovo Cimento 7, pp. 361–377, 1973, "F can be considered an upper bound on force",
and a further mention in H. -J. Treder, The planckions as largest elementary particles
and as smallest test bodies, Foundations of Physics 15, pp. 161–166, 1985. Then came the
dedicated paper V. de Sabbata & C. Sivaram, On limiting field strengths in gravita-
tion, Foundations of Physics Letters 6, pp. 561–570, 1993. It was followed by C. Massa,
Does the gravitational constant increase?, Astrophysics and Space Science 232, pp. 143–148,
1995, and by L. Kostro & B. Lange, Is 𝑐4 /𝐺 the greatest possible force in nature?, Physics
Essays 12, pp. 182–189, 1999. The next references are the paper by G. W. Gibbons, The
maximum tension principle in general relativity, Foundations of Physics 32, pp. 1891–1901,
2002, preprint at arxiv.org/abs/hep-th/0210109 – though he developed the ideas before that

Motion Mountain – The Adventure of Physics
date – and the older versions of the present text Christoph Schiller, Motion Moun-
tain – The Adventure of Physics, free pdf available at www.motionmountain.net. Then came
C. Schiller, Maximum force and minimum distance: physics in limit statements, preprint
at arxiv.org/abs/physics/0309118, and C. Schiller, General relativity and cosmology de-
rived from principle of maximum power or force, International Journal of Theoretical Phys-
ics 44, pp. 1629–1647, 2005, preprint at arxiv.org/abs/physics/0607090. See also R. Beig,
G. W. Gibbons & R. M. Schoen, Gravitating opposites attract, Classical and Quantum
Gravity 26, p. 225013, 2009. preprint at arxiv.org/abs/09071193.
In 2016, Gary Gibbons was not yet convinced maximum force or power can be seen as
a fundamental physical principle from which general relativity can be deduced – though he

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
sees it as a promising conjecture. Cited on pages 108, 113, 118, 124, 133, and 148.
110 See the fundamental paper by A. DiSessa, Momentum flow as an alternative perspective
in elementary mechanics, 48, p. 365, 1980, and A. DiSessa, Erratum: “Momentum flow
as an alternative perspective in elementary mechanics” [Am. J. Phys. 48, 365 (1980)], 48,
p. 784, 1980. Also the excellent physics textbook by Friedrich Herrmann, The Karls-
ruhe Physics Course, makes this point extensively; it is free to download in English, Span-
ish, Russian, Italian and Chinese at www.physikdidaktik.uni-karlsruhe.de/index_en.html.
Cited on page 110.
111 C. Schiller, Maximum force and minimum distance: physics in limit statements, preprint
at arxiv.org/abs/physics/0309118; the ideas are also part of the sixth volume of this text,
which is freely downloadable at www.motionmountain.net. Cited on pages 111, 113, 124,
and 133.
112 The analysis of the first detected gravitational wave event, called GW150914, is presented
in B.P. Abbott & al., (LIGO Scientific Collaboration and Virgo Collaboration) Ob-
servation of gravitational waves from a binary black hole merger, Physical Review Letters
116, p. 061102, 2016, also available for free download at journals.aps.org/prl/pdf/10.1103/
PhysRevLett.116.061102. Additional, more detailed papers on the event and the ones that
followed in 2015 and 2017 can be found via the website www.ligo.caltech.edu. A recent ex-
ample is arxiv.org/abs/1706.01812. Cited on pages 113 and 182.
113 H. C. Ohanian & Remo Ruffini, Gravitation and Spacetime, W.W. Norton & Co.,
1994. Another textbook that talks about the power limit is Ian R. Kenyon, General
Relativity, Oxford University Press, 1990. The maximum power is also discussed in
332 bibliography

L. Kostro, The quantity 𝑐5 /𝐺 interpreted as the greatest possible power in nature, Physics
Essays 13, pp. 143–154, 2000. Cited on pages 113, 123, 124, 127, 131, 318, 329, and 340.
114 An overview of the literature on analog model of general relativity can be found on Matt
Visser’s website www.physics.wustl.edu/~visser/Analog/bibliography.html. Cited on page
113.
115 See for example Wolf gang Rindler, Relativity – Special, General and Cosmological,
Oxford University Press, 2001, p. 70 ff, or Ray d ’ Inverno, Introducing Einstein’s Re-
lativity, Clarendon Press, 1992, p. 36 ff. Cited on page 115.
116 See for example A. Ashtekar, S. Fairhust & B. Krishnan, Isolated horizons:
Hamiltonian evolution and the first law, arxiv.org/abs/gr-qc/0005083. Cited on page 115.
117 T. Jacobson, Thermodynamics of spacetime: the Einstein equation of state, Physical Re-
view Letters 75, pp. 1260–1263, 1995 or arxiv.org/abs/gr-qc/9504004. Cited on page 116.
118 See for example Ekkehart Kröner, Kontinuumstheorie der Versetzungen und Eigen-
spannungen, Springer, 1958, volume 5 of the series ‘Ergebnisse der angewandten Mathem-
atik’. Kröner shows the similarity between the equations, methods and results of solid-state
continuum physics and those of general relativity, including the Ricci formalism. Cited on

Motion Mountain – The Adventure of Physics
pages 119 and 209.
119 See the excellent book Edwin F. Taylor & John A. Wheeler, Spacetime Physics – In-
troduction to Special Relativity, second edition, Freeman, 1992. Cited on pages 120 and 328.
120 This counter-example was suggested by Steve Carlip. Cited on page 122.
121 E. R. Caianiello, Lettere al Nuovo Cimento 41, p. 370, 1984. Cited on page 124.
122 J. D. Barrow & G. W. Gibbons, Maximum tension: with and without a cosmological
constant, Monthly Notices of the Royal Astronomical Society 446, pp. 3874–3877, 2014, pre-
print at arxiv.org/abs/1408.1820. Cited on page 125.
123 A notable exception is the physics teching group in Karlsruhe, who has always taught

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
force in the correct way. See F. Herrmann, Mengenartige Größen im Physikunterricht,
Physikalische Blätter 54, pp. 830–832, September 1998. See also the lecture notes on general
introductory physics on the website www.physikdidaktik.uni-karlsruhe.de/skripten. Cited
on page 129.
124 R. Penrose, Naked singularities, Annals of the New York Academy of Sciences 224,
pp. 125–134, 1973. Cited on page 130.
125 G. Huisken & T. Ilmanen, The Riemannian Penrose inequality, International Mathem-
atics Research Notices 59, pp. 1045–1058, 1997. S. A. Hay ward, Inequalities relating area,
energy, surface gravity and charge of black holes, Physical Review Letters 81, pp. 4557–4559,
1998. Cited on page 130.
126 C. Will, The Confrontation between General Relativity and Experiment, Living Reviews
in Relativity 17, 2014, available freely at www.livingreviews.org/lrr-2014-4. An older and
more extensive reference is Clifford M. Will, Was Einstein Right? – Putting General
Relativity to the Test, Oxford University Press, 1993. See also his paper arxiv.org/abs/gr-qc/
9811036. Cited on pages 131 and 336.
127 The measurement results by the WMAP satellite are summarized on the website map.
gsfc.nasa.gov/m_mm.html; the papers are available at lambda.gsfc.nasa.gov/product/map/
current/map_bibliography.cfm. Cited on page 132.
128 The simplest historical source is Albert Einstein, Sitzungsberichte der Preussischen
Akademie der Wissenschaften II pp. 844–846, 1915. It is the first explanation of the general
theory of relativity, in only three pages. The theory is then explained in detail in the famous
bibliography 333

article Albert Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der
Physik 49, pp. 769–822, 1916. The historic references can be found in German and English
in John Stachel, ed., The Collected Papers of Albert Einstein, Volumes 1–9, Princeton
University Press, 1987–2004.
Below is a selection of English-language textbooks for deeper study, in ascending order
of depth and difficulty:

— An entertaining book without any formulae, but nevertheless accurate and detailed, is
the paperback by Igor Novikov, Black Holes and the Universe, Cambridge University
Press, 1990.
— Almost no formulae, but loads of insight, are found in the enthusiastic text by
John A. Wheeler, A Journey into Gravity and Spacetime, W.H. Freeman, 1990.
— An excellent presentation is Edwin F. Taylor & John A. Wheeler, Exploring
Black Holes: Introduction to General Relativity, Addison Wesley Longman, 2000.
— Beauty, simplicity and shortness are the characteristics of Malcolm Ludvigsen,
General Relativity, a Geometric Approach, Cambridge University Press, 1999.
— Good explanation is the strength of Bernard Schutz, Gravity From the Ground Up,

Motion Mountain – The Adventure of Physics
Cambridge University Press, 2003.
— A good overview of experiments and theory is given in James Foster &
J. D. Nightingale, A Short Course in General Relativity, Springer Verlag, 2nd
edition, 1998.
— A pretty text is Sam Lilley, Discovering Relativity for Yourself, Cambridge University
Press, 1981.
— A modern text is by Ray d ’ Inverno, Introducing Einstein’s Relativity, Clarendon
Press, 1992. It includes an extended description of black holes and gravitational radi-
ation, and regularly refers to present research.
— A beautiful, informative and highly recommended text is H. C. Ohanian &

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Remo Ruffini, Gravitation and Spacetime, W.W. Norton & Co., 1994.
— A well written and modern book, with emphasis on the theory, by one of the great mas-
ters of the field is Wolf gang Rindler, Relativity – Special, General and Cosmolo-
gical, Oxford University Press, 2001.
— A classic is Steven Weinberg, Gravitation and Cosmology, Wiley, 1972.
— The passion of general relativity can be experienced also in John Klauder, ed., Magic
without Magic: John Archibald Wheeler – A Collection of Essays in Honour of His Sixtieth
Birthday, W.H. Freeman & Co., 1972.
— An extensive text is Kip S. Thorne, Black Holes and Time Warps – Einstein’s Out-
rageous Legacy, W.W. Norton, 1994.
— The most mathematical – and toughest – text is Robert M. Wald, General Relativity,
University of Chicago Press, 1984.
— Much information about general relativity is available on the internet. As a good starting
point for US-American material, see the math.ucr.edu/home/baez/physics/ website.

There is still a need for a large and modern textbook on general relativity, with colour ma-
terial, that combines experimental and theoretical aspects. For texts in other languages, see
the next reference. Cited on pages 136, 162, 164, 202, and 203.

129 A beautiful German teaching text is the classic G. Falk & W. Ruppel, Mechanik, Re-
lativität, Gravitation – ein Lehrbuch, Springer Verlag, third edition, 1983.
A practical and elegant booklet is Ulrich E. Schröder, Gravitation – Einführung
in die allgemeine Relativitätstheorie, Verlag Harri Deutsch, Frankfurt am Main, 2001.
334 bibliography

A modern reference is Torsten Fliessbach, Allgemeine Relativitätstheorie,
Akademischer Spektrum Verlag, 1998.
Excellent is Hubert Goenner, Einführung in die spezielle und allgemeine Relativität-
stheorie, Akademischer Spektrum Verlag, 1996.
In Italian, there is the beautiful, informative, but expensive H. C. Ohanian &
Remo Ruffini, Gravitazione e spazio-tempo, Zanichelli, 1997. It is highly recommended.
A modern update of that book would be without equals. Cited on pages 136, 162, 164, 179,
181, 203, and 338.
130 P. Mohazzabi & J. H. Shea, High altitude free fall, American Journal of Physics 64,
pp. 1242–1246, 1996. As a note, due to a technical failure Kittinger had his hand in (near) va-
cuum during his ascent, without incurring any permanent damage. On the consequences of
human exposure to vacuum, see the www.sff.net/people/geoffrey.landis/vacuum.html web-
site. Cited on page 137.
131 This story is told by W. G. Unruh, Time, gravity, and quantum mechanics, preprint avail-
able at arxiv.org/abs/gr-qc/9312027. Cited on page 137.
132 H. B ondi, Gravitation, European Journal of Physics 14, pp. 1–6, 1993. Cited on page 138.

Motion Mountain – The Adventure of Physics
133 J. W. Brault, Princeton University Ph.D. thesis, 1962. See also J. L. Snider, Physical Re-
view Letters 28, pp. 853–856, 1972, and for the star Sirius see J.L. Greenstein & al.,
Astrophysical Journal 169, p. 563, 1971. Cited on pages 140 and 290.
134 See the detailed text by Jeffrey Crelinsten, Einstein’s Jury – The Race to Test Relativity,
Princeton University Press, 2006, which covers all researchers involved in the years from
1905 to 1930. Cited on page 140.
135 The famous paper is R. V. Pound & G. A. Rebka, Apparent weight of photons, Phys-
ical Review Letters 4, pp. 337–341, 1960. A higher-precision version was published by
R. V. Pound & J. L. Snider, Physical Review Letters 13, p. 539, 1964, and R. V. Pound

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
& J. L. Snider, Physical Review B 140, p. 788, 1965. Cited on pages 140 and 290.
136 R.F.C. Vessot & al., Test of relativistic gravitation with a space-borne hydrogen maser,
Physical Review Letters 45, pp. 2081–2084, 1980. The experiment was performed in 1976;
there are more than a dozen co-authors involved in this work, which involved shooting a
maser into space with a scout missile to a height of c. 10 000 km. Cited on page 140.
137 L. Briatore & S. Leschiutta, Evidence for Earth gravitational shift by direct atomic-
time-scale comparison, Il Nuovo Cimento 37B, pp. 219–231, 1977. Cited on page 140.
138 More information about tides can be found in E. P. Clancy, The Tides, Doubleday, New
York, 1969. Cited on page 142.
139 The expeditions had gone to two small islands, namely to Sobral, north of Brazil, and to
Principe, in the gulf of Guinea. The results of the expedition appeared in The Times before
they appeared in a scientific journal. Today this would be called unprofessional. The res-
ults were published as F. W. Dyson, A. S. Eddington & C. Davidson, Philosophical
Transactions of the Royal Society (London) 220A, p. 291, 1920, and Memoirs of the Royal
Astronomical Society 62, p. 291, 1920. Cited on page 143.
140 D. Kennefick, Testing relativity from the 1919 eclipse – a question of bias, Physics Today
pp. 37–42, March 2009. This excellent article discusses the measurement errors in great
detail. The urban legend that the star shifts were so small on the negatives that they implied
large measurement errors is wrong – it might be due to a lack of respect on the part of some
physicists for the abilities of astronomers. The 1979 reanalysis of the measurement confirm
that such small shifts, smaller than the star image diameter, are reliably measurable. In fact,
bibliography 335

the 1979 reanalysis of the data produced a smaller error bar than the 1919 analysis. Cited
on page 143.
141 A good source for images of space-time is the text by G. F. R. Ellis & R. Williams, Flat
and Curved Space-times, Clarendon Press, Oxford, 1988. Cited on page 144.
142 J. Droste, Het veld van een enkel centrum in Einstein’s theorie der zwaartekracht, en de
beweging van een stoffelijk punt, Verslag gew. Vergad. Wiss. Amsterdam 25, pp. 163–180,
1916. Cited on page 146.
143 The name black hole was introduced in 1967 at a pulsar conference, as described in his
autobiography by John A. Wheeler, Geons, Black Holes, and Quantum Foam: A Life
in Physics, W.W. Norton, 1998, pp. 296–297: ‘In my talk, I argued that we should consider
the possibility that at the center of a pulsar is a gravitationally completely collapsed object.
I remarked that one couldn’t keep saying “gravitationally completely collapsed object”
over and over. One needed a shorter descriptive phrase. “How about black hole?” asked
someone in the audience. I had been searching for just the right term for months, mulling
it over in bed, in the bathtub, in my car, whenever I had quiet moments. Suddenly, this
name seemed exactly right. When I gave a more formal ... lecture ... a few weeks later on,

Motion Mountain – The Adventure of Physics
on December 29, 1967, I used the term, and then included it into the written version of
the lecture published in the spring of 1968 ... I decided to be casual about the term “black
hole”, dropping it into the lecture and the written version as if it were an old familiar
friend. Would it catch on? Indeed it did. By now every schoolchild has heard the term.’
The widespread use of the term began with the article by R. Ruffini &
J. A. Wheeler, Introducing the black hole, Physics Today 24, pp. 30–41, January 1971.
In his autobiography, Wheeler also writes that the expression ‘black hole has no hair’
was criticized as ‘obscene’ by Feynman. This is a bizarre comment, given that Feynman
used to write his papers in topless bars. Cited on pages 146, 262, 263, and 269.
144 L. B. Kreuzer, Experimental measurement of the equivalence of active and passive gravita-

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
tional mass, Physical Review 169, pp. 1007–1012, 1968. With a clever experiment, he showed
that the gravitational masses of fluorine and of bromine are equal. Cited on page 147.
145 A good and accessible book on the topic is David Blair & Geoff McNamara, Ripples
on a cosmic sea, Allen & Unwin, 1997. Cited on page 147.
146 That bodies fall along geodesics, independently of their mass, the so-called weak equival-
ence principle, has been checked by many experiments, down to the 10−13 level. The most
precise experiments use so-called torsion balances. See, for example, the website of the Eőt-
Wash group at www.npl.washington.edu/eotwash/experiments/experiments.html. Cited
on page 151.
147 So far, the experiments confirm that electrostatic and (strong) nuclear energy fall like mat-
ter to within one part in 108 , and weak (nuclear) energy to within a few per cent. This is
summarized in Ref. 151. Cited on page 151.
148 J. Soldner, Berliner Astronomisches Jahrbuch auf das Jahr 1804, 1801, p. 161. Cited on
page 152.
149 See for example K. D. Olum, Superluminal travel requires negative energies, Physical Re-
view Letters 81, pp. 3567–3570, 1998, or M. Alcubierre, The warp drive: hyper-fast travel
within general relativity, Classical and Quantum Gravity 11, pp. L73–L77, 1994. See also
Chris Van Den Broeck, A warp drive with more reasonable total energy requirements,
Classical and Quantum Gravity 16, pp. 3973–3979, 1999. Cited on page 155.
150 See the Astronomical Almanac, and its Explanatory Supplement, H.M. Printing Office, Lon-
don and U.S. Government Printing Office, Washington, 1992. For the information about
336 bibliography

various time coordinates used in the world, such as barycentric coordinate time, the time
at the barycentre of the solar system, see also the tycho.usno.navy.mil/systime.html web
page. It also contains a good bibliography. Cited on page 155.
151 An overview is given in Clifford Will, Theory and Experiment in Gravitational Physics,
chapter 14.3, revised edition, Cambridge University Press, 1993. Despite being a standard
reference, Will’s view of the role of tides and the role of gravitational energy within the
principle of equivalence has been criticised by other researchers. See also Ref. 126. Cited
on pages 156, 162, and 335.
152 The calculation omits several smaller effects, such as rotation of the Earth and red-shift.
For the main effect, see Edwin F. Taylor, ‘The boundaries of nature: special and general
relativity and quantum mechanics, a second course in physics’ – Edwin F. Taylor’s acceptance
speech for the 1998 Oersted Medal presented by the American Association of Physics Teachers,
6 January 1998, American Journal of Physics 66, pp. 369–376, 1998. Cited on page 157.
153 A. G. Lindh, Did Popper solve Hume’s problem?, Nature 366, pp. 105–106, 11 November
1993, Cited on page 157.
154 See the paper P. Kaaret, S. Piraino, P. F. Bloser, E. C. Ford, J. E. Grindlay,

Motion Mountain – The Adventure of Physics
A. Santangelo, A. P. Smale & W. Zhang, Strong Field Gravity and X-Ray Obser-
vations of 4U1820-30, Astrophysical Journal 520, pp. L37–L40, 1999, or at arxiv.org/abs/
astro-ph/9905236. The beautiful graphics at the research.physics.uiuc.edu/CTA/movies/
spm website illustrate this star system. Cited on page 157.
155 R. J. Nemiroff, Visual distortions near a black hole and a neutron star, American Journal
of Physics 61, pp. 619–632, 1993. Cited on page 157.
156 The equality was first tested with precision by R. von Eötvös, Annalen der Physik &
Chemie 59, p. 354, 1896, and by R. von Eötvös, V. Pekár, E. Fekete, Beiträge
zum Gesetz der Proportionalität von Trägheit und Gravität, Annalen der Physik 4, Leipzig
68, pp. 11–66, 1922. Eötvös found agreement to 5 parts in 109 . More experiments were

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
performed by P. G. Roll, R. Krotkow & R. H. Dicke, The equivalence of inertial
and passive gravitational mass, Annals of Physics (NY) 26, pp. 442–517, 1964, one of
the most interesting and entertaining research articles in experimental physics, and by
V. B. Braginsky & V. I. Panov, Soviet Physics – JETP 34, pp. 463–466, 1971. Modern
results, with errors less than one part in 1012 , are by Y. Su & al., New tests of the universal-
ity of free fall, Physical Review D50, pp. 3614–3636, 1994. Several future experiments have
been proposed to test the equality in space to less than one part in 1016 . Cited on pages 158
and 290.
157 Nigel Calder, Einstein’s Universe, Viking, 1979. Weizmann and Einstein once crossed
the Atlantic on the same ship. Cited on page 160.
158 L. Lerner, A simple calculation of the deflection of light in a Schwarzschild gravitational
field, American Journal of Physics 65, pp. 1194–1196, 1997. Cited on page 161.
159 A. Einstein, Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes, Annalen
der Physik 35, p. 898, 1911. Cited on page 162.
160 O. Titov, Testing of general relativity with geodetic VLBI, preprint at https://arxiv.org/abs/
1702.06647. Cited on pages 162 and 163.
161 I. I. Shapiro, & al., Fourth test of general relativity, Physical Review Letters 13, pp. 789–
792, 1964. Cited on page 163.
162 I. I. Shapiro, & al., Fourth test of general relativity: preliminary results, Physical Review
Letters 20, pp. 1265–1269, 1968. Cited on page 163.
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163 J. H. Taylor, Pulsar timing and relativistic gravity, Proceedings of the Royal Society, Lon-
don A 341, pp. 117–134, 1992. Cited on pages 164 and 166.
164 B. Bertotti, I. Ciufolini & P. L. Bender, New test of general relativity: measurement
of De Sitter geodetic precession rate for lunar perigee, Physical Review Letters 58, pp. 1062–
1065, 1987. Later it was confirmed by I.I. Shapiro & al., Measurement of the De Sitter
precession of the moon: a relativistic three body effect, Physical Review Letters 61, pp. 2643–
2646, 1988. Cited on pages 167 and 290.
165 The Thirring effect was predicted in H. Thirring, Über die Wirkung rotierender ferner
Massen in der Einsteinschen Gravitationstheorie, Physikalische Zeitschrift 19, pp. 33–39,
1918, and in H. Thirring, Berichtigung zu meiner Arbeit: “Über die Wirkung rotierender
Massen in der Einsteinschen Gravitationstheorie”, Physikalische Zeitschrift 22, p. 29, 1921.
The Thirring–Lense effect was predicted in J. Lense & H. Thirring, Über den Einfluß
der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Ein-
steinschen Gravitationstheorie, Physikalische Zeitschrift 19, pp. 156–163, 1918. Cited on page
169.
166 W. de Sitter, On Einstein’s theory of gravitation and its astronomical consequences,

Motion Mountain – The Adventure of Physics
Monthly Notes of the Royal Astrononmical Society 77, pp. 155–184, p. 418E, 1916. For a dis-
cussion of De Sitter precession and Thirring–Lense precession, see also B. R. Holstein,
Gyroscope precession in general relativity, American Journal of Physics 69, pp. 1248–1256,
2001. Cited on page 169.
167 The work is based on the LAGEOS and LAGEOS II satellites and is told in I. Ciufolini,
The 1995–99 measurements of the Thirring–Lense effect using laser-ranged satellites, Classical
and Quantum Gravity 17, pp. 2369–2380, 2000. See also I. Ciufolini & E. C. Pavlis,
A confirmation of the general relativistic prediction of the Lense–Thirring effect, Nature 431,
pp. 958–960, 2004. See, however, the next reference. Cited on pages 170 and 290.
168 See the interesting, detailed and disturbing discussion by L. Iorio, On some critical issues

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
of the LAGEOS-based tests of the Lense–Thirring effect, Journal of Modern Physics 2, pp. 210–
218, 2011, preprint available at arxiv.org/abs/1104.4464. Cited on pages 170 and 290.
169 On the Gravity Probe B satellite experiment, see the web page einstein.stanford.edu/
highlights/status1.html and the papers cited there. Cited on pages 167 and 170.
170 The detection of the Thirring–Lense effect in binary pulsars is presented in
R. D. Blandford, Lense–Thirring precession of radio pulsars, Journal of Astrophys-
ics and Astronomy 16, pp. 191–206, 1995. Cited on page 170.
171 G. Holzmüller, Zeitschrift für Mathematik und Physik 15, p. 69, 1870, F. Tisserand,
Comptes Rendus 75, p. 760, 1872, and Comptes Rendus 110, p. 313, 1890. Cited on page 170.
172 B. Mashhoon, Gravitoelectromagnetism: a brief review, arxiv.org/abs/gr-qc/0311030, and
B. Mashhoon, Gravitoelectromagnetism, arxiv.org/abs/gr-qc/0011014. See also its extens-
ive reference list on gravitomagnetism. Cited on page 171.
173 A. Tartaglia & M. L. Ruggiero, Gravito-electromagnetism versus electromagnetism,
European Journal of Physics 25, pp. 203–210, 2004. Cited on page 171.
174 D. Bedford & P. Krumm, On relativistic gravitation, American Journal of Physics 53,
pp. 889–890, 1985, and P. Krumm & D. Bedford, The gravitational Poynting vector and
energy transfer, American Journal of Physics 55, pp. 362–363, 1987. Cited on pages 172
and 179.
175 M. Kramer & al., Tests of general relativity from timing the double pulsar, preprint at arxiv.
org/abs/astro-ph/0609417. Cited on pages 174 and 290.
338 bibliography

176 The discussion of gravitational waves by Poincaré is found in H. Poincaré, Sur la
dynamique de l’électron, Comptes Rendus de l’Académie des Sciences 140, pp. 1504–
1508, 1905, which can be read online at www.academie-sciences.fr/pdf/dossiers/Poincare/
Poincare_pdf/Poincare_CR1905.pdf. Einstein’s prediction from an approximation of gen-
eral relativity, eleven years later, is found in A. Einstein, Näherungsweise Integration der
Feldgleichungen der Gravitation, Sitzungsberichte der Königlich-Preußischen Akademie
der Wissenschaften pp. 688–696, 1916. The first fully correct prediction of gravitational
waves is A. Einstein & N. Rosen, On gravitational waves, Journal of the Franklin In-
stitute 223, pp. 43–54, 1937. Despite heated discussions, until his death, Nathan Rosen con-
tinued not to believe in the existence of gravitational waves. On the story about the er-
rors of Einstein and Rosen about the reality of the waves, see also en.wikipedia.org/wiki/
Sticky_bead_argument. Cited on page 174.
177 The history of how the existence gravitational was proven in all its conceptual details is
told in the excellent paper C. D. Hill & P. Nurowski, How the green light was given for
gravitational wave search, preprint at arxiv.org/abs/1608.08673. Cited on page 174.
178 This is told in John A. Wheeler, A Journey into Gravity and Spacetime, W.H. Freeman,
1990. Cited on page 174.

Motion Mountain – The Adventure of Physics
179 See, for example, K. T. McDonald, Answer to question #49. Why 𝑐 for gravita-
tional waves?, American Journal of Physics 65, pp. 591–592, 1997, and section III of
V. B. Braginsky, C. M. Caves & K. S. Thorne, Laboratory experiments to test re-
lativistic gravity, Physical Review D 15, pp. 2047–2068, 1992. Cited on page 176.
180 A proposal to measure the speed of gravity is by S. M. Kopeikin, Testing the relativistic ef-
fect of the propagation of gravity by Very Long Baseline Interferometry, Astrophysical Journal
556, pp. L1–L5, 2001, and the experimental data is E. B. Formalont & S. M. Kopeikin,
The measurement of the light deflection from Jupiter: experimental results, Astrophysical
Journal 598, pp. 704–711, 2003. See also S. M. Kopeikin, The post-Newtonian treatment

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
of the VLBI experiment on September 8, 2002, Physics Letters A 312, pp. 147–157, 2003,
or arxiv.org/abs/gr-qc/0212121. Several arguments against the claim were published, such
as C. M. Will, Propagation speed of gravity and the relativistic time delay, arxiv.org/abs/
astro-ph/0301145, and S. Samuel, On the speed of gravity and the 𝑣/𝑐 corrections to the
Shapiro time delay, arxiv.org/abs/astro-ph/0304006. The discussion went on, as shown in
S. M. Kopeikin & E. B. Formalont, Aberration and the fundamental speed of gravity
in the Jovian deflection experiment, Foundations of Physics 36, pp. 1244–1285, 2006, pre-
print at arxiv.org/abs/astro-ph/0311063. Both sides claim to be right: the experiment claims
to deduce the speed of gravity from the lack of a tangential component of the light deflec-
tion by the gravity of Jupiter, and the critical side claims that the speed of gravity does not
enter in this measurement. If we compare the situation with analogous systems in transpar-
ent fluids or solids, which also show no tangential deflection component, we conclude that
neither the measurement nor the proposal allow us to deduce information on the speed of
gravity. A similar conclusion, but based on other arguments, is found on physics.wustl.edu/
cmw/SpeedofGravity.html. Cited on page 177.
181 For an introduction to gravitational waves, see B. F. Schutz, Gravitational waves on the
back of an envelope, American Journal of Physics 52, pp. 412–419, 1984. Cited on page 177.
182 The quadrupole formula is explained clearly in the text by Goenner. See Ref. 129. Cited on
page 179.
183 The beautiful summary by Daniel Kleppner, The gem of general relativity, Physics
Today 46, pp. 9–11, April 1993, appeared half a year before the authors of the cited work,
Joseph Taylor and Russel Hulse, received the Nobel Prize in Physics for the discovery of
bibliography 339

millisecond pulsars. A more detailed review article is J. H. Taylor, Pulsar timing and re-
lativistic gravity, Philosophical Transactions of the Royal Society, London A 341, pp. 117–
134, 1992. The original paper is J. H. Taylor & J. M. Weisberg, Further experimental
tests of relativistic gravity using the binary pulsar PSR 1913+16, Astrophysical Journal 345,
pp. 434–450, 1989. See also J. M. Weisberg, J. H. Taylor & L. A. Fowler, Pulsar PSR
1913+16 sendet Gravitationswellen, Spektrum der Wissenschaft, pp. 53–61, December 1981.
Cited on page 180.
184 D. R. Lorimer, Binary and millisecond pulsars, in www.livingreviews.org/lrr-2005-7, and
J. M. Weisberg & J. H. Taylor, The relativistic binary pulsar B1913+16: thirty years of
observations and analysis, pp. 25–31, in F. A. Rasio & I. H. Stairs, editors, Binary Radio
Pulsars, Proceedings of a meeting held at the Aspen Center for Physics, USA, 12 Janaury –
16 January 2004, volume 328 of ASP Conference Series, Astronomical Society of the Pacific,
2005. Cited on page 180.
185 W. B. B onnor & M. S. Piper, The gravitational wave rocket, Classical and Quantum
Gravity 14, pp. 2895–2904, 1997, or arxiv.org/abs/gr-qc/9702005. Cited on page 184.
186 Wolf gang Rindler, Essential Relativity, Springer, revised second edition, 1977. Cited

Motion Mountain – The Adventure of Physics
on page 186.
187 This is told (without the riddle solution) on p. 67, in Wolf gang Pauli, Relativität-
stheorie, Springer Verlag, Berlin, 2000, the edited reprint of a famous text originally
published in 1921. The reference is H. Vermeil, Notiz über das mittlere Krümmungs-
maß einer n-fach ausgedehnten Riemannschen Mannigfaltigkeit, Göttinger Nachrichten,
mathematische–physikalische Klasse p. 334, 1917. Cited on page 187.
188 M. Santander, L. M. Nieto & N. A. Cordero, A curvature based derivation of the
Schwarzschild metric, American Journal of Physics 65, pp. 1200–1209, 1997. Cited on pages
191, 193, and 194.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
189 Michael H. Soffel, Relativity in Astronomy, Celestial Mechanics and Geodesy, Springer
Verlag, 1989. Cited on page 192.
190 Richard P. Feynman, Fernando B. Morinigo, William G. Wagner &
Brian Hatfield, Feynman Lectures on Gravitation, Westview Press, 1995. Cited on
page 192.
191 J. C. Baez & E. F. Bunn, The meaning of Einstein’s equation, American Journal of Physics
73, pp. 644–652, 2005. Cited on page 195.
192 Y. Wang & M. Tegmark, New dark energy constraints from supernovae, microwave back-
ground and galaxy clustering, Physical Review Letters 92, p. 241302, 2004, or arxiv.org/
astro-ph/0403292. Cited on page 199.
193 Arguments for the emptiness of general covariance are given by John D. Norton, Gen-
eral covariance and the foundations of general relativity, Reports on Progress in Physics 56,
pp. 791–858, 1993. The opposite point, including the discussion of ‘absolute elements’, is
made in the book by J. L. Anderson, Principles of Relativity Physics, chapter 4, Academic
Press, 1967. Cited on page 199.
194 For a good introduction to mathematical physics, see the famous three-women text in
two volumes by Yvonne Choquet-Bruhat, Cecile DeWitt-Morette & Mar-
garet Dillard-Bleick, Analysis, Manifolds, and Physics, North-Holland, 1996 and
2001. The first edition of this classic appeared in 1977. Cited on page 200.
195 C. G. Torre & I. M. Anderson, Symmetries of the Einstein equations, Physical Review
Letters 70, pp. 3525–3529, 1993, or arxiv.org/abs/gr-qc/9302033. Cited on page 202.
340 bibliography

196 H. Nicolai, Gravitational billiards, dualities and hidden symmetries, arxiv.org//abs/gr-qc/
0506031. Cited on page 202.
197 The original paper is R. Arnowitt, S. Deser & C. Misner, Coordinate invariance and
energy expressions in general relativity, Physical Review 122, pp. 997–1006, 1961. Cited on
page 203.
198 See for example H. L. Bray, Black holes, geometric flows, and the Penrose inequality in gen-
eral relativity, Notices of the AMS 49, pp. 1372–1381, 2002. Cited on page 203.
199 See for example R.A. Knop & al., New constraints on Ω𝑀 , ΩΛ , and 𝑤 from an independent
set of eleven high-redshift supernovae observed with HST, Astrophysical Journal 598, pp. 102–
137, 2003. Cited on page 203.
200 See for example the paper by K. Dalton, Gravity, geometry and equivalence, preprint to
be found at arxiv.org/abs/gr-qc/9601004, and L. Landau & E. Lif shitz, The Classical
Theory of Fields, Pergamon, 4th edition, 1975, p. 241. Cited on page 204.
201 A recent overview on the experimental tests of the universality of free fall is that by
R. J. Hughes, The equivalence principle, Contemporary Physics 4, pp. 177–191, 1993. Cited
on page 206.

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202 The equivalence of the various definitions of the Riemann tensor is explained in most texts
on general relativity; see Ref. 113. Cited on page 207.
203 K. Tangen, Can the Pioneer anomaly have a gravitational origin?, arxiv.org/abs/gr-qc/
0602089. Cited on page 208.
204 H. Dittus & C. Lämmerzahl, Die Pioneer-Anomalie, Physik Journal 5, pp. 25–31,
January 2006. Cited on page 209.
205 Black hole analogues appear in acoustics, fluids and several other fields. This is an ongo-
ing research topic. See, for example, M. Novello, S. Perez Bergliaffa, J. Salim,
V. De Lorenci & R. Klippert, Analog black holes in flowing dielectrics, preprint

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
at arxiv.org/abs/gr-qc/0201061, T. G. Philbin, C. Kuklewicz, S. Robertson,
S. Hill, F. Konig & U. Leonhardt, Fiber-optical analog of the event horizon, Sci-
ence 319, pp. 1367–1379, 2008, O. Lahav, A. Itah, A. Blumkin, C. Gordon &
J. Steinhauer, A sonic black hole in a density-inverted Bose–Einstein condensate, arxiv.
org/abs/0906.1337. Cited on page 209.
206 This famous quote is the first sentence of the final chapter, the ‘Beschluß’, of Im-
manuel Kant, Kritik der praktischen Vernunft, 1797. Cited on page 211.
207 About the myths around the stars and the constellations, see the text by G. Fasching,
Sternbilder und ihre Mythen, Springer Verlag, 1993. On the internet there are also the beau-
tiful www.astro.wisc.edu/~dolan/constellations/ and www.astro.uiuc.edu/~kaler/sow/sow.
html websites. Cited on page 211.
208 Aetius, Opinions, III, I, 6. See Jean-Paul Dumont, Les écoles présocratiques, Folio Es-
sais, Gallimard, 1991, p. 445. Cited on page 211.
209 A. Mellinger, A color all-sky panorama of the Milky Way, preprint at arxiv.org/abs/0908.
4360. Cited on page 212.
210 P. Jetzer, Gravitational microlensing, Naturwissenschaften 86, pp. 201–211, 1999. Meas-
urements using orbital speeds around the Galaxy gives agree with this value. Cited on
pages 214 and 221.
211 Dirk Lorenzen, Geheimnivolles Universum – Europas Astronomen entschleiern das
Weltall, Kosmos, 2002. See also the beautiful website of the European Southern Observat-
ory at www.eso.org. Cited on page 214.
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212 A beautiful introduction to modern astronomy was Paolo Maffei, I mostri del cielo,
Mondadori Editore, 1976. Cited on page 220.
213 See for example A. N. Cox, ed., Allen’s Astrophysical Quantities, AIP Press and Springer
Verlag, 2000. An overview of optical observations is given by the Sloan Digital Sky Survey
at skyserver.sdss.org. More details about the universe can be found in the beautiful text
by W. J. Kaufmann & R. A. Fredman, Universe, fifth edition, W.H. Freeman & Co.,
1999. The most recent discoveries are best followed on the sci.esa.int and hubble.nasa.gov
websites. Cited on page 220.
214 D. R. Lorimer, A. J. Faulkner, A. G. Lyne, R. N. Manchester, M. Kramer,
M. A. McLaughlin, G. Hobbs, A. Possenti, I. H. Stairs, F. Camilo,
M. Burgay, N. D ’ Amico, A. Corongiu & F. Crawford, The Parkes multibeam
pulsar survey: VI. Discovery and timing of 142 pulsars and a Galactic population analysis,
Monthly Notices of the Royal Astronomical Society preprint at arxiv.org/abs/astro-ph/
0607640. Cited on page 221.
215 D. Figer, An upper limit to the masses of stars, Nature 434, pp. 192–194, 2005. Cited on
page 222.

Motion Mountain – The Adventure of Physics
216 G. Basri, The discovery of brown dwarfs, Scientific American 282, pp. 77–83, April 2001.
Cited on page 222.
217 See the well-written paper by P. M. Woods & C. Thompson, Soft gamma repeaters and
anomalous X-ray pulsars: magnetar candidates, preprint at arxiv.org/abs/astro-ph/0406133.
Cited on page 222.
218 B. M. Gaensler, N. M. McClure-Griffiths, M. S. Oey, M. Haverkorn,
J. M. Dickey & A. J. Green, A stellar wind bubble coincident with the anomalous X-ray
pulsar 1E 1048.1-5937: are magnetars formed from massive progenitors?, The Astrophysical
Journal (Letters) 620, pp. L95–L98, 2005, or arxiv.org/abs/astro-ph/0501563. Cited on page

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
222.
219 Opposition to the cosmological principle is rare, as experimental data generally supports
it. Local deviations are discussed by various cosmologists; the issue is still open. See, for
example, D. Wiltshire, Gravitational energy and cosmic acceleration, preprint at arxiv.
org/abs/0712.3982, and D. Wiltshire, Dark energy without dark energy, preprint at arxiv.
org/abs/0712.3984. Cited on page 226.
220 C. Wirtz, Scientia 38, p. 303, 1925, and K. Lundmark, The motions and the distances
of the spiral nebulae, Monthly Notices of the Royal Astronomical Society 85, pp. 865–894,
1925. See also G. Stromberg, Analysis of radial velocities of globular clusters and non-
galactic nebulae, Astrophysical Journal 61, pp. 353–362, 1925. Cited on page 226.
221 G. Gamow, The origin of the elements and the separation of galaxies, Physical Review 74,
p. 505, 1948. Cited on page 227.
222 A. G. Doroshkevich, & I. D. Novikov, Dokl. Akad. Nauk. SSSR 154, p. 809, 1964. It
appeared translated into English a few months later. The story of the prediction was told by
Penzias in his Nobel lecture. Cited on page 228.
223 Arno A. Penzias & Robert W. Wilson, A measurement of excess antenna temperat-
ure at 4080 Mcs, Astrophysical Journal 142, pp. 419–421, 1965. Cited on page 228.
224 See for example, D. Prialnik, An Introduction to the Theory of Stellar Structure and Evolu-
tion, Cambridge University Press, 2000. Cited on page 229.
225 Star masses are explored in D. Figier, An upper limit to the masses of stars, Nature 434,
pp. 192–194, 2005. Cited on page 229.
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226 Macrobius, Somnium Scipionis, XIV, 19. See Jean-Paul Dumont, Les écoles préso-
cratiques, Folio Essais, Gallimard, 1991, p. 61. Cited on page 230.
227 On the remote history of the universe, see the excellent texts by G. B örner, The Early Uni-
verse – Facts & Fiction, Springer Verlag, 3rd edition, 1993, or Barry Parker, Creation –
The Story of the Origin and the Evolution of the Universe, Plenum Press, 1988. For an excel-
lent popular text, see M. Longair, Our Evolving Universe, Cambridge University Press,
1996. Cited on page 230.
228 The first oxygen seems to have appeared in the atmosphere, produced by microorganisms,
2.32 thousand million years ago. See A. Becker & al., Dating the rise of atmospheric oxy-
gen, Nature 427, pp. 117–120, 2003. Cited on page 232.
229 Gabriele Walker, Snowball Earth – The Story of the Great Global Catastrophe That
Spawned Life as We Know It, Crown Publishing, 2003. Cited on page 232.
230 K. Knie, Spuren einer Sternexplosion, Physik in unserer Zeit 36, p. 8, 2005. The first
step of this connection is found in K. Knie, G. Korschinek, T. Faestermann,
E. A. Dorfi, G. Rugel & A. Wallner, 60 Fe anomaly in a deep-sea manganese crust
and implications for a nearby supernova source, Physics Review Letters 93, p. 171103, 2004,

Motion Mountain – The Adventure of Physics
the second step in N. D. Marsh & H. Svensmark, Low cloud properties influenced
by cosmic rays, Physics Review Letters 85, pp. 5004–5007, 2000, and the third step in
P. B. de Menocal, Plio-Pleistocene African climate, Science 270, pp. 53–59, 1995. Cited
on page 233.
231 A. Friedman, Über die Krümmung des Raumes, Zeitschrift für Physik 10, pp. 377–386,
1922, and A. Friedmann, Über die Möglichkeit einer Welt mit konstanter negativer Krüm-
mung des Raumes, Zeitschrift für Physik 21, pp. 326–332, 1924. (In the Latin transliteration,
the author aquired a second ‘n’ in his second paper.) Cited on page 234.
232 H. Knutsen, Darkness at night, European Journal of Physics 18, pp. 295–302, 1997. Cited

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
on pages 240 and 242.
233 See for example P.D. Peşić, Brightness at night, American Journal of Physics 66, pp. 1013–
1015, 1998. Cited on page 242.
234 Paul Wesson, Olbers’ paradox and the spectral intensity of extra-galactic background
light, Astrophysical Journal 367, p. 399, 1991. Cited on page 242.
235 Steven Weinberg, Gravitation and Cosmology, John Wiley, 1972. An excellent book
written with a strong personal touch and stressing most of all the relation with experimental
data. It does not develop a strong feeling for space-time curvature, and does not address the
basic problems of space and time in general relativity. Excellent for learning how to actually
calculate things, but less for the aims of our present adventure. Cited on pages 242 and 281.
236 Supernova searches are being performed by many research groups at the largest optical
and X-ray telescopes. A famous example is the Supernova Cosmology project described at
supernova.lbl.gov. Cited on page 244.
237 The experiments are discussed in detail in the excellent review by D. Giulini &
N. Straumann, Das Rätsel der kosmischen Vakuumenergiedichte und die beschleunigte
Expansion des Universums, Physikalische Blätter 556, pp. 41–48, 2000. See also
N. Straumann, The mystery of the cosmic vacuum energy density and the accelerated
expansion of the universe, European Journal of Physics 20, pp. 419–427, 1999. Cited on
pages 244 and 291.
238 A. Harvey & E. Schucking, Einstein’s mistake and the cosmological contant, American
Journal of Physics 68, pp. 723–727, 2000. Cited on page 245.
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239 The author of the bible explains rain in this way, as can be deduced from its very first page,
Genesis 1: 6-7. Cited on page 245.
240 Up to his death, Fred Hoyle defended his belief that the universe is not described by the big
bang, but by a steady flow; see G. Burbidge, F. Hoyle & J. V. Narlikar, A different
approach to cosmology, Physics Today 52, pp. 38–44, 1999. The team has also written a book
with the same title, published in 2000. The newest book on the topic is J. V. Narlikar &
G. Burbidge, Facts and speculations in Cosmology, Cambridge University Press, 2008,
well worth reading because it is one of the rare books that are thought-provoking. Cited on
page 246.
241 Stephen W. Hawking & G. F. R. Ellis, The Large Scale Structure of Space-Time, Cam-
bridge University Press, Cambridge, 1973. Among other things, this reference text discusses
the singularities of space-time, and their necessity in the history of the universe. Cited on
pages 246, 283, and 346.
242 Augustine, Confessions, 398, writes in Book XI: ‘My answer to those who ask ‘What was
god doing before he made Heaven and Earth?’ is not ‘He was preparing Hell for people who
pry into mysteries’. This frivolous retort has been made before now, so we are told, in order

Motion Mountain – The Adventure of Physics
to evade the point of the question. But it is one thing to make fun of the questioner and
another to find the answer. So I shall refrain from giving this reply. [...] Before God made
heaven and earth, he did not make anything at all. [...] But if before Heaven and Earth there
was no time, why is it demanded what you [god] did then? For there was no “then” when
there was no time.’ (Book XI, chapter 12 and 13). Cited on page 248.
243 Stephen Hawking, A Brief History of Time – From the Big Bang to Black Holes, 1988.
Reading this bestseller is almost a must for any physicist, as it is a frequent topic at dinner
parties. Cited on page 248.
244 Star details are explained in many texts on stellar structure and evolution. See for example

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Rudolf Kippenhahn & Alfred Weigert, Stellar Structure and Evolution, Springer,
1990. Cited on page 250.
245 J. Pelt, R. Kayser, S. Ref sdal & T. Schramm, The light curve and the time delay of
QSO 0957+561, Astronomy and Astrophysics 305, p. 97, 1996. Cited on page 252.
246 F. Zwicky, Nebulae as gravitational lenses, Physical Review Letters 51, p. 290, and
F. Zwicky, On the probability to detect nebulae which act as gravitational lenses, p. 679,
1937. The negative view by Einstein is found in A. Einstein, Lens-like action of a star by
the deviatioin of light in the gravitational field, Science 84, pp. 506–507, 1936. A review on
gravitational lensing can even be found online, in the paper by J. Wambsganss, Gravit-
ational lensing in astronomy, Living Reviews in Relativity 1-12, pp. 1–80, 1998, to be found
on the www.livingreviews.org/Articles/Volume1/1998-12wamb website.
There is also the book by P. Schneider, J. Ehlers & E. E. Falco, Gravitational
Lenses, Springer Verlag, Berlin, 1992. Cited on page 252.
247 M. Lachièze-Rey & J. -P. Luminet, Cosmic topology, Physics Reports 254, pp. 135–
214, 1995. See also B. F. Roukema, The topology of the universe, arxiv.org/abs/astro-ph/
0010185 preprint. Cited on page 254.
248 Steve Carlip clarified this point. Cited on page 255.
249 G. F. R. Ellis & T. Rothman, Lost horizons, American Journal of Physics 61, pp. 883–
893, 1993. Cited on page 255.
250 A. Guth, Die Geburt des Kosmos aus dem Nichts – Die Theorie des inflationären Univer-
sums, Droemer Knaur, 1999. Cited on page 256.
344 bibliography

251 Entropy values for the universe have been discussed by Ilya Prigogine, Is Future
Given?, World Scientific, 2003. This was his last book. For a different approach, see
G. A. Mena Marugán & S. Carneiro, Holography and the large number hypothesis,
arxiv.org/abs/gr-qc/0111034. This paper also repeats the often heard statement that the uni-
verse has an entropy that is much smaller than the theoretical maximum. The maximum is
often estimated to be in the range of 10100 𝑘 to 10120 𝑘. Other authors give 1084 𝑘. In 1974,
Roger Penrose also made statements about the entropy of the universe. However, it is more
correct to state that the entropy of the universe is not a useful quantity, because the universe
is not a physical system. Cited on page 257.
252 C. L. Bennet, M. S. Turner & M. White, The cosmic rosetta stone, Physics Today 50,
pp. 32–38, November 1997. The cosmic background radiation differs from black hole radi-
ation by less than 0.005 %. Cited on page 258.
253 The lack of expansion in the solar system is explained in detail in E. F. Bunn &
D. W. Hogg, The kinematic origin of the cosmological redshift, American Journal of
Physics 77, pp. 688–694, 2009. Cited on page 258.
254 A pretty article explaining how one can make experiments to find out how the hu-

Motion Mountain – The Adventure of Physics
man body senses rotation even when blindfolded and earphoned is described by M. -
L. Mittelstaedt & H. Mittelstaedt, The effect of centrifugal force on the percep-
tion of rotation about a vertical axis, Naturwissenschaften 84, pp. 366–369, 1997. Cited on
page 259.
255 No dependence of inertial mass on the distribution of surrounding mass has ever been
found in experiments. See, for example, R. H. Dicke, Experimental tests of Mach’s prin-
ciple, 7, pp. 359–360, 1961. Cited on page 259.
256 The present status is given in the conference proceedings by Julian Barbour & Her-
bert Pfister, eds., Mach’s Principle: From Newton’s Bucket to Quantum Gravity,
Birkhäuser, 1995. Various formulations of Mach’s principle – in fact, 21 different ones

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
– are compared on page 530.
In a related development, in 1953, Dennis Sciama published a paper in which he ar-
gues that inertia of a particle is due to the gravitational attraction of all other matter
in the universe. The paper is widely quoted, but makes no new statements on the issue.
See D. W. Sciama, On the origin of inertia, Monthly Notices of the Royal Astronomical
Society 113, pp. 34–42, 1953. Cited on page 259.
257 Information on the rotation of the universe is given in A. Kogut, G. Hinshaw &
A. J. Banday, Limits to global rotation and shear from the COBE DMR four-year sky maps,
Physical Review D 55, pp. 1901–1905, 1997. Earlier information is found in J. D. Barrow,
R. Juszkiewicz & D. H. Sonoda, Universal rotation: how large can it be?, Monthly No-
tices of the Royal Astronomical Society 213, pp. 917–943, 1985. See also J. D. Barrow,
R. Juszkiewicz & D. H. Sonoda, Structure of the cosmic microwave background,
Nature 309, pp. 397–402, 1983, or E. F. Bunn, P. G. Fereira & J. Silk, How anisotropic
is the universe?, Physical Review Letters 77, pp. 2883–2886, 1996. Cited on page 260.
258 The issue has been discussed within linearized gravity by Richard Tolman, in his text-
book Relativity, Thermodynamics, and Cosmology, Clarendon Press, 1934, on pp. 272–290.
The exact problem has been solved by A. Peres, Null electromagnetic fields in gen-
eral relativity theory, Physical Review 118, pp. 1105–1110, 1960, and by W. B. B onnor,
The gravitational field of light, Commun. Math. Phys. 13, pp. 163–174, 1969. See also
N. V. Mitskievic & K. K. Kumaradtya, The gravitational field of a spinning pencil
of light, Journal of Mathematical Physics 30, pp. 1095–1099, 1989, and P. C. Aichelburg
& R. U. Sexl, On the gravitational field of a spinning particle, General Relativity and Grav-
bibliography 345

itation 2, pp. 303–312, 1971. Cited on page 260.
259 See the delightful popular account by Igor Novikov, Black Holes and the Universe,
Cambridge University Press, 1990. The consequences of light decay were studied by
M. Bronshtein, Die Ausdehnung des Weltalls, Physikalische Zeitschrift der Sowjetunion
3, pp. 73–82, 1933. Cited on pages 261 and 267.
260 C. L. Carilli, K. M. Menten, J. T. Stocke, E. Perlman, R. Vermeulen,
F. Briggs, A. G. de Bruyn, J. Conway & C. P. Moore, Astronomical constraints
on the cosmic evolution of the fine structure constant and possible quantum dimensions,
Physical Review Letters 85, pp. 5511–5514, 25 December 2000. Cited on page 261.
261 The observations of black holes at the centre of galaxies and elsewhere are summarised by
R. Blandford & N. Gehrels, Revisiting the black hole, Physics Today 52, pp. 40–46,
June 1999. Cited on pages 262, 273, and 274.
262 An excellent and entertaining book on black holes, without any formulae, but nevertheless
accurate and detailed, is the paperback by Igor Novikov, Black Holes and the Universe,
Cambridge University Press, 1990. See also Edwin F. Taylor & John A. Wheeler,
Exploring Black Holes: Introduction to General Relativity, Addison Wesley Longman 2000.

Motion Mountain – The Adventure of Physics
For a historical introduction, see the paper by R. Ruffini, The physics of gravitation-
ally collapsed objects, pp. 59–118, in Neutron Stars, Black Holes and Binary X-Ray Sources,
Proceedings of the Annual Meeting, San Francisco, Calif., February 28, 1974, Reidel Pub-
lishing, 1975. Cited on page 262.
263 J. Michell, On the means of discovering the distance, magnitude, etc of the fixed stars,
Philosophical Transactions of the Royal Society London 74, p. 35, 1784, reprinted in
S. Detweiler, Black Holes – Selected Reprints, American Association of Physics Teach-
ers, 1982. Cited on page 262.
264 The beautiful paper is R. Oppenheimer & H. Snyder, On continued gravitational con-
traction, Physical Review 56, pp. 455–459, 1939. Cited on page 265.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
265 R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special met-
rics, Physical Review Letters 11, pp. 237–238, 1963. Cited on page 269.
266 E. T. Newman, E. Couch, R. Chinnapared, A. Exton, A. Prakash &
R. Torrence, Metric of a rotating, charged mass, Journal of Mathematical Physics 6,
pp. 918–919, 1965. Cited on page 269.
267 For a summary, see P. O. Mazur, Black hole uniqueness theorems, pp. 130–157, in
M. A. H. MacCallum, editor, General Relativity and Gravitation, Cambridge Univer-
sity Press, 1987, or the update at arxiv.org/abs/hep-th/0101012. See also D. C. Robinson,
Four decades of black hole uniqueness theorems, available at www.mth.kcl.ac.uk/staff/
dc_robinson/blackholes.pdf Cited on page 269.
268 H. P. Künzle & A. K. M. Masood-ul-Al am, Spherically symmetric static SU(2)
Einstein-Yang-Mills fields, Journal of Mathematical Physics 31, pp. 928–935, 1990. Cited on
page 269.
269 An example of research that shows the tendency of gravitational radiation to pro-
duce spherical shapes when black holes collide is L. Rezzolla, R. P. Macedo &
J. L. Jaramillo, Understanding the “anti kick” in the merger of binary black holes, Phys-
ical Review Letters 104, p. 221101, 2010. Cited on pages 270 and 290.
270 R. Penrose & R. M. Floyd, Extraction of rotational energy from a black hole, Nature
229, pp. 177–179, 1971. Cited on page 271.
271 The mass–energy relation for a rotating black hole is due to D. Christodoulou,
Reversible and irreversible transformations in black hole physics, Physical Review Let-
346 bibliography

ters 25, pp. 1596–1597, 1970. For a general, charged and rotating black hole it is due to
D. Christodoulou & R. Ruffini, Reversible transformations of a charged black hole,
Physical Review D 4, pp. 3552–3555, 1971. Cited on page 272.
272 J. D. Bekenstein, Black holes and entropy, Physical Review D7, pp. 2333–2346, 1973.
Cited on page 272.
273 On the topic of black holes in the early universe, there are only speculative research papers,
as found, for example, on arxiv.org. The issue is not settled yet. Cited on page 273.
274 For information about black holes formation via star collapse, see the Wikipedia article at
en.wikipedia.org/wikie/Stellar_black_hole. Cited on page 273.
275 Frederick Lamb, APS meeting 1998 press conference: Binary star 4U1820-30, 20 000
light years from Earth, Physics News Update, April 27, 1998. Cited on page 274.
276 The first direct evidence for matter falling into a black hole was published in early 2001 by
NASA astronomers led by Joseph Dolan. Cited on page 274.
277 For a readable summary of the Penrose–Hawking singularity theorems, see J. Natàrio,
Relativity and singularities – a short introduction for mathematicians, preprint at arxiv.org/

Motion Mountain – The Adventure of Physics
abs/math.DG/0603190. Details can be found in Ref. 241. Cited on page 274.
278 For an overview of cosmic censorship, see T. P. Singh, Gravitational collapse, black
holes and naked singularities, arxiv.org/abs/gr-qc/9805066, or R. M. Wald, Gravitational
collapse and cosmic censorship, arxiv.org/abs/gr-qc/9710068. The original idea is due to
R. Penrose, Gravitational collapse: the role of general relativity, Rivista del Nuovo Ci-
mento 1, pp. 252–276, 1969. Cited on page 275.
279 The paradox is discussed in M. A. Abramowicz, Black holes and the centrifugal
force paradox, Scientific American 266, pp. 74–81, March 1993, and in the comment by
D. N. Page, Relative alternatives, Scientific American 266, p. 5, August 1993. See also
M. A. Abramowicz & E. Szuszkiewicz, The wall of death, American Journal of

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Physics 61, pp. 982–991, 1993, and M. A. Abramowicz & J. P. Lasota, On traveling
round without feeling it and uncurving curves, American Journal of Physics 54, pp. 936–
939, 1986. Cited on page 277.
280 J. Ehlers, Introduction – Survey of Problems, pp. 1–10, in J. Ehlers, editor, Sistemi
gravitazionali isolati in relatività generale, Rendiconti della scuola internazionale di fisica
“Enrico Fermi”, LXVIIo corso, Società Italiana di Fisica/North Holland, 1979. Cited on
page 278.
281 G. J. Stoney, On the physical units of nature, Philosophical Magazine 11, pp. 381–391, 1881.
Cited on page 282.
282 The geometrodynamic clock is discussed in D. E. Brahm & R. P. Gruber, Limitations
of the geometrodynamic clock, General Relativity and Gravitation 24, pp. 297–303, 1992. The
clock itself was introduced by R. F. Marzke, in his Ph.D. thesis The theory of measurement
in general relativity, 1959, with John Wheeler as thesis adviser. Cited on page 282.
283 R. Geroch, Einstein algebras, Commun. Math. Phys. 26, pp. 271–275, 1972. Cited on page
283.
284 A. Macdonald, Einstein’s hole argument, American Journal of Physics 69, pp. 223–225,
2001. Cited on page 284.
285 Roman U. Sexl, Die Hohlwelttheorie, Der mathematisch-naturwissenschaftliche Unter-
richt 368, pp. 453–460, 1983. Roman U. Sexl, Universal conventionalism and space-
time., General Relativity and Gravitation 1, pp. 159–180, 1970. See also Roman U. Sexl,
Die Hohlwelttheorie, in Arthur Scharmann & Herbert Schramm, editors, Physik,
bibliography 347

Theorie, Experiment, Geschichte, Didaktik – Festschrift für Wilfried Kuhn zum 60. Geburtstag
am 6. Mai 1983, Aulis Verlag Deubner, 1984, pp. 241–258. Cited on page 285.
286 T. Damour, Experimental tests of relativistic gravity, arxiv.org/abs/gr-qc/9904057. It is the
latest in a series of his papers on the topic; the first was T. Damour, Was Einstein 100 %
right?, arxiv.org/abs/gr-qc/9412064. Cited on pages 289 and 290.
287 H. Dittus, F. Everitt, C. Lämmerzahl & G. Schäfer, Die Gravitation im Test,
Physikalische Blätter 55, pp. 39–46, 1999. Cited on page 289.
288 For theories competing with general relativity, see for example the extensive and excellent
review by C. M. Will, The confrontation between general relativity and experiment, Living
Reviews of Relativity 2001-2014, electronic version at www.livingreviews.org/lrr-2001-4,
update at www.livingreviews.org/lrr-2006-3 and preprint at arxiv.org/abs/1403.7377. For
example, the absence of the Nordtvedt effect, a hypothetical 28-day oscillation in the Earth–
Moon distance, which was looked for by laser ranging experiments without any result,
eliminated several competing theories. This effect, predicted by Kenneth Nordtvedt, would
only appear if the gravitational energy in the Earth–Moon system would fall in a differ-
ent way than the Earth and the Moon themselves. For a summary of the measurements,

Motion Mountain – The Adventure of Physics
see J. Müller, M. Schneider, M. Soffel & H. Ruder, Testing Einstein’s theory of
gravity by analyzing lunar laser ranging data, Astrophysical Journal Letters 382, pp. L101–
L103, 1991. Cited on page 289.
289 See S. Bässler & al., Improved test of the equivalence principle for gravitational selfenergy,
Physical Review Letters 83, pp. 3585–3588, 1999. See also C. M. Will, Gravitational radi-
ation and the validity of general relativity, Physics Today 52, p. 38, October 1999. Cited on
page 290.
290 The inverse square dependence has been checked down to 60 μm, as reported by
E. Adelberger, B. Heckel & C. D. Hoyle, Testing the gravitational inverse-square
law, Physics World 18, pp. 41–45, 2005. Cited on page 290.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
291 Almost everything of importance in general relativity is published in the free and excel-
lent internet-based research journal Living Reviews in Relativity, to be found at the www.
livingreviews.org website. The other important journal in the field is Classical and Quantum
Gravity. In astrophysics, the central publication is Astronomy & Astrophysics. Cited on page
290.
292 The study of chaos in Einstein’s field equations is just beginning. See, e.g., L. B ombelli,
F. Lombardo & M. Castagnino, Chaos in Robertson-Walker cosmology, arxiv.org/abs/
gr-qc/9707051. Cited on page 291.
293 The ESA satellite called ‘Planck’ has measured the polarization of the cosmic microwave
background. This will provide more details on galaxy formation. Cited on page 291.
294 A good introduction to the topic of gamma-ray bursts is S. Klose, J. Greiner &
D. Hartmann, Kosmische Gammastrahlenausbrüche – Beobachtungen und Modelle, Teil
I und II, Sterne und Weltraum March and April 2001. Cited on page 291.
295 The field solution database is built around the work of A. Karlhede, which allows one to
distinguish between solutions with a limited amount of mathematical computation. Cited
on page 291.
296 Beautiful simulated images of wormholes are available, for example on the wonderful web-
site www.tempolimit-lichtgeschwindigkeit.de. However, quantum effects forbid their exist-
ence, so that no such image is included here. A basic approach is the one by T. Diemer
& M. Hadley, Charge and the topology of spacetime, Classical and Quantum Gravity 16,
pp. 3567–3577, 1999, or arxiv.org/abs/gr-qc/9905069 and M. Hadley, Spin half in clas-
348 bibliography

sical general relativity, Classical and Quantum Gravity 17, pp. 4187–4194, 2000, or arxiv.
org/abs/gr-qc/0004029. Cited on page 291.
297 An important formulation of relativity is A. Ashtekar, New variables for classical and
quantum gravity, Physical Review Letters 57, pp. 2244–2247, 1986. Cited on page 291.
298 For a review on inflation and early universe, see D. Baumann, TASI lectures on inflation,
preprint at arxiv.org/abs/0907.5424. Cited on page 291.
299 A well written text on the connections between the big bang and particle physics is by
I. L. Rozental, Big Bang – Big Bounce, How Particles and Fields Drive Cosmic Evolution,
Springer, 1988. For another connection, see M. Nagano & A. A. Watson, Observations
and implications of the ultrahigh energy cosmic rays, Reviews of Modern Physics 72, pp. 689–
732, 2000. Cited on page 291.
300 Teaching will benefit in particular from new formulations, from concentration on prin-
ciples and their consequences, as has happened in special relativity, from simpler descrip-
tions at the weak field level, and from future research in the theory of general relativity. The
newer textbooks cited above are all steps in these directions. Cited on page 292.
301 G. E. Prince & M. Jerie, Generalising Raychaudhuri’s equation, in Differential Geo-

Motion Mountain – The Adventure of Physics
metry and Its Applications, Proc. Conf., Opava (Czech Republic), August 27-31, 2001,
Silesian University, Opava, 2001, pp. 235–242. Cited on page 293.
302 Torsion is presented in R. T. Hammond, New fields in general relativity, Contemporary
Physics 36, pp. 103–114, 1995. Cited on page 293.
303 A well-known approach is that by Bekenstein; he proposes a modification of general relativ-
ity that modifies univesal, 1/𝑟2 gravity at galactic distances. This is done in order to explain
the hundreds of measured galactic rotation curves that seem to require such a modifica-
tion. (This approach is called modified Newtonian dynamics or MOND.) An introduction
is given by Jacob D. Bekenstein, The modified Newtonian dynamics – MOND – and its
implications for new physics, Contemporary Physics 47, pp. 387–403, 2006, preprint at arxiv.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
org/abs/astro-ph/0701848v2. Cited on page 293.
304 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de
Breteuil, Parc de Saint Cloud, 92310 Sèvres, France. All new developments concerning SI
units are published in the journal Metrologia, edited by the same body. Showing the slow
pace of an old institution, the BIPM launched a website only in 1998; it is now reachable at
www.bipm.fr. See also the www.utc.fr/~tthomass/Themes/Unites/index.html website; this
includes the biographies of people who gave their names to various units. The site of its
British equivalent, www.npl.co.uk/npl/reference, is much better; it provides many details
as well as the English-language version of the SI unit definitions. Cited on page 295.
305 The bible in the field of time measurement is the two-volume work by J. Vanier &
C. Audoin, The Quantum Physics of Atomic Frequency Standards, Adam Hilge, 1989. A
popular account is Tony Jones, Splitting the Second, Institute of Physics Publishing, 2000.
The site opdaf1.obspm.fr/www/lexique.html gives a glossary of terms used in the field.
For precision length measurements, the tools of choice are special lasers, such as mode-
locked lasers and frequency combs. There is a huge literature on these topics. Equally large
is the literature on precision electric current measurements; there is a race going on for the
best way to do this: counting charges or measuring magnetic forces. The issue is still open.
On mass and atomic mass measurements, see page 71. On high-precision temperature meas-
urements, see Volume I, on page 548. Cited on page 296.
306 The unofficial SI prefixes were first proposed in the 1990s by Jeff K. Aronson of the Uni-
versity of Oxford, and might come into general usage in the future. See New Scientist 144,
p. 81, 3 December 1994. Other, less serious proposals also exist. Cited on page 297.
bibliography 349

307 See the review by L. Ju, D. G. Blair & C. Zhao, The detection of gravitational waves,
Reports on Progress in Physics 63, pp. 1317–1427, 2000. Cited on page 299.
308 See the clear and extensive paper by G. E. Stedman, Ring laser tests of fundamental physics
and geophysics, Reports on Progress in Physics 60, pp. 615–688, 1997. Cited on page 299.
309 J. Short, Newton’s apples fall from grace, New Scientist 2098, p. 5, 6 September 1997. More
details can be found in R. G. Keesing, The history of Newton’s apple tree, Contemporary
Physics 39, pp. 377–391, 1998. Cited on page 300.
310 The various concepts are even the topic of a separate international standard, ISO 5725, with
the title Accuracy and precision of measurement methods and results. A good introduction is
John R. Taylor, An Introduction to Error Analysis: the Study of Uncertainties in Physical
Measurements, 2nd edition, University Science Books, Sausalito, 1997. Cited on page 300.
311 P. J. Mohr & B. N. Taylor, CODATA recommended values of the fundamental physical
constants: 1998, Reviews of Modern Physics 59, p. 351, 2000. This is the set of constants res-
ulting from an international adjustment and recommended for international use by the
Committee on Data for Science and Technology (CODATA), a body in the International
Council of Scientific Unions, which brings together the International Union of Pure and

Motion Mountain – The Adventure of Physics
Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry (IUPAC)
and other organizations. The website of IUPAC is www.iupac.org. Cited on page 302.
312 Some of the stories can be found in the text by N. W. Wise, The Values of Precision,
Princeton University Press, 1994. The field of high-precision measurements, from which
the results on these pages stem, is a world on its own. A beautiful introduction to it
is J. D. Fairbanks, B. S. Deaver, C. W. Everitt & P. F. Michaelson, eds., Near
Zero: Frontiers of Physics, Freeman, 1988. Cited on page 302.
313 For details see the well-known astronomical reference, P. Kenneth Seidelmann, Ex-
planatory Supplement to the Astronomical Almanac, 1992. Cited on page 307.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
314 See the corresponding reference in the first volume.
Note that little is known about the basic properties of some numbers; for example, it is
Challenge 429 r still not known whether π + 𝑒 is a rational number or not! (It is believed that it is not.) Do
Challenge 430 s you want to become a mathematician? Cited on page 309.
C R E DI T S

Acknowled gements
Many people who have kept their gift of curiosity alive have helped to make this project come
true. Most of all, Peter Rudolph and Saverio Pascazio have been – present or not – a constant
reference for this project. Fernand Mayné, Ata Masafumi, Roberto Crespi, Serge Pahaut, Luca
Bombelli, Herman Elswijk, Marcel Krijn, Marc de Jong, Martin van der Mark, Kim Jalink, my
parents Peter and Isabella Schiller, Mike van Wijk, Renate Georgi, Paul Tegelaar, Barbara and

Motion Mountain – The Adventure of Physics
Edgar Augel, M. Jamil, Ron Murdock, Carol Pritchard, Richard Hoffman, Stephan Schiller, Franz
Aichinger and, most of all, my wife Britta have all provided valuable advice and encouragement.
Many people have helped with the project and the collection of material. In particular, I thank
Steve Carlip, Corrado Massa, Tom Helmond, Gary Gibbons, Ludwik Kostro, Heinrich Neumaier,
Peter Brown and David Thornton for interesting discussions on maximum force. Most useful was
the help of Mikael Johansson, Bruno Barberi Gnecco, Lothar Beyer, the numerous improvements
by Bert Sierra, the detailed suggestions by Claudio Farinati, the many improvements by Eric Shel-
don, the detailed suggestions by Andrew Young, the continuous help and advice of Jonatan Kelu,
the corrections of Elmar Bartel, and in particular the extensive, passionate and conscientious
help of Adrian Kubala.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Important material was provided by Bert Peeters, Anna Wierzbicka, William Beaty, Jim Carr,
John Merrit, John Baez, Frank DiFilippo, Jonathan Scott, Jon Thaler, Luca Bombelli, Douglas
Singleton, George McQuarry, Tilman Hausherr, Brian Oberquell, Peer Zalm, Martin van der
Mark, Vladimir Surdin, Julia Simon, Antonio Fermani, Don Page, Stephen Haley, Peter Mayr,
Allan Hayes, Norbert Dragon, Igor Ivanov, Doug Renselle, Wim de Muynck, Steve Carlip, Tom
Bruce, Ryan Budney, Gary Ruben, Chris Hillman, Olivier Glassey, Jochen Greiner, squark, Mar-
tin Hardcastle, Mark Biggar, Pavel Kuzin, Douglas Brebner, Luciano Lombardi, Franco Bagnoli,
Lukas Fabian Moser, Dejan Corovic, Paul Vannoni, John Haber, Saverio Pascazio, Klaus Finken-
zeller, Leo Volin, Jeff Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo
Uguzzoni, Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo Thies, Vi-
taliy Solomatin, Carl Offner, Nuno Proença, Elena Colazingari, Paula Henderson, Daniel Darre,
Wolfgang Rankl, John Heumann, Joseph Kiss, Martha Weiss, Antonio González, Antonio Mar-
tos, André Slabber, Ferdinand Bautista, Zoltán Gácsi, Pat Furrie, Michael Reppisch, Enrico Pasi,
Thomas Köppe, Martin Rivas, Herman Beeksma, Tom Helmond, John Brandes, Vlad Tarko, Na-
dia Murillo, Ciprian Dobra, Romano Perini, Harald van Lintel, Andrea Conti, François Belfort,
Dirk Van de Moortel, Heinrich Neumaier, Jarosław Królikowski, John Dahlman, Fathi Namouni,
Paul Townsend, Sergei Emelin, Freeman Dyson, S.R. Madhu Rao, David Parks, Jürgen Janek,
Daniel Huber, Alfons Buchmann, William Purves, Pietro Redondi, Sergei Kopeikin, Damoon
Saghian, Zach Joseph Espiritu, David Thornton, Miles Mutka, Fabrizio Bònoli, plus a number of
people who wanted to remain unnamed.
The software tools were refined with extensive help on fonts and typesetting by Michael Zedler
and Achim Blumensath and with the repeated and valuable support of Donald Arseneau; help
credits 351

came also from Ulrike Fischer, Piet van Oostrum, Gerben Wierda, Klaus Böhncke, Craig Up-
right, Herbert Voss, Andrew Trevorrow, Danie Els, Heiko Oberdiek, Sebastian Rahtz, Don Story,
Vincent Darley, Johan Linde, Joseph Hertzlinger, Rick Zaccone, John Warkentin, Ulrich Diez,
Uwe Siart, Will Robertson, Joseph Wright, Enrico Gregorio, Rolf Niepraschk and Alexander
Grahn.
The typesetting and book design is due to the professional consulting of Ulrich Dirr. The
typography was much improved with the help of Johannes Küster and his Minion Math font.
The design of the book and its website also owe much to the suggestions and support of my wife
Britta.
I also thank the lawmakers and the taxpayers in Germany, who, in contrast to most other
countries in the world, allow residents to use the local university libraries.
From 2007 to 2011, the electronic edition and distribution of the Motion Mountain text was
generously supported by the Klaus Tschira Foundation.

Film credits
The impressive film of a light pulse bouncing of a mirror on page 21 is copyright and courtesy

Motion Mountain – The Adventure of Physics
of Wang Lihong and Washington University at St. Louis. The beautiful animation of a dice fly-
ing at relativistic speed, on page 57, is copyright and courtesy by Ute Kraus. It can be found
on her splendid website www.tempolimit-lichtgeschwindigkeit.de, which provides many other
films of relativistic motions and the related publications. The beautiful animation of an observer
accelerating in a desert, on page 91, is copyright Anthony Searle and Australian National Univer-
sity, and courtesy of Craig Savage. It is from the wonderful website at www.anu.edu.au/Physics/
Savage/TEE. Also the equally beautiful animation of an observer accelerating between houses, on
page 93, is copyright Anthony Searle and Australian National University, and courtesy of Craig
Savage. It is from the equally wonderful website at www.anu.edu.au/Physics/Searle. The spectac-
ular animation on page 263 is courtesy and copyright of the European Southern Observatory
ESO and found on its website www.eso.org/public/news/eso0846/.

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Image credits
The photograph of the east side of the Langtang Lirung peak in the Nepalese Himalayas, shown
on the front cover, is courtesy and copyright by Kevin Hite and found on his blog thegettingthere.
com. The photograph of the night sky on page 14 is copyright and courtesy of Anthony Ayiomam-
itis; it is found on his wonderful website www.perseus.gr. The photograph of the reconstruction
of Fizeau’s experiment on page 20 is copyright by AG Didaktik und Geschichte der Physik, Uni-
versität Oldenburg, and courtesy of Jan Frercks, Peter von Heering and Daniel Osewold. The
photograph of a light pulse on page 20 is courtesy and copyright of Tom Mattick. On page 24
the photographs of electrical devices are courtesy Miele and EasyGlide. On page 30, the lithium
images are courtesy and copyright of the TSR relativity team at the Max Planck Gesellschaft.
On page 32 the Doppler images are copyright and courtesy of Maurice Gavin and NASA. On
page 33, the Doppler sonar system images are copyright and courtesy of Wikimedia, Hörmann
AG and Medison. On page 34, the wave graphic is copyright Pbroks13 and courtesy Wikime-
dia. On page 41, the image of the historical Michelson experiment is courtesy and copyright of
the Astrophysikalisches Institut Potsdam, and the images of the modern high-precision experi-
ment are copyright and courtesy of Stephan Schiller. The relativistic views on page 55 and 57 are
courtesy and copyright of Daniel Weiskopf. The relativistic images of the travel through the sim-
plified Stonehenge on page 56 are copyright of Nicolai Mokros and courtesy of Norbert Dragon.
On page 68, the photograph of the HARP experiment is courtesy and copyright of CERN. On
page 69, the photographs about the bubble chamber are courtesy and copyright of CERN. The
352 credits

stalactite photograph on page 108 is courtesy and copyright of Richard Cindric and found on the
website www.kcgrotto.org; the photograph of Saturn is courtesy NASA. The illustration of spatial
curvature on page 145 is courtesy and copyright of Farooq Ahmad Bhat. On page 150, the volcano
photograph is copyright and courtesy of Marco Fulle and found on the wonderful website www.
stromboli.net. On page 181, the VIRGO photographs are courtesy and copyright of INFN. On
page 168, the photographs about lunar reflectors are copyright and courtesy NASA and Wikime-
dia; the photograph of the Nice observatory is courtesy and copyright of Observatoire de la Côte
d’Azur. The figures of galaxies on pages 214, 212, 215, 215, 213, 219, 216, 243, 253 and 254 are cour-
tesy of NASA. The photo of the night sky on page 212 is copyright and courtesy of Axel Mellinger;
more details on the story of this incredible image is found on his website at home.arcor.de/axel.
mellinger. The picture of the universe on page 213 is courtesy of Thomas Jarret, IPAC and Caltech,
and is found on the spider.ipac.caltech.edu/staff/jarret/lss/index.html website. The photograph
of the molecular cloud on page 216 is courtesy and copyright of the European Southern Obser-
vatory ESO; it was also featured on the antwrp.gsfc.nasa.gov/apod/ap030202.html website. On
page 217, the photopgraphs of the Very Large Telescopes are copyright and courtesy of ESO. On
page 218, the photographs of the XMM-Newton satellite and of the Planck satellite are copyright
and courtesy of ESA and found on the fascinting website www.esa.int. The Hubble diagram on

Motion Mountain – The Adventure of Physics
page 227 is courtesy of Saul Perlmutter and the Supernova Cosmology Project. The maps of the
universe on page 225 and the Hertzsprung–Russell diagram on page 229 are courtesy and copy-
right of Richard Powell, and taken from his websites www.anzwers.org/free/universe and www.
atlasoftheuniverse.com. On page 230, the photograph of M15 is copyright and courtesy of ESA
and NASA. The photograph on page 240 is courtesy and copyright of Wally Pacholka and found
on the wonderful website www.twanlight.org that collects pictures of the world at night. On page
page 241, the tree image is copyright of Aleks G and courtesy of Wikimedia, whereas the Hubble
deep sky image is courtesy of NASA and ESA and found at apod.nasa.gov/apod/ap140605.html.
On page 243, the Planck data map is courtesy and copyright of Planck/ESA. The simulated view of
a black hole on page 264 is copyright and courtesy of Ute Kraus and can be found on her splendid

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
website www.tempolimit-lichtgeschwindigkeit.de. On page 285, the image of the hollow Earth
theory is courtesy of Helmut Diel and was drawn by Isolde Diel. On page 312, the drawing of
the Fraunhofer lines is copyright and courtesy of Fraunhofer Gesellschaft. The photograph on
the back cover, of a basilisk running over water, is courtesy and copyright by the Belgian group
TERRA vzw and found on their website www.terravzw.org. All drawings are copyright by Chris-
toph Schiller. If you suspect that your copyright is not correctly given or obtained, this has not
been done on purpose; please contact me in this case.
NA M E I N DE X

A A Augustine of Hippo 247 Bergliaffa, S. Perez 340
Abramowicz Abbott, B.P. 331 Australian National Bergson, Henri 25
Abdo, A.A. 322 University 91, 93, 351 Bertotti, B. 337
Abramowicz, M.A. 346 Ayiomamitis, Anthony 16, 351 Bessel, Friedrich Wilhelm 240
Adelberger, E. 347 Besso, Michele 76

Motion Mountain – The Adventure of Physics
Adenauer, Konrad 129 B Beyer, Lothar 350
Adler, C.G. 330 Babinet, Jacques Biggar, Mark 350
Aetius 211, 340 life 296 Bilaniuk, O.M. 328
Ahmad Bhat, Farooq 145, 352 Bachem, Albert 140 Birkhoff 201
Ahmad, Q.R. 325 Baez, John 339, 350 Bladel, Jean van 323
Aichelburg, P.C. 344 Baggett, N. 323 Blair, David 335, 349
Alanus de Insulis 262 Bagnoli, Franco 350 Blandford, R. 345
Alcubierre, M. 335 Bailey, J. 326, 327 Blandford, R.D. 337
Aleks G 241, 352 Bailey, J.M. 322 Blau, Steven 47, 326
Allen, Woody 223 Banday, A.J. 344 Bliokh, K.Y. 316

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Alspector, J. 323 Barberi Gnecco, Bruno 350 Bloser, P.F. 336
Alväger, T. 322, 323 Barbour, Julian 344 Blumensath, Achim 350
Anderson, I.M. 202, 339 Barrow, J.D. 332, 344 Blumkin, A. 340
Anderson, J.D. 328 Bartel, Elmar 350 Bohr, Niels 25
Anderson, J.L. 339 Bartocci, Umberto 76, 329 Bombelli, Luca 347, 350
Antonini, P. 324, 326 Basri, G. 341 Bondi, Hermann 324, 334
Arago, François 40 Bateman, H. 325 Bonnor, W.B. 184, 339, 344
Archytas of Tarentum 255 Baumann, D. 348 Boone, Roggie 350
Aristarchus of Samos 321 Bautista, Ferdinand 350 Born, Max 325
life 19 Baylis, W.E. 326 Boughn, S.P. 327
Aristotle 321 Beaty, William 350 Boyce, K.R. 328
Arnowitt, Richard 203, 340 Becker, A. 342 Brace, Dewitt 41
Aronson, Jeff K. 348, 350 Bedford, D. 337 Bradley, James 17, 18
Arseneau, Donald 350 Beeckman, Isaac 16 Braginsky, V.B. 336, 338
Ashtekar, A. 332, 348 Beeksma, Herman 350 Brahm, D.E. 346
Astrophysikalisches Institut Behroozi, C.H. 324 Brandes, John 350
Potsdam 41, 351 Beig, R. 331 Brault, J.W. 334
Ata Masafumi 350 Bekenstein, Jacob 272, 346, Braxmeier, C. 324
Audoin, C. 348 348 Bray, H.L. 340
Augel, Barbara 350 Belfort, François 350 Brebner, Douglas 350
Augel, Edgar 350 Bender, P.L. 337 Brecher, K. 322
Augustine 343 Bennet, C.L. 344 Brehme, R.W. 330
354 name index

Brewer, Sydney G. 321 Conti, Andrea 350 Dittus, H. 340, 347
Briatore, L. 140, 334 Conway, J. 345 Dobra, Ciprian 350
Briggs, F. 345 Copernicus, Nicolaus 19 Dolan, Joseph 346
Broeck, Chris Van Den 335 Cordero, N.A. 339 Doppler, Christian
Bronshtein, Matvei 8 Cornell, E.A. 324 life 33
Bronshtein, Matvey 261, 345 Corongiu, A. 341 Dorfi, E.A. 342
Brown, J.M. 328 Corovic, Dejan 350 Doroshkevich, A.G. 228, 341
Brown, Peter 350 Costa, S.S. 328 Dragon, Norbert 54, 56, 350,
Bruce, Tom 350 Costella, J.P. 328 351
Bruyn, A.G. de 345 Couch, E. 345 Droste, J. 335
Buchmann, Alfons 350 Cox, A.N. 341 Droste, Johannes 146
B Budney, Ryan 350
Bunn, E.F. 339, 344
Crawford, F. 341
Crelinsten, Jeffrey 334
Duff, M.J. 330
Duguay, M.A. 322
Burbidge, G. 343 Crespi, Roberto 350 Dumont, Jean-Paul 321, 340,
Brewer Burgay, M. 341 Crowe, Michael J. 321 342
Bäßler, S. 347 Currie, D.G. 329 Dutton, Z. 324
Bònoli, F. 321 Dyson, F.W. 334

Motion Mountain – The Adventure of Physics
Bònoli, Fabrizio 350 D Dyson, Freeman 350
Böhncke, Klaus 351 D’Amico, N. 341
Börner, G. 342 Dahlman, John 350 E
Börner, H.G. 328 Dalton, K. 340 EasyGlide 24, 351
Damour, Thibault 289, 347 Eckstein, G. 325
C Danecek, Petr 350 Eddington, A.S. 334
Caianiello, E.R. 332 Darley, Vincent 351 Ehlers, J. 343, 346
Calder, Nigel 336 Darre, Daniel 350 Ehlers, Jürgen 278
Caltech 213, 352 Davidson, C. 334 Ehrenfest, P. 329
Camilo, F. 341 Deaver, B.S. 349 Eichenwald, Alexander 40

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Cannoni, M. 328 Democritus of Abdera 99 Einstein, A. 338
Carilli, C.L. 345 Descartes, René 16 Einstein, Albert 24, 25, 27, 28,
Carlip, Steve 332, 343, 350 Deser, Stanley 203, 340 45, 62, 71, 76, 136, 140, 143,
Carneiro, S. 344 Deshpande, V.K. 328 152, 160, 174, 198, 200, 233,
Carr, Jim 350 Deslattes, R.D. 328 284, 289, 290, 293, 323, 329,
Carter, Brandon 269 Desloge, E.A. 330 332, 333, 336, 343
Cassini, Giovanni 16 Desloge, Edward A. 330 life 25–26
Castagnino, M. 347 Detweiler, S. 345 Einstein, Eduard 143
Caves, C.M. 338 Dewey, M.S. 328 Eisele, Ch. 326
CERN 68, 69, 351 DeWitt-Morette, Cecile 339 Ellis, George 283, 335, 343
Cheseaux, Jean Philippe Loÿs Dicke, R.H. 336, 344 Els, Danie 351
de 241 Dickey, J.M. 341 Elswijk, Herman B. 350
Chinnapared, R. 345 Diehl, Helmut 285 Emelin, Sergei 350
Choquet-Bruhat, Yvonne 339 Diel, Helmut 352 Empedocles 15
Christodoulou, D. 345, 346 Diel, Isolde 352 Eötvös, Roland von 158, 336
Cindric, Richard 108, 352 Diemer, T. 347 ESA 218, 230, 241, 352
Ciufolini, Ignazio 170, 337 Dietze, H. 325 Eshelby, J. 325
Clancy, E.P. 334 Diez, Ulrich 351 ESO 216, 217, 351, 352
Clausius, Rudolph 256, 257 DiFilippo, Frank 328, 350 Espiritu, Zach Joseph 350
Cohen, M.H. 327 Dillard-Bleick, Margaret 339 Euler, Leonhard 187
Colazingari, Elena 350 Dirr, Ulrich 351 European Southern
Columbus 259 DiSessa, A. 331 Observatory 351, 352
name index 355

Everitt, C.W. 349 G Hall, D.B. 326
Everitt, F. 347 Gabuzda, D.C. 327 Halley, Edmund 17
Ewing, Anne 263 Gaensler, B.M. 341 Hamilton, J.D. 330
Exton, A. 345 Galilei, Galileo 16 Hammond, R.T. 348
Gallisard de Marignac, Jean Hardcastle, Martin 350
F Charles 76 Harris, S.E. 324
Faestermann, T. 342 Gamow, George 341 Hartmann, D. 347
Fairbanks, J.D. 349 life 227 Harvey, A. 327, 342
Fairhust, S. 332 Gauß, Carl-Friedrich Hasenöhrl, Friedrich 76, 329
Falco, E.E. 343 life 189 Hatfield, Brian 339
Falk, G. 333 Gavin, Maurice 32, 351 Hausherr, Tilman 350
E Farinati, Claudio 350
Farley, F.J.M. 322
Gearhart, R. 325
Gehrels, N. 345
Haverkorn, M. 341
Hawking, Stephen 131, 248,
Fasching, G. 340 Georgi, Renate 350 272, 274, 283, 343
Everit t Faulkner, A.J. 341 Geroch, Robert 283, 346 Hawking, Stephen W. 343
Fekete, E. 336 Gibbons, G.W. 332 Hayes, Allan 350
Fereira, P.G. 344 Gibbons, Gary 125, 331, 350 Hayward, S.A. 332

Motion Mountain – The Adventure of Physics
Fermani, Antonio 350 Gibbs, J. Willard 108, 330 Heckel, B. 347
Feynman, Richard P. 339 Gide, André 194 Heering, Peter von 351
Figer, D. 341 Giltner, D.M. 324 Helmond, Tom 350
Figier, D. 341 Giulini, D. 342 Henderson, Paula 350
Finkenzeller, Klaus 350 Glassey, Olivier 350 Heracles 214
Fischer, Ulrike 351 Goenner, Hubert 179, 334 Heraclitus of Ephesus 230
Fishman, G.J. 322 González, Antonio 350 Herrmann, F. 332
Fitzgerald, George 44 Good, R.H. 330 Herrmann, Friedrich 331
Fizeau, Hippolyte 19, 40 Gordon, C. 340 Herschel, John 242
Fließbach, Torsten 334 Gould, Andrew 163 Hertz, Heinrich 110

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Floyd, R.M. 345 Grahn, Alexander 351 Hertzlinger, Joseph 351
Ford, E.C. 336 Grebe, Leonhard 140 Hesiod 226
Formalont, E.B. 338 Green, A.J. 341 Hestenes, David 326
Foster, James 333 Greenstein, J.L. 334 Heumann, John 350
Fowler, E.C. 323 Gregorio, Enrico 351 Hilbert, David 194, 201
Fowler, L.A. 339 Greiner, Jochen 347, 350 Hill, C.D. 338
Frank, F.C. 325 Grindlay, J.E. 336 Hill, S. 340
Fraunhofer Gesellschaft 312, Gruber, Christian 53, 327 Hillman, Chris 350
352 Gruber, R.P. 346 Hinshaw, G. 344
Fredman, R.A. 341 Gualandi, A. 321 Hipparchus 19
French, A.P. 330 Guiragossian, Z.G.T. 325 Hirth, J.P. 325
Frenkel, J. 325 Gutfreund, Hanoch 323 Hite, Kevin 351
Frercks, Jan 19, 321, 351 Gutfreund, Henoch 323 Hobbs, G. 341
Fresnel, Augustin 40 Guth, Alan 256, 343 Hoek, Martin 40
Friedmann, Aleksander 342 Gácsi, Zoltán 350 Hoffman, Richard 350
life 234 Göklü, E. 324 Hogg, D.W. 344
Frisch, D.H. 326 Holstein, B.R. 337
Fukuda, Y. 325 H Holzmüller, G. 170, 337
Fulle, Marco 150, 352 Haber, John 350 Horace, in full Quintus
Furrie, Pat 350 Hadley, M. 347 Horatius Flaccus 136
Fölsing, Albrecht 323 Hafele, J.C. 140, 326 Houtermans, Friedrich 228
Füllekrug, M. 322 Haley, Stephen 350 Hoyle, C.D. 347
356 name index

Hoyle, Fred 246, 343 life 211 Kumaradtya, K.K. 344
Hubble, Edwin Kapuścik, E. 325 Kuzin, Pavel 350
life 226 Karlhede, A. 347 Künzle, H.P. 345
Huber, Daniel 350 Kaufmann, W.J. 341 Küster, Johannes 351
Hughes, R.J. 340 Kayser, R. 343
Huisken, G. 332 Keating, Richard E. 140, 326 L
Hulse, Russel 338 Keesing, R.G. 349 Lachièze-Rey, M. 343
Huygens, Christiaan 16, 17 Kelu, Jonatan 350 Lahav, O. 340
Hörmann AG 33, 351 Kennedy, R.J. 324 Lakes, Rod S. 324
Kennefick, D. 334 Lamb, Frederick 157, 346
I Kenyon, Ian R. 331 Lambert, Johann
H Ilmanen, T. 332
INFN 181, 352
Kepler, Johannes 241
Kerr, Roy 269, 345
life 190
Lambourne, R. 324
Inverno, Ray d’ 332, 333 Kessler, E.G. 328 Landau, L. 340
Hoyle Iorio, L. 337 Kiefer, D. 324 Lange, B. 331
IPAC 213, 352 Kilmister, C.W. 329 Langevin, Paul 76
Israel, Werner 269 Kippenhahn, Rudolf 343 Laplace, Pierre 262

Motion Mountain – The Adventure of Physics
Itah, A. 340 Kiss, Joseph 350 Lasota, J.P. 346
Ivanov, Igor 350 Kittinger 334 Laue, Max von 85
Ives, H.E. 324 Kittinger, Joseph 136, 141 Leibfried, G. 325
Iyer, C. 327 Kjellman, J. 322 Lemaître, Georges A.
Klauder, John 333 life 234
J Klaus Tschira Foundation 351 Lense, Josef 169, 337
Jacobson, T. 332 Kleppner, Daniel 338 Leonhardt, U. 340
Jalink, Kim 350 Klippert, R. 340 Lerner, L. 336
Jamil, M. 350 Klose, S. 347 Leschiutta, S. 140, 334
Janek, Jürgen 350 Knie, K. 342 Leucippus of Elea 99

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Jaramillo, J.L. 345 Knop, R.A. 340 Levi-Civita, Tullio 195
Jarret, Thomas 213, 352 Knutsen, H. 342 Lewis, G.N. 328
Jenkins, Francis A. 321 Kogut, A. 344 Liebscher, Dierck-Ekkehard
Jentschel, M. 328 Konig, F. 340 324
Jerie, M. 348 Kontorowa, T. 325 Lifshitz, E. 340
Jetzer, P. 340 Kopeikin, Sergei 338, 350 LIGO 183
Johansson, Mikael 350 Korschinek, G. 342 Lille, Alain de 262
Johnson, Samuel Kostro, L. 331, 332 Lilleskov, E. 327
life 321 Kostro, Ludwik 330, 350 Lilley, Sam 333
Jones, Quentin David 350 Kramer, M. 337, 341 Linde, Johan 351
Jones, Tony 348 Kraus, Ute 54, 57, 264, 351, 352 Lindh, A.G. 336
Jong, Marc de 350 Kreuzer, L.B. 147, 335 Linfield, R.P. 327
Jordan, T.F. 329 Krijn, Marcel 350 Lintel, Harald van 53, 327, 350
Ju, L. 349 Krikalyov, Sergei 50 Liu, C. 324
Juszkiewicz, R. 344 Krisher, T.P. 328 Lodge, Oliver 41
Krishnan, B. 332 Logan, R.T. 328
K Krotkow, R. 336 Lombardi, Luciano 350
Köppe, Thomas 350 Krumm, P. 337 Lombardo, F. 347
Kaaret, P. 336 Królikowski, Jarosław 350 Longair, M. 342
Kalbfleisch, G.R. 323 Kröner, Ekkehart 209, 332 Lorenci, V. De 340
Kalckar, Jørgen 174 Kubala, Adrian 350 Lorentz, Hendrik Antoon 44,
Kant, Immanuel 211, 226, 340 Kuklewicz, C. 340 325, 326
name index 357

life 40 McDonald, K.T. 338 N
Lorenzen, Dirk 340 McGowan, R.W. 324 Nabarro, Frank R.N. 325
Lorimer, D.R. 339, 341 McKellar, B.H.J. 328 Nagano, M. 348
Lothe, J. 325 McLaughlin, M.A. 341 Nahin, Paul J. 326
Low, R.J. 330 McNamara, Geoff 335 Namouni, Fathi 350
Ludvigsen, Malcolm 333 McQuarry, George 350 Narlikar, J.V. 343
Luke, Lucky 37 Medison 33, 351 NASA 32, 168, 230, 241, 351,
Luminet, J.-P. 343 Mellinger, Axel 212, 340, 352 352
Lundmark, Knut 226, 341 Mena Marugán, G.A. 344 Natarajan, V. 328
Lutes, G.F. 328 Menocal, P.B. de 342 Natàrio, J. 346
Lyne, A.G. 341 Menten, K.M. 345 Nemiroff, R.J. 327, 336
L Lämmerzahl, C. 340, 347
Lévy, J.M. 329
Merrit, John 350
Michaelson, P.F. 349
Neumaier, Heinrich 350
Nevsky, A.Yu. 326
Michell, John 262, 345 Newcomb, Simon 325
Lorenzen M Michelson, Albert Abraham Newman, E.T. 345
MacCallum, M.A.H. 345 41, 112, 326 Newton 299
Macdonald, A. 346 life 40 Nicolai, H. 340

Motion Mountain – The Adventure of Physics
Macedo, R.P. 345 Miele 24, 351 Niepraschk, Rolf 351
Mach, Ernst 259 Minkowski, Hermann 45 Nieto, L.M. 339
Macrobius 342 life 45 Nietzsche, Friedrich 120
Madhu, Rao S.M. 350 Mirabel, I.F. 327 Nieuwpoort, Frans van 350
Madhu, Rao S.R. 314 Mishra, L. 100, 330 Nightingale, J.D. 333
Maeterlink, Maurice Misner, Charles 203, 330, 340 Nordström, Gunnar 269
life 258 Mitalas, R. 324 Nordtvedt, Kenneth 347
Maffei, Paolo 341 Mitskievic, N.V. 344 Nori, F. 316
Magueijo, João 330 Mittelstaedt, H. 344 Norton, John D. 339
Mahoney, Alan 350 Mittelstaedt, M.-L. 344 Novello, M. 340

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Mainwaring, S.R. 326 Mlynek, J. 324 Novikov, Igor 228, 333, 341,
Maleki, L. 328 Moffat, John 330 345
Manchester, R.N. 341 Mohazzabi, P. 334 Nurowski, P. 338
Manfredi, Eustachio 17, 18 Mohr, P.J. 349
Mark, Martin van der 350 Mokros, Nicolai 54, 56, 351 O
Marsh, N.D. 342 Moore, C.P. 345 Oberdiek, Heiko 351
Martos, Antonio 350 Moore, Henry 187 Oberquell, Brian 350
Marzke, R.F. 346 Moortel, Dirk Van de 314, 350 Observatoire de la Côte
Mashhoon, B. 337 Morinigo, Fernando B. 339 d’Azur 168, 352
Mason, W.P. 325 Morley, Edward 41, 326 Oey, M.S. 341
Masood-ul-Alam, A.K.M. 345 Moser, Lukas Fabian 350 Offner, Carl 350
Massa, Corrado 330, 331, 350 Murdock, Ron 350 Ohanian, Hans 315, 328, 331,
Matsas, G.E.A. 327, 328 Murillo, Nadia 350 333, 334
Matthews, W.N. 329 Murray, J.J. 325 Ohanian, Hans C. 323
Mattick, Tom 20, 322, 351 Musil, Rober 239 Okhapkin, M. 324, 326
Max Planck Gesellschaft 30, Mutka, Miles 350 Okun, Lev B. 329
351 Mutti, P. 328 Olbers, Wilhelm
Maxwell, James Clerk 44 Muynck, Wim de 350 life 240
Mayné, Fernand 350 Myers, E.G. 328 Olum, K.D. 335
Mayr, Peter 350 Møller, Christian 329 Oostrum, Piet van 351
Mazur, P.O. 269, 345 Müller, H. 324 Oppenheimer, Robert 345
McClure-Griffiths, N.M. 341 Müller, J. 347 life 265
358 name index

Osewold, Daniel 351 Pound, R.V. 140, 334 Robertson, H.P. 234
Osserman, Bob 254 Powell, Richard 225, 229, 352 Robertson, S. 340
Ovidius, in full Publius Prabhu, G.M. 327 Robertson, Will 351
Ovidius Naro 22 Pradl, O. 324 Robinson, D.C. 269, 345
Prakash, A. 345 Rodríguez, L.F. 327
P Preston, S. Tolver 329 Roll, P.G. 336
Pacholka, Wally 240, 352 Preston, Tolver 76 Rømer, Ole C. 321
Page, Don 346, 350 Pretto, O. De 329 life 16
Pahaut, Serge 350 Pretto, Olinto De 76, 323 Rosen, N. 338
Panov, V.I. 336 Prialnik, D. 341 Rosen, Nathan 174
Papapetrou, A. 329 Prigogine, Ilya 344 Rossi, B. 326
O Parker, Barry 342
Parks, David 350
Primas, L.E. 328
Prince, G.E. 348
Rothbart, G.B. 325
Rothenstein, B. 325
Pascazio, Saverio 350 Pritchard, Carol 350 Rothman, T. 343
Osewold Pasi, Enrico 350 Pritchard, D.E. 328 Rottmann, K. 330
Paul, W. 328 Pritchard, David 71 Roukema, B.F. 343
Pauli, Wolfgang 62, 339 Proença, Nuno 350 Rozental, I.L. 348

Motion Mountain – The Adventure of Physics
Pavlis, E.C. 337 Pryce, M.H.L. 329 Ruben, Gary 350
Pbroks13 34, 351 Purves, William 350 Ruder, Hanns 54, 347
Pearson, T.J. 327 Pythagoras of Samos 325 Rudolph, Peter 350
Peeters, Bert 350 Ruffini, Remo 269, 331,
Pekár, V. 336 R 333–335, 345, 346
Pelt, Jaan 252, 343 Rahtz, Sebastian 351 Rugel, G. 342
Penrose, Roger 131, 271, 274, Rainville, S. 328 Ruggiero, M.L. 330, 337
327, 332, 344–346 Rankl, Wolfgang 350 Ruppel, W. 333
Penzias, Arno 228, 341 Rasio, F.A. 339 Russell, Bertrand 87
Peres, A. 344 Rauscher, E.A. 331 Russo, J.G. 329

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Perini, Romano 350 Rawlinson, A.A. 328 Rybicki, G.R. 327
Perkins, D.H. 326 Raymond, D.J. 330 Röntgen, Wilhelm 40
Perlman, E. 345 Readhead, A.C.S. 327
Perlmutter, Saul 227, 238, 352 Rebka, G.A. 140, 334 S
Perot, Alfred 140 Recami, E. 328 Sabbata, V. de 331
Peşić, P.D. 342 Rector, T.A. 224 Sabbata, Venzo de 148, 330
Peters, A. 324 Redondi, Pietro 350 Saghian, Damoon 350
Pfister, Herbert 344 Refsdal, S. 343 Sagnac, Georges 41
Philbin, T.G. 340 Reinhardt, S. 324 Salim, J. 340
Philpott, R.J. 330 Reissner, Hans 269 Samuel, S. 338
Physical Review Letters 183 Renn, Jürgen 323 Santander, M. 339
Piper, M.S. 184, 339 Renselle, Doug 350 Santangelo, A. 336
Piraino, S. 336 Reppisch, Michael 350 Sastry, G.P. 327
Planck, Max 25, 62, 63, 82, 88, Rezzolla, L. 345 Savage, Craig 54, 351
111 Ricci-Cubastro, Gregorio Scarcelli, G. 325
Planck/ESA 243, 352 life 195 Schaefer, B.E. 322
Plato 246 Riemann, Bernhard Scharmann, Arthur 346
Poincaré, H. 338 life 207 Schiller, Britta 350, 351
Poincaré, Henri 42, 44, 76, Rindler, Wolfgang 324, 326, Schiller, Christoph 330, 331,
136, 155, 174, 329 327, 332, 333, 339 352
life 27 Ritz 322 Schiller, Isabella 350
Possenti, A. 341 Rivas, Martin 350 Schiller, Peter 325, 350
name index 359

Schiller, Stephan 41, 324, 326, Smith, J.B. 326 Thompson, C. 341
350, 351 Snider, J.L. 334 Thompson, J.K. 328
Schmidt, Herbert Kurt 326 Snyder, Hartland 265, 345 Thompson, R.C. 328
Schneider, M. 347 Soffel, Michael H. 339, 347 Thorndike, E.M. 324
Schneider, P. 343 Soldner, Johann 152, 161, 162, Thorne, Kip 330, 333, 338
Schoen, R.M. 331 335 Thornton, David 350
Schramm, Herbert 346 Solomatin, Vitaliy 350 Tisserand, F. 337
Schramm, T. 343 Sonoda, D.H. 344 Tisserand, Félix 170
Schröder, Ulrich E. 329, 333 Stachel, John 333 Titov, O. 336
Schucking, E. 327, 342 Stairs, I.H. 339, 341 Tolman, Richard 328, 344
Schutz, Bernard 333, 338 Stark, Johannes Torre, C.G. 202, 339
S Schwarzschild, Karl 140, 193
life 146
life 34
Stedman, G.E. 326, 349
Torrence, R. 345
Townsend, P.K. 329
Schweiker, H. 224 Steinhauer, J. 340 Townsend, Paul 350
S chiller Schwinger, Julian 323 Stephenson, G. 329 Treder, H.-J. 331
Schäfer, G. 347 Stephenson, G.J. 328 Trevorrow, Andrew 351
Sciama, Dennis 344 Stilwell, G.R. 324 Trout, Kilgore 255

Motion Mountain – The Adventure of Physics
Scott, David 151 Stocke, J.T. 345 Tschira, Klaus 351
Scott, Jonathan 350 Stodolsky, Leo 325 TSR relativity team 30, 351
Searle, Anthony 54, 91, 93, 351 Stoney, G.J. 346 Tuinstra, F. 321, 324
Seeger, A. 325 Story, Don 351 Tuppen, Lawrence 350
Seidelmann, P. Kenneth 349 Straumann, N. 342 Turner, M.S. 344
Seielstad, G.A. 327 Stromberg, Gustaf 226, 341
Selig, Carl 323 Strutt Rayleigh, John 41 U
Seneca, Lucius Annaeus 134, Su, Y. 336 Uguzzoni, Arnaldo 350
280 Sudarshan, George 328, 329 Ulfbeck, Ole 174
Sexl, Roman 285, 326, 344, 346 Supplee, J.M. 327 Unruh, William 137, 334

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
Shapiro, I.I. 337 Surdin, Vladimir 350 Unwin, S.C. 327
Shapiro, Irwin 163, 336 Svensmark, H. 342 Upright, Craig 351
Shaw, R. 327 Synge, J.L. 328
Shea, J.H. 334 Szuszkiewicz, E. 346 V
Sheldon, Eric 326, 350 Valencia, A. 325
Shih, Y. 325 T Vanier, J. 348
Short, J. 349 Tangen, K. 340 Vannoni, Paul 350
Siart, Uwe 351 Tarko, Vlad 350 Vergilius, Publius 158
Sierra, Bert 350 Tartaglia, A. 337 Vermeil, H. 187, 339
Silk, J. 344 Taylor, B.N. 349 Vermeulen, R. 345
Sills, K.R. 324 Taylor, Edwin F. 323, 330, 332, Vessot, R.F.C. 140, 334
Simon, Julia 350 333, 336, 345 Vestergaard Hau, L. 324
Simon, R.S. 327 Taylor, John R. 349 Voigt, Woldemar
Singh, T.P. 346 Taylor, Joseph 180, 299, life 44
Singleton, Douglas 350 337–339 Volin, Leo 350
Sitter, W. de 337 Tegelaar, Paul 350 Voltaire 299
Sitter, Willem de 238, 322 Tegmark, M. 339 Voss, Herbert 351
Sitter, Willem de 23, 41 Terrell, J. 327
life 166 Thaler, Jon 350 W
Sivaram, C. 148, 330, 331 Thies, Ingo 350 Wagner, William G. 339
Slabber, André 350 Thirring, Hans 169, 337 Wald, Robert M. 333, 346
Smale, A.P. 336 Thomas, Llewellyn 62 Walker, A.G. 234
360 name index

Walker, Gabriele 342 323, 330, 332, 333, 335, 338, Wright, Joseph 351
Walker, R.C. 327 345, 346 Wright, Steven 292
Wallin, I. 322 life 263
Wallner, A. 342 White, Harvey E. 321 Y
Wambsganss, J. 343 White, M. 344 Yearian, M.R. 325
Wang Lihong 21, 351 Whitney, A.R. 327 Young, Andrew 350
Wang, Y. 339 Wierda, Gerben 351
Warkentin, John 351 Wierzbicka, Anna 350 Z
Washington University at St. Wijk, Mike van 350 Zaccone, Rick 351
Louis 21, 351 Wikimedia 33, 168, 250, 351, Zalm, Peer 350
Watson, A.A. 348 352 Zedler, Michael 350
W Weigert, Alfred 343
Weinberg, Steven 325, 333, 342
Wilken, T. 325
Will, C.M. 322, 328, 332, 336,
Zeeman, Pieter 40
Zel’dovich, Yakov 228
Weisberg, J.M. 339 338, 347 Zensus, J.A. 327
Walker Weiskopf, Daniel 54, 55, 57, Williams, R. 335 Zeus 214
351 Wilson, Harold 40 Zhang Yuan Zhong 323
Weiss, Martha 350 Wilson, Robert 228, 341 Zhang, W. 157, 336

Motion Mountain – The Adventure of Physics
Weizmann, Chaim 160 Wiltshire, D. 341 Zhao, C. 349
Wertheim, Margaret 325 Wirtz, Carl 226, 341 Zhong, Q. 327
Wesson, Paul 242, 342 Wise, N.W. 349 Zwicky, Fritz 252, 343
Westra, M.T. 317 Woodhouse, Nick M.J. 323
Wheeler, John 263, 269, 282, Woods, P.M. 341

Symbols proper 80 antigravity device, patent for
Cerenkov radiation 28 space-time illustration 95 153
3-vector 77 uniform 94–96 antimatter 70, 206, 245
4-acceleration 79–81 accretion 273 in collisions 74
4-angular momentum 85 discs 220 aphelion 307

Motion Mountain – The Adventure of Physics
4-coordinate 45 accuracy 300 apogee 306
4-coordinates limits to 301 Apollo 167, 310
definition 77 action 87–89 apparent lifetime 51
4-force 83–84 meaning of 87 apple
4-jerk 80, 94 principle of least 104, 202, fall of 158–159
4-momentum 81–83 288 floating 154
4-vector action, quantum of, ℏ Newton tree DNA 300
example 77 physics and 8 standard 300
general definition 79 active galactic nuclei 262 Archaeozoicum 232
introduction 76–78 ADM mass 203 archean 232

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
magnitude 77 aether arms, human 258
4-velocity 78–79 and general relativity 119, Ashtekar variables 291
144 astronaut see cosmonaut
A age astronomical unit 307
α-rays 15 of universe 237 atmosphere
a (year) 233 age of universe 76 illustration of
aberration 17–19, 54–55 agoraphobics 236 transmittance 249
constant 18 air 249 pressure 306
acausality 47 cannot fill universe 245 atom formation 231
accelerated observer Aldebaran 251 atomic mass unit 305
illustration 92 Alluvium 233 atomism is wrong 99
accelerating frames 93 Alnilam 251 atto 297
acceleration 326 Alnitak 251 Augustine of Hippo 247
and length limit 101–102 alpha decay 227 average curvature 195
and colour change 99–100 Altair 251 Avogadro’s number 302
and force 83 ampere azoicum 232
comoving 80 definition 295
composition 100–101 Andromeda nebula 211, 226 B
composition theorem 100 photograph 214 β-rays 15
composition, illustration angular momentum as a B1938+666 252
101 tensor 85 background radiation 228,
constant 94–96 annihilation 245 233, 246
362 subject index

illustration of fluctuations illustration of matter brown dwarf 222, 251, 252
243 orbits 267 brute force approach 120
spectrum 228 intermediate 273 bubble chamber
bags, plastic 318 Kerr 269 figures about 69
ball micro 273 bucket experiment, Newton’s
illustration of puzzle 153 primordial 273 259
barycentric coordinate time process, irreversible 272 bulb
336 process, reversible 272 light, superluminal 60
barycentric dynamical time radiation 344 Bureau International des
155 rotating 270 Poids et Mesures 295
baryon number density 308 Schwarzschild 269 burst
B base units 295
becquerel 297
stellar 273
supermassive 273
𝛾-ray 22
gamma-ray 322
Beetle 187 type table 273 bus 138
bags beginning black hole attempt 126 best seat in 55–58
of the universe 226 black paint 242
of time 226 black vortex 270 C

Motion Mountain – The Adventure of Physics
Bellatrix 251 black-body radiation 250 cable-car attempt 125
Betelgeuse 251 blue-shift Caenozoicum 233
big bang 228, 238, 240, 246 definition 34 Cambrian 232
consequences 246 body candela
not a singularity 131 rigid 85, 102 definition 296
not big nor a bang 247 solid 102 Canopus 251
billiards 67 Bohr magneton 304 capture
BIPM 295 Bohr radius 304 of light 269
bird appearance 232 Boltzmann constant 𝑘 302 capture, gravitational 268
bit invariance 63 Carboniferous 232

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
to entropy conversion 305 physics and 8 caress 84
black hole 98, 262–279, 335 bomb 70 cat’s eye
see also Schwarzschild boost see Lorentz boost, 113 see also retroreflectors
radius and force limit 122 Cat’s-eye, lunar 168
analogous to universe? 279 concatenation 62 cathode ray tube 38
and force limit 108 definition 44 cathode rays 15
and gravitational waves 182 relations 44 causality 46
collisions 274 boost attempt 122 and maximum speed 47
definition 146 bottom quark cause and effect 46
does not exist 276 mass 303 cenozoic 233
entropy of 272 boxes 101 censorship, cosmic 130
extremal 270 bradyon centi 297
first law of 115 definition 74 centre of mass 74
halo 278 braking in relativity 75
horizon 262 attempt 121 centrifugal effect 277
illustration of 264 limit 121 Čerenkov radiation 183
illustration of ergosphere Brans–Dicke ‘theory’ 209 CERN 322, 326
270 brick CGPM 296
illustration of light orbits attempt 122 chair
268 tower 122 as time machine 50
illustration of local light brick tower, infinitely high 122 challenge
cones 265 Bronshtein cube 8 classification 9
subject index 363

change Conférence Générale des aging 50
quantum of, precise value Poids et Mesures 295, 299 coulomb 297
302 conformal group 90 coupling
channel rays 15 conformal invariance 90 principle of minimal 199
charge conformal transformation courage 28
differs from mass 184 89–90 covariance
elementary 𝑒, physics and Conférence Générale des principle of general 199
8 Poids et Mesures 296 covariance, general 202
positron or electron, value conic sections 165 crackpots 37, 325
of 302 connection creation 249
charm quark causal 46 of particles 70
C mass 303
chemical mass defect 71
constant
cosmological see
Cretaceous 232
cube
chocolate 242 cosmological constant Bronshtein 8
change Christoffel symbols of the constants physics 8
second kind 206 table of astronomical 305 curvature 143, 144, 147
CL0024+1654 253 table of basic physical 302 see also space-time

Motion Mountain – The Adventure of Physics
claustrophobics 236 table of cosmological 308 and stiffness 188
clock table of derived physical average 189
at Equator, puzzle 51 304 extrinsic 185
graph of time dilation 30 constellations 211 extrinsic, definition 185
height dependence 141 container 46 Gaussian 186, 188
matter and 281 contraction 195, 208 illustration of extremal
paradox 49 relativistic 43 directions 187
synchronization 29, 36 Convention du Mètre 295 illustration of signs 186,
cloud 268 conveyor belt 138 188
frequent in universe 219 coordinate illustration with tidal

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
in the night sky, relativistic transformation effects 191
photograph 216 44 in three dimensions 188
CODATA 349 coordinate systems 202 intrinsic 185–192
coherence corkscrew 178 intrinsic, definition 185
definition 310 cosmic background radiation measured by Gauss 190
collapsars 265 240, 245, 344 measured on Earth 190
collapse 273 illustration of fluctuations near mass 193
collision 243 sectional 191
definition 72 spectrum 228 tensor 189
in relativity 66 cosmic censorship 130, 275, curve
space-time diagram 73 346 closed time-like 283
colour shift 34 cosmic radiation cyclotron frequency 304
coloured constellation 250 and evolution 75 Cygnus bubble
comb muons in 50 photograph 224
frequency 36 cosmological constant 195, Cygnus X-1 273
comets 221 199, 201, 203, 244, 246, 291,
Commission Internationale 308 D
des Poids et Mesures 295 cosmological principle 226 dark energy see cosmological
composition theorem for cosmology constant, 71, 223, 288, 291
accelerations 100 agoraphobic 236 dark matter 71, 223, 252, 274,
Compton wavelength 304 claustrophobic 236 287, 291, 293
conductance quantum 304 cosmonaut 42, 137, 154 problem 244
364 subject index

darkness definition 32 see field equations
of night sky 242 figures about 32 Einstein, Albert
speed of 58–61 figures of uses 33 life 25–26
day for sound 36–37 Einstein–Cartan theory 293
sidereal 305 gravitational 140 elasticity 147
time unit 297 illustration of 34 electricity, start of 233
death 19 rotational 86 electrodynamics 282
deca 297 transversal 35 electromagnetism 84
decay of photons 261 use of 35 electron 15
deceleration parameter 237 Doppler red-shift 250 classical radius 304
deci 297 Doppler shift g-factor 304
D degree
angle unit 297
rotational 86
down quark
magnetic moment 304
mass 303
degree Celsius 297 mass 303 electron size 102
darkness density Draconis, Gamma 18 electron volt
perturbations 231 duality value 305
proper 197 space-time 286 ellipse 166

Motion Mountain – The Adventure of Physics
dependence on 1/𝑟2 290 dust model 197 energy 69
detection of gravitational concentrated 70
waves 182 E dark see cosmological
deviation Earth constant
standard, illustration 301 age 306 kinetic, definition 70
Devonian 232 average density 306 no free 71
dice equatorial radius 306 no undiscovered 71
film of relativistic 57 flattening 306 potential 82
diet 71 gravitational length 306 relativistic kinetic 82
diffeomorphism invariance hollow 285 relativistic potential 82

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
202, 280, 284, 285 length contraction 52 energy density
diffraction 205 mass 306 negative 201
dilations 89 normal gravity 306 energy is bounded 82
Diluvium 233 radius 306 energy of the universe 256
dimension ring around 220 energy–mass equivalence
fourth 45, 46 Earth formation 232 68–71
dimensionless 304 Earth’s rotation 298 energy–momentum
dinosaurs 232 Earth, hollow and space-time diagram 81
dislocations 36 illustration 285 energy–momentum 4-vector
dispersion relation 177 eccentricity 166 82
distance eccentrics 285 energy–momentum tensor
rod 93 ecliptic 18 116, 197
distribution effects engine
Gaussian 300 acausal 47 maximum power of 112
normal 300 Ehrenfest paradox 316 Enlightenment 211
divergence attempt 122 Ehrenfest’s paradox 85 entropy 257
DNA 300 Einstein relativistic transformation
donate error 51 63
to this book 10 his mistakes 26 relativistic transformation
door sensors 35 Einstein algebra 283 of 63
Doppler effect 31–37, 54, 99 Einstein tensor 195 to bit conversion 305
and films 54–55 Einstein’s field equations entropy of black hole 272
subject index 365

Eocene 233 303 of atomic transition 299
equilibrium first law frequency comb 36
thermal, and relativistic of black hole mechanics 115 full width at half maximum
observers 63 of horizon mechanics 115 300
equivalence principle 199, 290 flatness fungi 232
ergosphere 270, 271 asymptotic 203
error measurement of 15 G
Einstein’s 51 flow of time 284 γ-rays 15
in measurements 300 food-excrement mass galaxies and black holes 262
random 300 difference 71 galaxy 213, 259
relative 300 force 204 collision photograph 215
E systematic 300
total 300
addition 124
definition 83, 110
distant, photograph 216
photograph 215
escape velocity 262 horizon 114 galaxy formation 231
Eo cene ether, also called luminiferous is relative 110, 120 galaxy image
ether 326 maximum see force limit, multiple, photograph 254
European Space Agency 319 maximum Galilean satellites 16

Motion Mountain – The Adventure of Physics
event horizon 97 maximum, conditions 119 gamma-ray burst 322
event space 45 minimum in nature gamma-ray bursters 262
evolution 75 134–135 gamma-ray bursts 221, 254,
evolution, marginal 235 Planck see force limit, 322, 347
Exa 297 maximum gas constant, universal 304
excrements 71 force limit 110 Gaussian curvature 188
exoplanet 36 and universe’s age 132 Gaussian distribution 300
extrasolar planets 252 and universe’s size 132 Gedanken experiment 120
force, maximum 107 general covariance 202
F change cannot be general relativity

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
fall 157 measured 292 see also field equations
permanent 265 hidden 128 correct description of
fall, free 137 illustration of equivalence gravitation 136
farad 297 with general relativity 114 first half 147
Faraday’s constant 304 table of logic 109 logic of 109
faster than light 154 force, perfect 287 orbits 164–166
and acceleration 100 forest origin of name 27
examples 58–61 and sky analogy 241 second half 152
illustration of 59 Foucault pendulum 169 simplicity of 107
velocity 84 fourth dimension 45, 46 single principle 108
faster than light motion frame dragging 163, 170, 173 table of logic 109
in accelerated frame 100 frame of reference table of tests 290
in collisions 73 accelerated 93–94 weak field summary 184
femto 297 general, definition 94 without space-time 283
fence 44 inertial 42 general relativity in one
Fermi coupling constant 302 frame-dragging 169 paragraph 192
Fermi-Walker transport 319 Franz Aichinger 350 general relativity in ten points
Ferrari Fraunhofer lines 140, 311 287
length contraction 52 illustration 312 general relativity, accuracy of
field equations free fall 289
summary of 210 permanent 265 general relativity, statements
fine-structure constant 302, frequency of 148
366 subject index

genius 149 benefits 184 henry 297
definition 26 detector 184 hertz 297
geocaching 157 detector details 181 Hertzsprung–Russell diagram
geodesic illustration of binary 228
definition 110 pulsar evidence 180 illustration 229
light-like 149 illustration of effects 178 Higgs mass 303
time-like 149 illustration of necessity 174 Hilbert action 201–202
geodesic deviation 207 no plane 131 hole argument 284
geodesic effect 167, 290 speed 183 illustration of 284
illustration of 167 gravitational waves 174 hole paradox 284
geometric phase 319 made of particles 182 hollow Earth
G geometrodynamic clock 282
Giga 297
gravitational waves, detection
of 182
illustration 285
hollow Earth hypothesis 285
globular clusters 221 gravitational waves, speed of Hollywood films 89
genius gluon 303 177, 182 Holocene 233
gods 198, 257 gravitodynamics 176 Homo sapiens appears 233
Gondwana 232 gravitomagnetic field 172 Homo sapiens sapiens 233

Motion Mountain – The Adventure of Physics
GPS, global positioning gravitomagnetism 290 horizon 227, 264, 266, 314
system 156 illustration of 172 see also black hole
grass 44 gravitons 182 and acceleration 121
grass, appearance of 233 gravity see gravitation, 84, and maximum force 113
gravitation see also universal 129, 137 as mixtures of space and
gravitation bends light 161–163 particles 98
see also general relativity, weak 160 energy flow across 114
quantum gravity Gravity Probe B 170 equation 115
as braking mechanism 112 gravity wave emission delay event, definition 97
photographs of effects 108 290 first law of 115

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
weak 160 gravity waves 174 force 117
gravitation, universal see gravity waves, spin of 176 illustration of 98
universal gravitation gray 297 importance of 98–99
gravitational and inertial grey hair 58 in special relativity 96–99
mass identity 199 group is a surface 97
gravitational constant conformal 90, 91 is black 97
geocentric 306 group 4-velocity 86 limits 117
heliocentric 306 group velocity mechanics 114
gravitational constant 𝐺 302 above 𝑐 60 moving faster than light 59
see also force limit, GUT epoch 231 of a black hole 262
maximum, see also power, gyromagnetic ratio 277 power 117
upper limit space-time illustration 97
as limit value 132 H horizon equation
is constant 131 hadrons 231 general 116
physics and 8 hair, grey 58 horizon force 114
gravitational energy 198, 204 hand 74 horizons 107
gravitational field 171 in vacuum 334 horizons as limit systems 287
gravitational lensing 252, 274 harmonic wave 86 horsepower
gravitational red-shift 251 HARP 68 maximum value of 108
gravitational wave hecto 297 hour 297
and rockets 184 Helios II satellite 19 Hubble constant 226
beam 184 helium 15, 231, 246 Hubble diagram
subject index 367

graph of 227 conformal 90 37
Hubble parameter 308 larger 68
of the speed of light 22–25,
hurry 88 29 laser distance measurement
hydrogen fusion 231 invariants of curvature tensor of Moon 321
hyperbola 166 208 Laurasia 232
hyperbolas 267 inversion 89 law of cosmic laziness 87
hyperbolic cosine 95 inversion symmetry 90 learning
hyperbolic secant 97 Io 16 best method for 9
hyperbolic sine 95 irreducible mass 272 without markers 9
hyperbolic tangent 97 irreducible radius 272 without screens 9
hyperboloid IUPAC 349 least action
H circular 188, 310
hypernova 220
IUPAP 349 and proper time 87
principle 87–89
hypersurfaces 92 J length
Hubble Jarlskog invariant 303 is relative 42
I jet length contraction 30, 44,
Icarus 166, 290 photographs of 219 52–54, 327

Motion Mountain – The Adventure of Physics
ice age 233 jets 220 and submarine 54
imaginary mass 73 jewel textbook 329 illustration with plane and
impact John Barrow 125 barn 52
definition 72 Josephson frequency ratio 304 illustration with trap and
impact parameter 162 joule 297 snowboard 53
impact parameters 268 Jupiter 204 puzzle with glider 53
in all directions 260 properties 306 puzzle with rope 53
incandescence 249 Jurassic 232 length limit
indeterminacy relation acceleration and 101–102
relativistic 102 K LEP 37

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
inertial frame 93 k-calculus 29 life appearance 232
inertial frame of reference 42 kaleidoscope 255 lifetime
infinite number of SI prefixes kelvin apparent 51
299 definition 295 relativistic changes of 50
inflation 231, 255, 256, 291 Kepler’s relation 179 light
after the big bang 256 kilo 297 acceleration 36
inflaton field 256 kilogram acceleration of 31
infrared rays 15 definition 295 bending by masses 161–163
initial conditions 231, 248 kilogram, prototype 289 capture 269
interaction, is gravity an 204 kinematics, relativistic 42 faster than 154
interferogram 310 kiss 84 longitudinal polarization
interferometers 299 Klitzing, von – constant 304 31
intermediate black holes 274 massive 31
International Astronomical L moving 205
Union 307 ladder the unstoppable 31
International Commission on and time dilation 31 weighing of 71–72
Stratigraphy 233 LAGEOS 337 light bending
International Earth Rotation LAGEOS satellites 170 illustration of gravitational
Service 298 Lagrangian 152 161
International Geodesic Union lamp light bulb
307 many, attempt 127 superluminal 60
invariance Large Electron Positron ring light cone 78
368 subject index

future 46 lumen 297 mass flow limit
past 46 lunar retroreflector 168 and universe’s size 132
light deflection 290 Lunokhod 167, 310 mass ratio
light motion lux 297 muon–electron 304
table of properties 22 Lyman-𝛼 311 neutron–electron 305
light path neutron–proton 305
from dense body, M proton–electron 305
illustration 276 M15 cluster mass–energy equivalence
light pulses photograph 230 68–71
circling each other 155 M31 211 history of 76
light source M51 215 mass-defect
L moving 23
light speed
Mach’s principle 199, 259
Magellanic clouds 212
nuclear 71
material systems 104
finite 242 magnetar 222 matter
light light year 306, 307 magnetic flux quantum 304 metastable 265
light-like magneton, nuclear 305 matter domination 231
interval definition 78 magnitude of a 4-vector 77 mattress 144, 174, 175, 177–179

Motion Mountain – The Adventure of Physics
vector 47 mammals illustration of space model
lightlike geodesics 206 appearance of 232 144, 145
lightning appearance of large 233 maximal ageing 89
speed of 21 man, wise old 87 maximum force
lightning puzzle 48 manifold see force limit, maximum
lightning, colour of 311 see also space, space-time as measurement unit 111
limit concept 276 of events 45 experiments 112–113,
limit size marker 133–134
of physical system 130 bad for learning 9 hidden 128
limit speed Mars 166, 299 implies general relativity

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
and computers 38 maser 140 113
and massless entities 39 mass implies horizons 113
table of checks 26 as concentrated energy 70 is fascinating 110
limits centre of 74 late discovery 128
to precision see precision, decrease while thinking 70 principle 108
301 definition 65–67 series of arguments 109
Linux 20 equality of inertial and value 108
liquid 197 gravitational 158 maximum power
litre 297 flow, maximum 109 and exhausts 112
Lorentz boosts 90 gravitational 144 experiments 112–113,
Lorentz factor 43 has only one sign 184 133–134
Lorentz symmetry imaginary 73 implies general relativity
see Lorentz invariance inertial 144 113
Lorentz transformations of maximum rate change 109 principle 108
space and time 44 spherical 193 value 108
Loschmidt’s number 304 total, in general relativity maximum speed
lottery 47 203 see also speed of light 𝑐,
loudspeaker mass defect limit speed
and speed of light measurement of chemical measurement
invariance 24 71 comparison 298
Lucky Luke mass density definition 295, 298
figure of 37 critical 235 error definition 300
subject index 369

irreversibility 298 dynamics 348 is nonsense 258
meaning 298 molar volume 304 muon
of space and time 282 molecule 155 g-factor 305
precision see precision momenergy 82 time dilation illustration
process 298 momentum 81 50
mechanics conservation 65–67 muon experiment 50
not possible in relativity 83 definition 67 muon magnetic moment 304
relativistic 65 MOND 348 muon mass 303
Mega 297 Moon muons 326
megaparsec 226 and geodesic effect 290 music record 58
Megrez 251 density 306 myths and stars 211
M memory 47
mesozoic 232
formation 232
properties 306
Mössbauer effect 140

Messier object listing 211 Moon, laser distance N
mechanics meteorites 221 measurement 321 naked singularities 275
metre moons, Galilean 16 nano 297
bar 141, 281 mother-daughter puzzle NASA 299

Motion Mountain – The Adventure of Physics
definition 295 49–51 natural unit 304
metric 78, 88 motion necklace of pearls 55
metric connection 206 and measurement units Neogene 233
Michelson’s experiment 296 neutrino 39, 231, 311, 325
figures about 41 does not exist 46 masses 303
micro 297 hyperbolic 95 PMNS mixing matrix 303
microscopic motion 289 ideal 15 neutron
microwave background illustration with proper Compton wavelength 305
temperature 308 time 87 magnetic moment 305
mile 298 is fundamental 296 mass 305

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
milk 20, 214 is usually slow 75–76 New Galactic Catalogue 215
Milky Way 211 opposite of free fall 137 newton 297
age 307 relativistic 104 NGC 205 215
mass 307 superluminal 58, 60 night sky
mass measurement 214 understanding 281 and power limit 132
rotation 35 undisturbed 15 colours of 240
size 307 unstoppable, i.e., light 31 galaxy distance illustration
milli 297 Motion Mountain 213
Minion Math font 351 aims of book series 7 infrared 212
Minkowski space-time 45 helping the project 10 photograph 212
Mintaka 251 supporting the project 10 X-ray 213
minute 297 motor, electric night sky, darkness of 242
definition 307 and speed of light no-interaction theorem 329
Miocene 233 invariance 24 noise
mirror motorbike 96, 102 reduction in gravitational
puzzle 48 mountain 74 detectors 182
relativistic 35 and photons 74 Nordtvedt effect 131, 347
mirror invariance 202 mountain attempt 125 North Pole 142, 248
mixing matrix to exceed maximum force nova 220, 228
CKM quark 303 126 nuclear magneton 305
PMNS neutrino 303 multiverse nuclei 231
modified Newtonian is gibberish 258 nucleosynthesis 231
370 subject index

null paint, black 242 permittivity
geodesics 206 Paleocene 233 vacuum 304
vector 47, 78, 79, 86 Paleogene 233 perturbation calculations 281
number paleozoic 232 Peta 297
imaginary 73 Pangaea 232 phase
nutshell parabola 166, 267 geometric 319
general relativity in a 287 paraboloid of wave, definition 86
special relativity in a 104 hyperbolic 188 phase 4-velocity 87
paradox phase velocity
O about maximum force above 𝑐 60
objects 120–125 photon
N real 73
virtual 73
about maximum power
125–128
decay 261
mass 32, 303
observer clock 49 number density 308
null accelerated 91–93 Ehrenfest’s 85 sphere 269
accelerated, illustration 92 length contraction 53 virtual 74
comoving 80 Olbers’ 241 physics

Motion Mountain – The Adventure of Physics
free-floating 42 pearl necklace 55 map of 8
inertial 42 twin 49 start of 233
inertial, illustration 43 twin, illustration 49 physics cube 8
odometer 78 parallax 18 pico 297
ohm 297 parallel transport 319 pilot puzzle 48
Olbers’ paradox 132, 240 parallelism Pioneer anomaly 208
Olbers, Wilhelm in relativity 62 pizza
life of 240 is relative 53 slice, best way of holding
Oligocene 233 parity invariance 104, 288 188
orbit parsec 226, 306 Planck area

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
illustration of gravitational particle corrected 135
164 size of elementary 102 Planck constant
in general relativity ultrarelativistic 82 value of 302
164–166 virtual 72–74 Planck force 111
smallest circular around pascal 297 Planck force 𝑐4 /4𝐺
mass 165 path see force limit, maximum
summary in general rosetta 166 Planck length 282
relativity 184 pearl necklace paradox 55 see also Planck scales,
orbits 205 Penning traps 71 Planck energy
order Penrose inequality 130 Planck scale
time is a partial 46 Penrose–Hawking singularity see also Planck units
Ordovician 232 theorems 274, 346 Planck speed 𝑐 see speed of
Orion 72, 250 periastron light 𝑐
oscilloscope 60 definition 166 Planck value
oven, hot shift 166 see Planck units, see also
photograph of colours 250 perigee 306 natural units
Oxford 287 perihelion 166, 307 Planck’s natural length unit
oxygen, appearance in perihelion shift 290 282
atmosphere 342 permanent free fall 265 plane gravity wave 177
permeability planet formation 232
P vacuum 304 plant
π = 3.141592... 85 Permian 232 appearance 232
subject index 371

plasma 220 force limit, maximum and hadrons 231
Pleiades star cluster 232 of minimal coupling 199 mixing matrix 303
Pleistocene 233 of relativity 42 quasar 60, 229, 262, 273
Pliocene 233 Procyon 251 jet 75
point proper distance 78 Quaternary 233
isotropic 188 proper force 83
particle, size of 276 proper length 52 R
polders 40 proper time 45, 77, 78 rabbit puzzle 234
pool and least action 88 radar 35
game of 67 illustration with motion radian 296
positron charge possibilities 87 radiation 15
P specific 304
value of 302
proper velocity 48
proterozoic 232
and force limit 122
speed of 104
post-Newtonian formalism proton radius
plasma 156 Compton wavelength 305 excess 189
potential energy g factor 305 rain, falling 17–19
in relativity 82 gyromagnetic ratio 305 rainbow 314

Motion Mountain – The Adventure of Physics
power magnetic moment 305 rapidity 39
and 4-force 83 mass 305 ray days 15
maximum 107 specific charge 305 rays 15
maximum in nature 278 prototype kilogram 289 reaction
maximum, conditions 119 PSR 1913+16 174, 180 chemical 71
paradox 127 PSR J0737-3039 174 recombination 231
power limit pulsar 214 rectilinear 94
and night sky 132 and gravitational waves red-shift 261
power–force 4-vector 83 180 definition 34
Poynting vector 179 binary 156, 164, 166, 174 Doppler 250

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
PPN, parametrized Crab nebula 22 gravitational 140
post-Newtonian frame dragging and 170 illustration of effects 139
formalism 156 pulsars 290 large values 311
precession 167 puzzle mechanisms 261
precision 300 ball 153 number 35
bad example 37 lightning 48 tests 290
limits to 301 mirror 48 reference frame
prefixes 297, 348 mother-daughter 49–51 preferred in universe 237
SI, table 297 pilot 48 reflection 205
prefixes, SI 297 rabbit 234 refraction 205
primates, appearance of 233 travel distance 48 vacuum index of 162
Principe, island of 334 Regulus 251
principle Q Reissner–Nordström black
correspondence 199 Q0957+561 252 holes 269
equivalence 199 quadrupole relativistic correction 43
of equivalence 137, 201, 290 definition 179 relativistic kinematics 42
of general covariance 199 radiation 179 relativistic mass 82
of general relativity 199 quantum of action 88 relativistic velocity 78
of least action 87, 104, 201, precise value 302 relativity
202, 288 quantum of circulation 304 alternatives to 292
of maximal ageing 89 quantum physics 282, 294 breakdown of special 105
of maximum force see quark general see general
372 subject index

relativity satellite Silurian 232
is classical 202 around Jupiter 204 singularities 130, 201, 343
limits of 293–294 experiments 156, 289, 290 singularities, dressed 275
of parallelism 53 Helios II 19 singularities, naked 275
origin of name 27 Hipparcos 163 Sirius 251, 334
special see special LAGEOS 170 size limit 130
relativity, 22 LAGEOS, photograph of size of electron 102
theory of 27 170 sky
without space-time 283 photograph of 218 darkness at night 239–242
rest 136, 137 Saturn 108 night infrared 212
definition 136 scale factor 89, 234, 242 night photograph 212
R is relative 42
rest energy 72
scale symmetry 202
Schwarzschild black hole
Sloan Digital Sky Survey 341
slow motion 76
rest mass 82 see black hole is usual 75–76
rest retroreflectors definition 267 smartphone
see also cat’s eye Schwarzschild metric 146, 266 bad for learning 9
Ricci scalar 192, 195 Schwarzschild radius 146, 263 snooker 67–68

Motion Mountain – The Adventure of Physics
Ricci tensor 117, 194–195 see also black hole figure for non-relativistic
Riemann curvature tensor Schwarzschild solution 193 67
206 science fiction 70 figure for relativistic 68
Riemann tensor 207 scissors 58 snowboarder
Riemann-Christoffel search engines 322 relativistic 52
curvature tensor 206 searchlight effect 54 Sobral, island of 334
Riemannian manifold 207 second 297 solid body 102
Riemannian space-times 45 definition 295, 307 acceleration and length
Rigel 251 semimajor axis 166 limit 102
rigid bodies do not exist in shadows 60 sound

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
nature 102 and light motion 15 versus light 36–37
rigid coordinate system 94 and radiation 15 wave 35
rigidity not parallel 316 south-pointing carriage 208
and relativity 53 speed of 22, 37, 58–61 space
ring interferometers 299 shape as elastic material 119
Robertson–Walker solutions and relativity 53 bending of, and body
234 shape of universe 254 shape 145
rocket 271 shear modulus 119 distinction from time 286
rod distance 93 definition 119 is not absolute 42
rope attempt 121 shear stress, theoretical 119 tearing apart 119
rosetta path 166, 267, 269 ships space and time, differences
illustration of 267 and the speed of light 18 between 280
rotation SI space of life 280
illustration of observers 85 prefixes 299 space probe
in relativity 84–86 table of 297 and time delay 163
of the Earth 298 units 295, 302 space-like
Rydberg constant 304 SI units convention 78
definition 295 interval definition 78
S prefixes 297 vector 47
sailing supplementary 296 space-time see also curvature,
and the speed of light 18 siemens 297 45, 152
Saiph 251 sievert 297 as background 46
subject index 373

as container 46 effects on a lighthouse multiple image 253
collision diagram 73 beam 21 speed measurement 35
diagram 43, 47 experiments showing star myths 211
distance, definition 77 invariance 40 stardust 233
Galilean 46 figure of Rømer’s Stark effect 34
interval 45 measurement 17 stars
interval, definition 77 finite 242 double 23
introduction of 45–46 Fizeau’s measurement 20 start of physics 233
made of particles 182 independent of frequency static limit 270
Minkowski 45 22 Stefan–Boltzmann black body
tachyon diagram 61 invariance 22–25 radiation constant 305
S tearing 131
space-time diagram
invariance of 29, 90
meaning ‘in vacuum’ 28
stellar black hole 273
steradian 296
and energy–momentum 81 measurement, illustration stiffness
space-time with light cone 77 64 and curvature 188
space-time interval 77 one-way 63, 315 stone
special conformal pulse moving through illustration of curvature of

Motion Mountain – The Adventure of Physics
transformations 89 milk 20 flying 150
special relativity 22 rainwalker’s measurement throw 89
all in one drawing 28–31 18 stones 74, 149, 150, 265
breakdown of 105 surfer’s measurement 18 straightness
definition 104 two-way 63 figure puzzle on checking
in four sentences 104 value 19 it 16
origin of name 27 speed of shadow 58–61 measurement of 15
principle 40–44 speed of sound strain 146
table of logic 26 values 101 strange quark
speed spin mass 303

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
is usually slow 75–76 and classical wave stretch factor 43
of darkness 58–61 properties 177 strong coupling constant 302
of light 𝑐 of a wave 176 strong field effects 289
physics and 8 of gravity waves 176 submarine
of shadows 60 spin–orbit coupling 167 paradox 54
perfect 16, 104, 287 spin–spin coupling 167 relativistic 54
speed is relative 120 sponsor Sun 211, 232, 251
speed limit this book 10 distance of 19
consequences 26–28 squark 350 edge colours 35
is the speed of light 25 staff of Archytas 255 is not a black hole 250
speed of gravitational waves stalactite 108 Sun’s age 307
177, 182 stalagmite 19 Sun’s lower photospheric
speed of light standard apple 299, 300 pressure 307
and sailing 18 standard deviation 300 Sun’s luminosity 306
and ships 18 illustration 301 Sun’s mass 306
as limit speed 25 star Sun’s motion
change cannot be and soul 230 around galaxy 214
measured 105 classes 250, 251 Sun’s surface gravity 307
conjectures with variable colour, table of 251 superluminal
104 distance-speed diagram motion 58
consequences of 227 speed 255
invariance 26–28 generations 231 superluminal motion
374 subject index

in accelerated frame 100 Tera 297 illustration of 30
supermassive black holes 273 terrestrial dynamical time 155 illustration with ladders 31
supernova 212, 220 Tertiary 233 muon experiment 51
definition 228 tesla 297 time independence of 𝐺 290
support tests Time magazine 143
this book 10 of general relativity, table time travel 46–51
surface 290 to the future 50
physical 110 Thames 18 time-like
surface gravity of black hole theorem, no-interaction 329 closed curves 283
264 theory of relativity convention 78
surface, physical 127 origin of name 27 interval definition 78
S suspenders 318
synchronization
thermodynamic equilibrium
265
vector 47
TNT energy content 305
of clocks 29, 36 thermodynamics tonne, or ton 297
supermassive Système International second principle of 47 toothbrush 275
d’Unités (SI) 295 Thirring effect 169 top quark
illustration 169 mass 303

Motion Mountain – The Adventure of Physics
T Thirring–Lense effect 167, topology of the universe 254
tachyon 61, 72–74 169, 268 torque 173
definition 61, 73 Thomas precession 62, 167 torsion
energy 73 thought experiment 122 balance 335
mass 73 tidal effects 142, 191, 207, 268 in general relativity 204,
momentum 73 tides 184, 334 292
space-time diagram 61 and curvature, illustration train 138
Tarantula nebula 212 191 illustration of accelerating
tau mass 303 illustration of effects 142 138
tax collection 295 time relativistic circular 85

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
tea 70 and light cone 47 transformation
teaching as a partial order 46 conformal 55, 89–90
best method for 9 beginning of 226 scaling 89
general relativity 292 delay near mass 163–164 translation 89
teleportation 65 distinction from space 286 travel
telescope 213 is not absolute 42 into the past 47
largest 214 is relative 42 puzzle about maximum
on satellites 214 machine 50 distance 48
photograph 217 proper 45, 87 time 47
temperature relativity of 29 tree
experimental challenge 63 stopping, minimum 121 and photons 74
relativistic transformation wristwatch 45, 145 and star analogy 241
62–63 time delay 290 appearance 232
tennis illustration of binary apple fall from 299
and speed of light 22–25 pulsar measurement 180 colour 99
relativistic 35 illustration of gravitational colour shift 140
tennis ball 122 164 triangle
tensor time dilation 29, 30, 44, 51 sum of angles 163
energy–momentum 116 experiments 30 Triassic 232
of curvature 189 factor 29 tropical year 305
Ricci 117, 194–195 illustration about muons tunnel 60
trace 191 50 twin
subject index 375

accelerated 55–58 illustration of bad for learning 9
twin paradox 49 measurements 238 virtual particle 72–74, 319
illustration 49, 313 illustrations of options 239 visualization
Unix 20 of relativistic motion 55–57
U Unruh effect see Volkswagen 187
udeko 297 Fulling–Davies–Unruh volt 297
Udekta 297 effect vortex, black 270
ultrarelativistic particle 82 Unruh radiation see
ultraviolet rays 15 Fulling–Davies–Unruh W
umbrellas 18 effect W boson
uncertainty up quark mass 303
T relative 300
total 300
mass 303
UTC 155
walking
Olympic 58
unit speed record puzzle 58
t win astronomical 306 V walking speed record
natural 304 vacuum see also space, 90 puzzle about 59
units 295 cleaner 24 water

Motion Mountain – The Adventure of Physics
non-SI 298 curvature 195 cannot fill universe 245
provincial 298, 299 definition 283 on Earth 232
SI, definition 295 dragging 173 watt 297
universal gravity 173 hand in 334 wave 4-vector 86
deviation from 245 impedance 304 wave motion
universal time coordinate 155, permeability 304 in relativity 86–87
298 permittivity 304 phase 86
universe 260 variance 300 wave, gravitational
a black hole? 279 Vavilov–Čerenkov radiation spectrum table 175
age of 237 28, 183 weak energy condition 155

copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
air-filled 245 velocity weak equivalence principle
believed 223 angular 85 335
curved 236 composition 39–40 weak mixing angle 302
energy of 256 composition formula 40 weber 297
finite age 227 composition, graph of 39 weight 158
flat 236 faster than light 84 weko 297
full 224 graph on relation with Wekta 297
illustration of parameters momentum 38 white dwarfs 222, 251
236 illustration of locality 84 Wien’s displacement constant
not static 234, 237 measurement 90 305
observable 223 perfect 287 wind 18
preferred frame 237 proper 314 window frame 58
shape of 254 proper, definition 48 wise old man 87
slow motion in 76 relative 74, 84 WMAP 133
state of 257 relative - undefined 193 women 37, 269
topology 236 velocity of light World Geodetic System 307
topology of 254 one-way 63, 315 world-line 46, 47
transparency of 245–246 two-way 63 wristwatch time 45, 145
water-filled 245 vendeko 297 written texts 233
universe’s atlas Vendekta 297
illustration 225 Venus 166 X
universe’s evolution video X-rays 15
376 subject index

xenno 297 effect 50, 51 zepto 297
Xenta 297 retaining 151 Zetta 297
Yucatan impact 233 Zwicky ring
Y photograph of 253
yocto 297 Z
Yotta 297 Z boson
youth mass 303

X
xenno

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–August 2023 free pdf ﬁle available at www.motionmountain.net
MOTION MOUNTAIN
The Adventure of Physics – Vol. II
Relativity and Cosmology

What is the most fantastic voyage possible?
Are shadows faster than light?
Can light be accelerated?
How does empty space bend and how do we measure it?
How stiff is empty space?
What are black holes?
What can we see and discover with the best telescopes?
What is the history of the universe?
What are the maximum force and power values in nature?

Answering these and other questions on motion,
this series gives an entertaining and mind-twisting
introduction into modern physics – one that is
surprising and challenging on every page.
Starting from everyday life, the adventure provides
an overview of modern results in mechanics,
heat, electromagnetism, relativity,
quantum physics and unification.

Christoph Schiller, PhD Université Libre de Bruxelles,
is a physicist and physics popularizer. He wrote this
book for his children and for all students, teachers and
readers interested in physics, the science of motion.

Pdf file available free of charge at
www.motionmountain.net