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

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

”
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.

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.

“
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).

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

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?

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

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.

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

Motion Mountain – The Adventure of Physics
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

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

— 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

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

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

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

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

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
today’s
scale
Scale 1.0 0
a

redshift
a(t) 0.5

ed
The expansion

rat
d
0.5 ate 1
either... ler

ele
e
ac c

dec
l Planck 1.5
hen 2
d, t

s
e

way
t 3
lera
ce

... or al
Quantum de
st past present future
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 con