GRAVITATION AND
GAUGE SYMMETRIES
Series in High Energy Physics, Cosmology and Gravitation
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SERIES IN HIGH ENERGY PHYSICS,
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GRAVITATION AND

GAUGE SYMMETRIES
Milutin Blagojevi
´
c
Institute of Physics,
Belgrade, Yugoslavia
INSTITUTE OF PHYSICS PUBLISHING
BRISTOL AND PHILADELPHIA
c
IOP Publishing Ltd 2002
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN 0 7503 0767 6
Library of Congress Cataloging-in-Publication Data are available
Cover image: A burst of outgoing gravitational radiation originating
from a grazing merger of two black holes. (Courtesy Konrad-Zuse-
Zentrum fuer Informationstechnik Berlin (ZIB) and Max-Planck-Institut fuer
Gravitationsphysik (Albert-Einstein-Institut, AEI). Simulation: AEI Numerical
Relativity Group in cooperation with WashU and NCSA (Ed Seidel, Wai-Mo Suen
et al). Visualization: Werner Benger.)
Commissioning Editor: James Revill
Production Editor: Simon Laurenson
Production Control: Sarah Plenty
Cover Design: Victoria Le Billon

Marketing Executive: Laura Serratrice
Published by Institute of Physics Publishing, wholly owned by The Institute of
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A
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Printed in the UK by MPG Books Ltd, Bodmin, Cornwall
Contents
Preface xi
1 Space, time and gravitation 1
1.1 Relativity of space and time 1
Historical introduction 1
Relativity of motion and the speed of light 3
From space and time to spacetime 6
1.2 Gravitation and geometry 9
The principle of equivalence 9
Physics and geometry 11
Relativity, covariance and Mach’s ideas 14
Perspectives of further developments 18
2 Spacetime symmetries 20
2.1 Poincar´e symmetry 21
Poincar´e transformations 21
Lie algebra and its representations 22
Invariance of the action and conservation laws 24
2.2 Conformal symmetry 27

Conformal transformations and Weyl rescaling 27
Conformal algebra and finite transformations 29
Conformal symmetry and conserved currents 32
Conformal transformations in D = 235
Spontaneously broken scale invariance 37
Exercises 39
3 Poincar
´
e gauge theory 42
3.1 Poincar´e gauge invariance 43
Localization of Poincar´e symmetry 43
Conservation laws and field equations 47
On the equivalence of different approaches 50
3.2 Geometric interpretation 51
Riemann–Cartan space U
4
51
Geometric and gauge structure of PGT 61
vi
Contents
The principle of equivalence in PGT 62
3.3 Gravitational dynamics 65
Einstein–Cartan theory 65
Teleparallel theory 68
General remarks 72
Exercises 75
4 Weyl gauge theory 78
4.1 Weyl gauge invariance 79
Localization of Weyl symmetry 79
Conservation laws and field equations 83

Conformal versus Weyl gauge symmetry 85
4.2 Weyl–Cartan geometry 86
Conformal transformations in Riemann space 86
Weyl space W
4
89
Weyl–Cartan space Y
4
92
4.3 Dynamics 94
Weyl’s theory of gravity and electrodynamics 95
Scalar fields and the improved energy–momentum tensor 96
Goldstone bosons as compensators 100
General remarks 102
Exercises 105
5 Hamiltonian dynamics 107
5.1 Constrained Hamiltonian dynamics 108
Introduction to Dirac’s theory 108
Generators of gauge symmetries 119
Electrodynamics 123
5.2 The gravitational Hamiltonian 125
Covariance and Hamiltonian dynamics 125
Primary constraints 128
The (3 + 1) decomposition of spacetime 129
Construction of the Hamiltonian 131
Consistency of the theory and gauge conditions 134
5.3 Specific models 136
Einstein–Cartan theory 136
The teleparallel theory 141
Exercises 148

6 Symmetries and conservation laws 152
6.1 Gauge symmetries 153
Constraint algebra 153
Gauge generators 154
6.2 Conservation laws—EC theory 157
Asymptotic structure of spacetime 158
Contents
vii
Improving the Poincar´e generators 160
Asymptotic symmetries and conservation laws 163
6.3 Conservation laws—the teleparallel theory 168
A simple model 168
The Poincar´e gauge generators 170
Asymptotic conditions 171
The improved Poincar´e generators 173
Conserved charges 176
6.4 Chern–Simons gauge theory in D = 3 179
Chern–Simons action 180
Canonical analysis 183
Symmetries at the boundary 187
Exercises 191
7 Gravity in flat spacetime 195
7.1 Theories of long range forces 196
Scalar field 196
Vector field 197
The symmetric tensor field 202
The sign of the static interaction 206
7.2 Attempts to build a realistic theory 207
Scalar gravitational field 207
Symmetric tensor gravitational field 210

Can the graviton have a mass? 214
The consistency problem 219
Exercises 220
8 Nonlinear effects in gravity 222
8.1 Nonlinear effects in Yang–Mills theory 222
Non-Abelian Yang–Mills theory 222
Scalar electrodynamics 226
8.2 Scalar theory of gravity 228
8.3 Tensor theory of gravity 231
The iterative procedure 231
Formulation of a complete theory 232
8.4 The first order formalism 237
Yang–Mills theory 237
Einstein’s theory 239
Exercises 243
9 Supersymmetry and supergravity 245
9.1 Supersymmetry 246
Fermi–Bose symmetry 246
Supersymmetric extension of the Poincar´e algebra 251
The free Wess–Zumino model 255
viii
Contents
Supersymmetric electrodynamics 257
9.2 Representations of supersymmetry 259
Invariants of the super-Poincar´e algebra 259
Massless states 261
Massive states 264
Supermultiplets of fields 267
Tensor calculus and invariants 269
The interacting Wess–Zumino model 271

9.3 Supergravity 273
The Rarita–Schwinger field 273
Linearized theory 276
Complete supergravity 278
Algebra of local supersymmetries 281
Auxiliary fields 283
General remarks 286
Exercises 290
10 Kaluza–Klein theory 293
10.1 Basic ideas 294
Gravity in five dimensions 294
Ground state and stability 298
10.2 Five-dimensional KK theory 303
Five-dimensional gravity and effective theory 303
Choosing dynamical variables 307
The massless sector of the effective theory 310
Dynamics of matter and the fifth dimension 312
Symmetries and the particle spectrum 315
10.3 Higher-dimensional KK theory 320
General structure of higher-dimensional gravity 320
The massless sector of the effective theory 325
Spontaneous compactification 329
General remarks 331
Exercises 335
11 String theory 338
11.1 Classical bosonic strings 339
The relativistic point particle 339
Action principle for the string 341
Hamiltonian formalism and symmetries 344
11.2 Oscillator formalism 346

Open string 347
Closed strings 349
Classical Virasoro algebra 350
11.3 First quantization 353
Quantum mechanics of the string 353
Contents
ix
Quantum Virasoro algebra 355
Fock space of states 356
11.4 Covariant field theory 358
Gauge symmetries 358
The action for the free string field 361
Electrodynamics 362
Gravity 364
11.5 General remarks 366
Exercises 369
Appendices 373
A Local internal symmetries 373
B Differentiable manifolds 379
C De Sitter gauge theory 390
D The scalar–tensor theory 396
E Ashtekar’s formulation of GR 402
F Constraint algebra and gauge symmetries 410
G Covariance, spin and interaction of massless particles 415
H Lorentz group and spinors 421
I Poincar
´
e group and massless particles 433
J Dirac matrices and spinors 443
K Symmetry groups and manifolds 451

L Chern–Simons gravity in three dimensions 473
M Fourier expansion 487
Bibliography 489
Notations and conventions 513
Index 517

Preface
The concept of a unified description of the basic physical interactions has evolved
in parallel with the development of our understanding of their dynamical struc-
ture. It has its origins in Maxwell’s unification of electricity and magnetism in the
second half of the nineteenth century, matured in Weyl’s and Kaluza’s attempts
to unify gravity and electromagnetism at the beginning of the last century, and
achieved its full potential in the 1970s, in the process of unifying the weak and
electromagnetic and also, to some extent, the strong interactions. The biggest bar-
rier to this attractive idea comes from the continual resistance of gravity to join the
other basic interactions in the framework of a unified, consistent quantum theory.
As the theory of electromagnetic, weak and strong interactions developed,
the concept of (internal) gauge invariance came of age and established itself as an
unavoidable dynamical principle in particle physics. It is less well known that the
principle of equivalence, one of the prominent characteristics of the gravitational
interaction, can also be expressed as a (spacetime) gauge symmetry. This book
is intended to shed light upon the connection between the intrinsic structure of
gravity and the principle of gauge invariance, which may lead to a consistent
unified theory.
The first part of this book, chapters 1–6, gives a systematic account of the
structure of gravity as a theory based on spacetime gauge symmetries. Some basic
properties of space, time and gravity are reviewed in the first, introductory chapter.
Chapter 2 deals with elements of the global Poincar´e and conformal symmetries,
which are necessary for the exposition of their localization; the structure of
the corresponding gauge theories is explored in chapters 3 and 4. Then, in

chapters 5 and 6, we present the basic features of the Hamiltonian dynamics
of Poincar´e gauge theory, discuss the relation between gauge symmetries and
conservation laws and introduce the concept of gravitational energy and other
conserved charges. The second part of the book treats the most promising
attempts to build a unified field theory containing gravity, on the basis of the
gauge principle. In chapters 7 and 8 we discuss the possibility of constructing
gravity as a field theory in flat spacetime. Chapters 9–11 yield an exposition
of the ideas of supersymmetry and supergravity, Kaluza–Klein theory and string
theory—these ideas can hardly be avoided in any attempt to build a unified theory
of basic physical interactions.
xi
xii
Preface
This book is intended to provide a pedagogical survey of the subject of
gravity from the point of view of particle physics and gauge theories at the
graduate level. The book is written as a self-contained treatise, which means
that I assume no prior knowledge of gravity and gauge theories on the part of the
reader. Of course, some familiarity with these subjects will certainly facilitate
the reader to follow the exposition. Although the gauge approach differs from the
more standard geometric approach, it leads to the same mathematical and physical
structures.
The first part of the book (chapters 2–6) has evolved from the material
covered in the one-semester graduate course Gravitation II, taught for about
20 years at the University of Belgrade. Chapters 9–11 have been used as the basis
for a one-semester graduate course on the unification of fundamental interactions.
Special features
The following remarks are intended to help the reader in an efficient use of the
book.
Examples in the text are used to illustrate and clarify the main exposition.
The exercises given at the end of each chapter are an integral part of the

book. They are aimed at illustrating, completing, applying and extending the
results discussed in the text.
Short comments on some specific topics are given at the end of each
chapter, in order to illustrate the relevant research problems and methods
of investigation.
The appendix consists of 13 separate sections (A–M), which have different
relationships with the main text.
– Technical appendices J and M (Dirac spinors, Fourier expansion) are
indispensable for the exposition in chapters 9 and 11.
– Appendices A, H and I (internal local symmetries, Lorentz and Poincar´e
group) are very useful for the exposition in chapters 3 (A) and 9 (H, I).
– Appendices C, D, E, F, G and L (de Sitter gauge theory, scalar–tensor
theory, Ashtekar’s formulation of general relativity, constraint algebra
and gauge symmetries, covariance, spin and interaction of massless
particles, and Chern–Simons gravity) are supplements to the main
exposition, and may be studied according to the reader’s choice.
– The material in appendices B and K (differentiable manifolds, symmetry
groups and manifolds) is not necessary for the main exposition. It gives
a deeper mathematical foundation for the geometric considerations in
chapters 3, 4 and 10.
The bibliography contains references that document the material covered in
the text. Several references for each chapter, which I consider as the most
suitable for further reading, are denoted by the symbol •.
Preface
xiii
Chapters 4, 7 and 8 can be omitted in the first reading, without influencing
the internal coherence of the exposition. Chapters 9–11 are largely
independent of each other, and can be read in any order.
Acknowledgments
The material presented in this book has been strongly influenced by the research

activities of the Belgrade group on Particles and Fields, during the last 20 years.
I would like to mention here some of my teachers and colleagues who, in one
or another way, have significantly influenced my own understanding of gravity.
They are: Rastko Stojanovi´c and Marko Leko, my first teachers of tensor calculus
and general relativity; Djordje
ˇ
Zivanovi´c, an inspiring mind for many of us who
were fascinated by the hidden secrets of gravity; Paul Senjanovi´c, who brought the
Dirac canonical approach all the way from City College in New York to Belgrade;
Ignjat Nikoli´c and Milovan Vasili´c, first my students and afterwards respected
collaborators; Branislav Sazdovi´c, who silently introduced supersymmetry and
superstrings into our everyday life; then, Dragan Popovi´c, the collaborator from
an early stage on our studies of gravity, and Djordje
ˇ
Sijaˇcki, who never stopped
insisting on the importance of symmetries in gravity.
For illuminating comments and criticisms on a preliminary version of the
manuscript, I wish to thank my colleagues Tatjana Vukaˇsinac, Milovan Vasili´c,
Djordje
ˇ
Sijaˇcki, Branislav Sazdovi´c, Nenad Manojlovi´c, Aleksandar Bogojevi´c,
Friedrich Hehl and Eckehard Mielke. I am especially grateful for their valuable
help to my students Olivera Miˇskovi´c and Dejan Stojkovi´c, who carefully studied
the first version of the manuscript and solved all the exercises with patience. I owe
special thanks to Ignjat Nikoli´c, coauthor of the manuscript The Theory of Gravity
II, written in 1986 for the graduate school of physics in Belgrade, for allowing me
to use a large part of my contribution to that text, and include some of his formu-
lations (in sections 5.2 and 8.4) and exercises. Finally, it is my pleasure to thank
Friedrich Hehl and Yuri Obukhov for their sincere support during this project.
The exposition of Poincar´e gauge theory and its Hamiltonian structure is

essentially based on the research done in collaboration with Ignjat Nikoli´cand
Milovan Vasili´c. The present book incorporates a part of the spirit of that
collaboration, which was born from our longlasting joint work on the problems
of gravity.
The sections on teleparallel theory, Chern–Simons gauge theory and three-
dimensional gravity have been written as an update to the first version of the
manuscript (published in 1997, in Serbian). I would like to thank Antun Balaˇs
and Olivera Miˇskovi´c for their assistance at this stage, and to express my gratitude
for the hospitality at the Primorska Institute for Natural Sciences and Technology,
Koper, Slovenia, where part of the update was completed.
Milutin Blagojevi
´
c
Belgrade, April 2001
Chapter 1
Space, time and gravitation
Theories of special and general relativity represent a great revolution in our
understanding of the structure of space and time, as well as of their role in
the formulation of physical laws. While special relativity (SR) describes the
influence of physical reality on the general properties of and the relation between
space and time, the geometry of spacetime in general relativity (GR) is connected
to the nature of gravitational interaction. Perhaps the biggest barrier to a
full understanding of these remarkable ideas lies in the fact that we are not
always ready to suspect the properties of space and time that are built into our
consciousness by everyday experience.
In this chapter we present an overview of some aspects of the structure
of space, time and gravitation, which are important for our understanding of
gravitation as a gauge theory. These aspects include:
the development of the principle of relativity from classical mechanics and
electrodynamics, and its influence on the structure of space and time; and

the formulation of the principle of equivalence, and the introduction of
gravitation and the corresponding geometry of curved spacetime.
The purpose of this exposition is to illuminate those properties of space, time and
gravitation that have had an important role in the development of GR, and still
have an influence on various attempts to build an alternative approach to gravity
(Sciama 1969, Weinberg 1972, Rindler 1977, Hoffmann 1983).
1.1 Relativity of space and time
Historical introduction
In order to get a more complete picture of the influence of relativity theories on
the development of the concepts of space and time, we recall here some of the
earlier ideas on this subject.
1
2
Space, time and gravitation
In Ancient Greece, the movement of bodies was studied philosophically.
Many of the relevant ideas can be found in the works of Aristotle (fourth century
BC) and other Greek philosophers. As an illustration of their conception of the
nature of movement, we display here the following two statements.
The speed of a body in free fall depends on its weight; heavy bodies fall
faster than lighter ones.
The earth is placed at the fixed centre of the universe.
While the first statement was so obvious that practically everyone believed in
it, different opinions existed about the second one. One of the earliest recorded
proposals that the earth might move belongs to the Pythagorean Philolaus (fifth
century BC). Two centuries later, the idea appeared again in a proposal of the
Greek astronomer Aristarchus (third century BC). However, it was not persuasive
for most of the ancient astronomers. From a number of arguments against the
idea of a moving earth, we mention Aristotle’s. He argued that if the earth were
moving, then a stone thrown straight up from the point A would fall at another
point B, since the original point A would ‘run away’ in the direction of earth’s

movement. However, since the stone falls back at the same point from which it is
thrown, he concluded that the earth does not move.
For a long time, the developments of physics and astronomy have been
closely connected. Despite Aristarchus, the ancient Greek astronomers continued
to believe that the earth is placed at the fixed centre of the universe. This
geocentric conception of the universe culminated in Ptolemy’s work (second
century AD). The Ptolemaic system endured for centuries without major
advances. The birth of modern astronomy started in the 16th century with the
work of Copernicus (1473–1543), who dared to propose that the universe is
heliocentric, thus reviving Aristarchus’ old idea. According to this proposal, it is
not the earth but the sun that is fixed at the centre of everything, while the earth and
other planets move around the sun. The Danish astronomer Brahe (1546–1601)
had his own ideas concerning the motion of planets. With the belief that the clue
for all answers lies in measurements, he dedicated his life to precise astronomical
observations of the positions of celestial bodies. On the basis of these data Kepler
(1571–1630) was able to deduce his well known laws of planetary motion. With
these laws, it became clear how the planets move; a search for an answer to the
question why the planets move led Newton (1643–1727) to the discovery of the
law of gravitation.
A fundamental change in the approach to physical phenomena was made
by the Italian scientist Galileo Galilei (1564–1642). Since he did not believe in
Aristotle’s ‘proofs’, he began a systematic analysis and experimental verification
of the laws of motion. By careful measurement of spatial distances and time
intervals during the motion of a body along an inclined plane, he found new
relations between distances, time intervals and velocities, that were unknown in
Aristotle’s time. What were Galileo’s answers to the questions of free fall and the
motion of the earth?
Relativity of space and time
3
Studying the problem of free fall, Galileo discovered that all bodies fall with

the same acceleration, no matter what their masses are nor what they are
made of. This is the essence of the principle of equivalence,whichwasused
later by Einstein (1879–1955) to develop GR.
Trying to understand the motion of the earth, Galileo concluded that the
uniform motion of the earth cannot be detected by means of any internal
mechanical experiment (thereby overturning Aristotle’s arguments about the
immobile earth). The conclusion about the equivalence of different (inertial)
reference frames moving with constant velocities relative to each other,
known as the principle of relativity, has been of basic importance for the
development of Newton’s mechanics and Einstein’s SR.
Let us mention one more discovery by Galileo. The velocity of a body
moving along an inclined plane changes in time. The cause of the change is
the gravitational attraction of that body and the earth. When the attraction is
absent and there is no force acting on the body, its velocity remains constant.
This is the well known law of inertia of classical mechanics.
The experiments performed by Galileo may be considered to be the origin of
modern physics. His methods of research and the results obtained show, by their
simplicity and their influence on future developments in physics, all the beauty
and power of the scientific truth. He studied the motion of bodies by asking where
and when something happens. Since then, measurements of space and time have
been an intrinsic part of physics.
The concepts of time and space are used in physics only with reference
to physical objects. What are these entities by themselves? ‘What is time—if
nobody asks me, I know, but if I want to explain it to someone, then I do not
know’ (St Augustine; a citation from J R Lucas (1973)). Time and space are
connected with change and things that change. The challenge of physics is not to
define space and time precisely, but to measure them precisely.
Relativity of motion and the speed of light
In Galileo’s experiments we find embryos of the important ideas concerning space
and time, ideas which were fully developed later in the works of Newton and

Einstein.
Newton’s classical mechanics, in which Galileo’s results have found a
natural place, is based on the following three laws:
1. A particle moves with constant velocity if no force acts on it.
2. The acceleration of a particle is proportional to the force acting on it.
3. The forces of action and reaction are equal and opposite.
Two remarks will clarify the content of these laws. First, the force appearing
in the second law originates from interactions with other bodies, and should be
known from independent considerations (e.g. Newton’s law of gravitation). Only
4
Space, time and gravitation
then can the second law be used to determine the acceleration stemming from
a given force. Second, physical quantities like velocity, acceleration, etc, are
defined always and only relative to some reference frame.
Galilean principle of relativity. The laws of Newtonian mechanics do not
always hold in their simplest form, as stated earlier. If, for instance, an observer is
placed on a disc rotating relative to the earth, he/she will sense a ‘force’ pushing
him/her toward the periphery of the disc, which is not caused by any interaction
with other bodies. Here, the acceleration is not a consequence of the usual force,
but of the so-called inertial force. Newton’s laws hold in their simplest form only
in a family of reference frames, called inertial frames. This fact represents the
essence of the Galilean principle of relativity (PR):
PR: The laws of mechanics have the same form in all inertial frames.
The concepts of force and acceleration in Newton’s laws are defined relative
to an inertial frame. Both of them have the same value in two inertial frames,
moving relative to each other with a constant velocity. This can be seen by
observing that the space and time coordinates† in two such frames S and S

(we
assume that S


moves in the x-direction of S with constant velocity v) are related
in the following way:
x

= x − vty

= yz

= zt

= t. (1.1)
These relations, called Galilean transformations, represent the mathematical
realization of the Galilean PR. If a particle moves along the x -axis of the frame S,
its velocities, measured in S and S

, respectively, are connected by the relations
u

1
= u
1
− v u

2
= u
2
u

3

= u
3
(1.2)
representing the classical velocity addition law (u
1
= dx/dt, etc). This law
implies that the acceleration is the same in both frames. Also, the gravitational
force m
1
m
2
/r
2
, for instance, has the same value in both frames.
Similar considerations lead us to conclude that there is an infinite set of
inertial frames, all moving uniformly relative to each other. What property singles
out the class of inertial frames from all the others in formulating the laws of
classical mechanics?
Absolute space. In order to answer this and other similar questions, Newton
introduced the concept of absolute space, that is given aprioriand independently
of the distribution and motion of matter. Each inertial frame moves with a constant
velocity relative to absolute space and inertial forces appear as a consequence of
the acceleration relative to this space.
† Standard coordinates in inertial frames are orthonormal Cartesian spatial coordinates (x, y, z) and
a time coordinate t.
Relativity of space and time
5
Trying to prove the physical relevance of acceleration relative to absolute
space, Newton performed the following experiment. He filled a vessel with water
and set it to rotate relative to the frame of distant, fixed stars (absolute space).

The surface of the water was at first flat, although the vessel rotated. Then, due
to the friction between the water and the vessel, the water also began to rotate, its
surface started to take a concave form, and the concavity increased until the water
was rotating at the same rate as the vessel. From this behaviour Newton drew the
conclusion that the appearance of inertial forces (measured by the concavity of
the surface of water) does not depend on the acceleration relative to other objects
(the vessel), but only on the acceleration relative to absolute space.
Absolute space did not explain the selected role of inertial frames; it only
clarified the problem. Introduction of absolute space is not consistent within
classical mechanics itself. The physical properties of absolute space are very
strange. Why can we only observe accelerated and not uniform motion relative to
absolute space? Absolute space is usually identified with the frame of fixed stars.
Well-founded objections against absolute space can be formulated in the form of
the following statements:
The existence of absolute space contradicts the internal logic of classical
mechanics since, according to Galilean PR, none of the inertial frames can
be singled out.
Absolute space does not explain inertial forces since they are related to
acceleration with respect to any one of the inertial frames.
Absolute space acts on physical objects by inducing their resistance to
acceleration but it cannot be acted upon.
Thus, absolute space did not find its natural place within classical mechanics, and
the selected role of inertial frames remained essentially unexplained.
The speed of light. Galilean PR holds for all phenomena in mechanics. In the
last century, investigation of electricity, magnetism and light aroused new interest
in understanding the PR. Maxwell (1831–79) was able to derive equations which
describe electricity and magnetism in a unified way. The light was identified
with electromagnetic waves, and physicists thought that it propagated through a
medium called the ether. For an observer at rest relative to the ether, the speed
of light is c = 3 × 10

10
cm s
−1
, while for an observer moving towards the light
source with velocity v, the speed of light would be c

= c + v, on the basis of
the classical velocity addition law. The ether was for light the same as air is for
sound. It was one kind of realization of Newton’s absolute space. Since it was
only in the reference frame of ether that the speed of light was c, the speed of
light could be measured in various reference frames and the one at rest relative
to the ether could be found. If there was one such frame, PR would not hold
for electromagnetism. The fate of absolute space was hidden in the nature of
electromagnetic phenomena.
6
Space, time and gravitation
Although the classical velocity addition law has many confirmations in
classical mechanics, it does not hold for the propagation of light. Many
experiments have shown that
c

= c.
The speed of light is the same in all inertial frames, at all times and in all
directions, independently of the motion of the source and/or the observer. This
fact represents a cornerstone of SR. It contradicts classical kinematics but must
be accepted on the basis of the experimental evidence. The constancy of the speed
of light made the ether unobservable and eliminated it from physics forever.
A convincing experimental resolution of the question of the relativity of light
phenomena was given by Michelson and Morley in 1887. They measured the
motion of the light signal from a source on the moving earth and showed that its

velocity is independent of the direction of motion. From this, we conclude that
since the motion of an observer relative to the ether is unobservable, the PR
also holds for light phenomena; and
the speed of light does not obey the classical velocity addition law, but has
the same value in all inertial frames.
Note that in the first statement the PR puts all inertial frames on an equal footing
without implying Galilean transformations between them, since these contain the
classical velocity addition law, which contradicts the second statement. Thus, it
becomes clear that the PR must take a new mathematical form, one that differs
from (1.1).
From space and time to spacetime
The results of previous considerations can be expressed in the form of the two
postulates on which SR is based.
The first postulate is a generalization of Galilean PR not only to light
phenomena, but to the whole of physics and is often called Einstein’s principle
of relativity.
P1. Physical laws have the same form in all inertial frames.
Although (P1) has a form similar to Galilean PR, the contents of the two are
essentially different. Indeed, Galilean PR is realized in classical mechanics in
terms of Galilean transformations and the classical velocity addition law, which
does not hold for light signals. The realization of (P1) is given in terms of Lorentz
transformations, as we shall soon see.
The second postulate is related to the experimental fact concerning the speed
of light.
P2. The speed of light is finite and equal in all inertial frames.
Relativity of space and time
7
The fact that the two postulates are not in agreement with the classical
velocity addition law cannot be explained within Newtonian mechanics. Einstein
found a simple explanation of this puzzle by a careful analysis of the space

and time characteristics of physical events. He came to the conclusion that the
concepts of time and space are relative, i.e. dependent on the reference frame of
an observer.
The moment at which an event happens (e.g. the flash of a bulb) may be
determined by using clocks. Let T
A
be a clock at point A; the time of an event
at A is determined by the position of the clock hands of T
A
at the moment of
the occurrence of that event. If the clock T
1
is at A
1
, and the bulb is placed at
some distant point A
2
,thenT
1
does not register the moment of the bulb flash at
A
2
, but the moment the signal arrives at A
1
. We can place another clock at A
2
which will measure the moment of the flash, but that is not enough. The clocks
T
1
and T

2
have to be synchronized: if the time of the flash at A
2
is t
2
, then the
time of the arrival of the signal at A
1
, according to T
1
,mustbet
1
= t
2
+ (the
travelling time of the signal). By this procedure we have defined the simultaneity
of distant events: taking into account the travelling time of the signal we know
which position of the clock hands at A
1
is synchronized with the bulb flash at A
2
.
A set of synchronized clocks T
1
, T
2
, , disposed at all points of reference
frame S, enables the measurement of time t of an arbitrary event in S. According
to this definition, the concept of simultaneity of two events is related to a given
inertial reference frame S. This notion of simultaneity is relative, as it depends on

the inertial frame of an observer.
Using similar arguments, we can conclude that space lengths and time
intervals are also relative quantities.
Lorentz transformations. Classical ideas about space and time, which are
expressed by Galilean transformations, have to be changed in accordance with
postulates (P1) and (P2). These postulates imply a new connection between two
inertial frames S and S

, which can be expressed by the Lorentz transformation of
coordinates:
x

=
x − vt

1 − v
2
/c
2
t

=
t − vx /c
2

1 − v
2
/c
2
y


= yz

= z.
(1.3)
In the limit of small velocities v, the Lorentz transformation reduces to the
Galilean one.
From (1.3) we obtain a new law for the addition of velocities:
u

1
=
u
1
− v
1 − u
1
v/c
2
u

2
= u
2
, u

3
= u
3
. (1.4)

The qualitative considerations concerning the relativity of space and time
can now be put into a precise mathematical form.
8
Space, time and gravitation
We begin by the relativity of lengths. Consider a rigid rod fixed in an inertial
frame S

, whose (proper) length is x

= x

2
− x

1
. The length of the rod in
another inertial frame S is determined by the positions of its ends at the same
moment of the S-time: x = x
2
(t) − x
1
(t). From expressions (1.3) it follows
that x = x


1 − v
2
/c
2
. The length of the moving rod, measured from S,is

less than its length in the rest frame S

, x <x

. This effect is called the length
contraction.
In order to clarify the relativity of time intervals, we consider a clock fixed in
S

. Its ‘tick’ and ‘tack’ can be described by coordinates (x

, t

1
) and (x

, t

2
).Using
Lorentz transformation (1.3), we obtain the relation t

= t

1 − v
2
/c
2
,which
shows that the time interval between two strikes of the clock is shortest in its rest

frame, t

<t. Since the rate assigned to a moving clock is always longer than
its proper rate, we talk about time dilatation. An interesting phenomenon related
to this effect is the so-called twin paradox.
Both length contraction and time dilatation are real physical effects.
Four-dimensional geometry. The connection between the space and time
coordinates of two inertial frames, moving with respect to one other with some
velocity v, is given by the Lorentz transformation (1.3). It is easily seen that the
general transformation between the two inertial frames includes spatial rotations
and translations as well as time translations. The resultant set of transformations is
known as the set of Poincar´e transformations. Since these transformations ‘mix’
space and time coordinates, it turns out that it is more natural to talk about four-
dimensional spacetime than about space and time separately. Of course, although
space and time have equally important roles in spacetime, there is a clear physical
distinction between them. This is seen in the form of the Lorentz transformation,
and this has an influence on the geometric properties of spacetime.
The invariance of the expression s
2
= c
2
t
2
− x
2
− y
2
− z
2
with respect

to the Poincar´e transformations represents a basic characteristic of the spacetime
continuum—the Minkowski space M
4
. The points in M
4
maybelabelledby
coordinates (t, x, y, z), and are called events. The expression s
2
has the role
of the squared ‘distance’ between the events (0, 0, 0, 0) and (t, x, y, z).Inthe
same way as the squared distance in the Euclidean space E
3
is invariant under
Galilean transformations, the expression s
2
in M
4
is invariant under Poincar´e
transformations. It is convenient to introduce the squared differential distance
between neighbouring events:
ds
2
= c
2
dt
2
− dx
2
− dy
2

− dz
2
. (1.5a)
This equation can be written in a more compact, tensor form:
ds
2
= η
µν
dx
µ
dx
ν
(1.5b)
where dx
µ
= (dt, dx , dy, dz), η
µν
= diag(+, −, −, −) is the metric of M
4
,and
a summation over repeated indices is understood.
Gravitation and geometry
9
Lorentz transformations have the form x
µ
= 
µ
ν
x
ν

, where the coefficients

µ
ν
are determined by equation (1.3). The set of four quantities which transform
according to this rule is called a vector of M
4
. The geometric formalism can be
further developed by introducing general tensors; Lorentz transformations can be
understood as ‘rotations’ in M
4
(since they do not change the ‘lengths’ of the
vectors), etc. The analogy with the related concepts of Euclidean geometry is
substantial, but not complete. While Euclidean metric is positive definite, i.e. ds
2
is positive, the Minkowskian metric is indefinite, i.e. ds
2
may be positive, negative
or zero. The distance between two points in M
4
maybezeroevenwhenthese
points are not identical. However, this does not lead to any essential difference in
the mathematical treatment of M
4
compared to the Euclidean case. The indefinite
metric is a mathematical expression of the distinction between space and time.
The geometric formulation is particularly useful for the generalization of this
theory and construction of GR.
1.2 Gravitation and geometry
The principle of equivalence

Clarification of the role of inertial frames in the formulation of physical laws is
not the end of the story of relativity. Attempts to understand the physical meaning
of the accelerated frames led Einstein to the general theory of space, time and
gravitation.
Let us observe possible differences between the inertial and gravitational
properties of a Newtonian particle. Newton’s second law of mechanics can be
written in the form F = m
i
a,wherem
i
is the so-called inertial mass, which
measures inertial properties (resistance to acceleration) of a given particle. The
force acting on a particle in a homogeneous gravitational field g has the form
F
g
= m
g
g,wherem
g
is the gravitational mass of the particle, which may
be regarded as the gravitational analogue of the electric charge. Experiments
have shown that the ratio m
g
/m
i
is the same for all particles or, equivalently,
that all particles experience the same acceleration in a given gravitational field.
This property has been known for a long time as a consequence of Galileo’s
experiments with particles moving along an inclined plane. It is also true in an
inhomogeneous gravitational field provided we restrict ourselves to small regions

of spacetime. The uniqueness of the motion of particles is a specific property of
the gravitational interaction, which does not hold for any other force in nature.
On the other hand, all free particles in an accelerated frame have the same
acceleration. Thus, for instance, if a train accelerates its motion relative to the
earth, all the bodies on the train experience the same acceleration relative to
the train, independently of their (inertial) masses. According to this property,
as noticed by Einstein, the dynamical effects of a gravitational field and an
accelerated frame cannot be distinguished. This is the essence of the principle
of equivalence (PE).
10
Space, time and gravitation
PE. Every non-inertial frame is locally equivalent to some gravitational
field.
The equivalence holds only locally, in small regions of space and time, where
‘real’ fields can be regarded as homogeneous.
Expressed in a different way, the PE states that a given gravitational field
can be locally compensated for by choosing a suitable reference frame—a freely
falling (non-rotating) laboratory. In each such frame, all the laws of mechanics or,
more generally, the laws of physics have the same form as in an inertial frame. For
this reason, each freely falling reference frame is called a locally inertial frame.
PE

. At every point in an arbitrary gravitational field we can choose a
locally inertial frame in which the laws of physics take the same form
as in SR.
We usually make a distinction between the weak and strong PE. If we restrict
this formulation to the laws of mechanics, we have the weak PE. On the other
hand, if ‘the laws of physics’ means all the laws of physics, we have Einstein’s
PE in its strongest form (sometimes, this ‘very strong’ version of the PE is
distinguished from its ‘medium-strong’ form, which refers to all non-gravitational

laws of physics) (for more details see, for instance, Weinberg (1972) and Rindler
(1977)).
In previous considerations we used Newtonian mechanics and gravitation
to illustrate the meaning of the (weak) PE. As previously mentioned, the first
experimental confirmation of the equality of m
i
and m
g
(in suitable units) was
given by Galileo. Newton tested this equality by experiments with pendulums of
equal length but different composition. The same result was verified later, with
a better precision, by E¨otv¨os (1889; with an accuracy of 1 part in 10
9
), Dicke
(1964; 1 part in 10
11
) and Braginsky and Panov (1971; 1 part in 10
12
). Besides,
all experimental evidence in favour of GR can be taken as an indirect verification
of the PE.
The PE and local Poincar
´
e symmetry. It is very interesting, but not widely
known, that the PE can be expressed, using the language of modern physics, as
the principle of local symmetry. To see this, we recall that at each point in a given
gravitational field we can choose a locally inertial reference frame S(x) (on the
basis of the PE). The frame S(x ) can be obtained from an arbitrarily fixed frame
S
0

≡ S(x
0
) by
translating S
0
, so as to bring its origin to coincide with that of S(x) and
performing Lorentz ‘rotations’ on S
0
, until its axes are brought to coincide
with those of S(x).
Four translations and six Lorentz ‘rotations’ are the elements of the Poincar´e
group of transformations, the parameters of which depend on the point x at which
Gravitation and geometry
11
the locally inertial frame S(x) is defined. Since the laws of physics have the
same form in all locally inertial frames (on the basis of the PR), these Poincar´e
transformations are symmetry transformations. Thus, an arbitrary gravitational
field is characterized by the group of local, x-dependent Poincar´e transformations,
acting on the set of all locally inertial frames. When the gravitational field
is absent, we return to SR and the group of global, x-independent Poincar´e
transformations.
Physics and geometry
The physical content of geometry. The properties of space and time cannot
be deduced by pure mathematical reasoning, omitting all reference to physics.
There are many possible geometries that are equally good from the mathematical
point of view, but not so many if we make use of the physical properties of
nature. We believed in Euclidean geometry for more than 2000 years, as it was
very convincing with regard to the description of physical reality. However, its
logical structure was not completely clear. Attempts to purify Euclid’s system
of axioms led finally, in the 19th century, to the serious acceptance of non-

Euclidean geometry as a logical possibility. Soon after that, new developments
in physics, which resulted in the discovery of SR and GR, showed that non-
Euclidean geometry is not only a mathematical discipline, but also part of physics.
We shall now try to clarify how physical measurements can be related to the
geometric properties of space and time.
For mathematicians, geometry is based on some elementary concepts (such
as a point, straight line, etc), which are intuitively more or less clear, and certain
statements (axioms), which express the most fundamental relations between these
concepts. All other statements in geometry can be proved on the basis of
some definite mathematical methods, which are considered to be true within a
given mathematical structure. Thus, the question of the truthfulness of a given
geometric statement is equivalent to the question of the ‘truthfulness’ of the
related set of axioms. It is clear, however, that such a question has no meaning
within the geometry itself.
For physicists, the space is such as is seen in experiments; that space is, at
least, the space relevant for physics. Therefore, if we assign a definite physical
meaning to the basic geometric concepts (e.g. straight line ≡ path of the light
ray, etc), then questions of the truthfulness of geometric statements become
questions of physics, i.e. questions concerning the relations between the relevant
physical objects. This is how physical measurements become related to geometric
properties.
Starting from physically defined measurements of space and time in SR,
we are naturally led to introduce the Minkowskian geometry of spacetime. To
illustrate what happens in GR, we shall consider the geometry on a flat disc,
rotating uniformly (relative to an inertial frame) around the axis normal to its
plane, passing through the centre of the disc. There is an observer on the disc,
12
Space, time and gravitation
a)
b)

Figure 1.1. Approximate realization (a) of a curved surface (b).
trying to test spacetime geometry by physical measurements. His/her conclusions
will also be valid locally for true gravitational fields, on the basis of the PE.
Assume that the observer sits at the centre of the disc, and has two identical
clocks: one of them is placed at the centre, the other at some point on the
periphery. The observer will see that the clock on the periphery is running
slower (time dilatation from SR). Consequently, clocks at various positions in the
gravitational field run faster or slower, depending on the local strength of the field.
There is no definition of time that is pertinent to the whole spacetime in general.
The observer will also conclude that the length of a piece of line orthogonal to the
radius of the disc will be shortened (length contraction from SR). Therefore, the
ratio of the circumference of the circle to its radius will be smaller than 2π,so
that the Euclidean geometry of space does not hold in GR.
Geometry of curved surfaces. We have seen that in spacetime, within a limited
region, we can always choose a suitable reference frame, called the local inertial
frame. Taking spacetime apart into locally inertial components, we can apply the
laws of SR in each such component and derive various dynamical conclusions.
Reconstruction of the related global dynamical picture, based on the PE (and
some additional, simple geometric assumptions), gives rise to GR.
This procedure for dissecting spacetime into locally inertial components,
out of which we can reconstruct, using the PE, the global structure of spacetime
containing the gravitational field, can be compared geometrically with an attempt
to build a curved surface, approximately, from a lot of small, plane elements
‘continuously’ bound to each other (figure 1.1). A locally flat surface is a
geometric analogue of locally inertial spacetime.
GR makes essential use of curved spaces (or, more accurately curved
spacetimes). Since the curved, four-dimensional spacetime cannot be visualized,
let us try to understand the basic features of curved spaces by considering a
two-dimensional surface X
2

. We shall focus our attention on those geometric
properties of X
2
, which could be determined by an intelligent, two-dimensional
being entirely confined to live and measure in the surface, a being that does
not know anything about the embedding Euclidean space E
3
. Such properties
determine the intrinsic geometry on X
2
.