Relativity This page intentionally left blank Relativity
SPECIAL, GENERAL, AND COSMOLOGICAL
SECOND EDITION
Wolfgang Rindler Professor of Physics The University of Texas at Dallas
3 3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c W. Rindler, 2006 The moral rights of the author have been asserted Database right Oxford University Press (maker) First edition first published 2001 Second edition first published 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd., King’s Lynn, Norfolk
ISBN 0–19–856731–6 978–0–19–856731–8 ISBN 0–19–856732–4 978–0–19–856732–5
10987654321 To my wife Linda the most generous person Ihaveeverknown This page intentionally left blank Make Physics as simple as possible, but no simpler. Albert Einstein
The ideal is to reach proofs by comprehension rather than by computation. Bernhard Riemann
Preface
My earlier book, Essential Relativity, aimed to provide a quick if thoughtful intro- duction to the subject at the level of advanced undergraduates and beginning graduate students, while ‘containing enough new material and simplifications of old arguments so as not to bore the expert teacher.’ But general relativity has by now robustly entered the mainstream of physics, in particular astrophysics, new discoveries in cosmology are routinely reported in the press, while ‘wormholes’ and time travel have made it into popular TV. Students thus want to know more than the bare minimum. The present book offers such an extension, in which the style, the general philosophy, and the mathematical level of sophistication have nevertheless remained the same. Any- one who knows the calculus up to partial differentiation, ordinary vectors to the point of differentiating them, and that most useful method of approximation, the binomial theorem, should be able to read this book. But instead of the earlier nine chapters there are now eighteen, and instead of 167 exercises, now there are more than 300; above all, tensors are introduced without apology and then thoroughly used. Einstein’s special and general relativity, the theories of flat and curved spacetime and of the physics therein, and relativistic cosmology, with its geometry and dynamics for the entire universe, not only seem necessary for a scientist’s balanced view of the world, but also offer some of the greatest intellectual thrills of modern physics. Perhaps the chief motivation in writing this book has been once more the desire to convey that thrill, as well as some of the insights that long preoccupation with a subject inevitably yields. It is true that many aspects of general relativity have still not been tested experimentally. Nevertheless enough have been tested to justify the view that all of relativity is by now well out of the tentative stage. That is also the reason why the introductory chapter contains an overview of all of relativity and cosmology, so that the student can appreciate from the very beginning the local character of special relativity and how it fits into the general scheme. The three main parts that follow deal extensively with special relativity, general relativity, and cosmology. In each I have tried to report on the most important crucial experiments and observations, both historical and modern, but stressing concepts rather than experimental detail. In fact, the emphasis throughout is on understanding the concepts and making the ideas come alive. But an equal value is put on developing the mathematical formalism rigorously, and on guiding the student to use both concepts and mathematics in conjunction with viii Preface the tricks of the trade to become an expert problem solver. A vital part in this process should be played by the exercises, which have been put together rather carefully, and which are mostly of the ‘thinking’ variety. Though their full solution often requires some ingenuity, they should at least be looked at, as a supplement to the text, for the extra information they contain. No book ever has enough diagrams. That is one of the luxuries that classroom teaching has over a book. So readers should constantly draw their own, especially since relativity is a very geometric subject in which the facility to think geometrically is a great asset. Readers should also constantly make up their own problems, however trivial: what would happen if ...? In an initially paradoxical subject like relativity, it is often the most skeptical student who is the most successful. Each of the three parts could well be cut short drastically so that the book might serve as a text for a one-semester course. To present it fully will take two semesters, probably with material to spare. But apart from its serving as an introductory text for a formal course, I also envisage the book as having some use for the general scientist who might wish to browse in it, and for the more advanced graduate student in search of greener pastures, as a change from the rocky pinnacles of more severe texts. At the end of the book there is an Appendix on curvature components for diagonal metrics (in a little more generality than the old ‘Dingle formulae’), which could be useful even to workers in the field who have not read the rest of the book. And finally a word of warning: in many sections, as is the custom in relativity, the units are chosen so as to make the speed of light unity, and later even to make Newton’s constant of gravitation unity, which must be borne in mind when comparing formulae; where the dimensions seem wrong, c’s or G’s are missing. I owe much to many modern authors (Sexl and Urbantke, Misner, Thorne and Wheeler, Ohanian and Ruffini, Woodhouse, etc.), though an exact assignment of debt would be difficult at this stage. I have also benefitted from the many searching questions of my students over the years, among whom I might perhaps single out James Gilson and Jack Denur. But the greatest debt I owe, as so often before, to my friend Jurgen¨ Ehlers—discussion partner, scientific conscience, font of knowledge without peer.
Dallas, Texas W.R. January 2001 Preface to the Second Edition
It has been most gratifying to see the favorable reception of the first edition of this book, to the extent that a second edition is already in order. In this second edition the last three chapters on cosmology, in particular, have been updated and revised. But there are additions, improvements, and some new exercises throughout. It is my hope that readers of the book will enjoy and give an equal welcome to this amended version of it.
Dallas, Texas W.R. January 2006 This page intentionally left blank Contents
Introduction 1
1 From absolute space and time to influenceable spacetime: an overview 3 1.1 Definition of relativity 3 1.2 Newton’s laws and inertial frames 4 1.3 The Galilean transformation 5 1.4 Newtonian relativity 6 1.5 Objections to absolute space; Mach’s principle 7 1.6 The ether 9 1.7 Michelson and Morley’s search for the ether 9 1.8 Lorentz’s ether theory 10 1.9 Origins of special relativity 12 1.10 Further arguments for Einstein’s two postulates 14 1.11 Cosmology and first doubts about inertial frames 15 1.12 Inertial and gravitational mass 16 1.13 Einstein’s equivalence principle 18 1.14 Preview of general relativity 20 1.15 Caveats on the equivalence principle 22 1.16 Gravitational frequency shift and light bending 24 Exercises 1 27
I Special Relativity 31
2 Foundations of special relativity; The Lorentz transformation 33 2.1 On the nature of physical theories 33 2.2 Basic features of special relativity 34 2.3 Relativistic problem solving 36 2.4 Relativity of simultaneity, time dilation and length contraction: a preview 38 2.5 The relativity principle and the homogeneity and isotropy of inertial frames 39 2.6 The coordinate lattice; Definitions of simultaneity 41 2.7 Derivation of the Lorentz transformation 43 xii Contents
2.8 Properties of the Lorentz transformation 47 2.9 Graphical representation of the Lorentz transformation 49 2.10 The relativistic speed limit 54 2.11 Which transformations are allowed by the relativity principle? 57 Exercises 2 58
3 Relativistic kinematics 61 3.1 Introduction 61 3.2 World-picture and world-map 61 3.3 Length contraction 62 3.4 Length contraction paradox 63 3.5 Time dilation; The twin paradox 64 3.6 Velocity transformation; Relative and mutual velocity 68 3.7 Acceleration transformation; Hyperbolic motion 70 3.8 Rigid motion and the uniformly accelerated rod 71 Exercises 3 73
4 Relativistic optics 77 4.1 Introduction 77 4.2 The drag effect 77 4.3 The Doppler effect 78 4.4 Aberration 81 4.5 The visual appearance of moving objects 82 Exercises 4 85
5 Spacetime and four-vectors 89 5.1 The discovery of Minkowski space 89 5.2 Three-dimensional Minkowski diagrams 90 5.3 Light cones and intervals 91 5.4 Three-vectors 94 5.5 Four-vectors 97 5.6 The geometry of four-vectors 101 5.7 Plane waves 103 Exercises 5 105
6 Relativistic particle mechanics 108 6.1 Domain of sufficient validity of Newtonian mechanics 108 6.2 The axioms of the new mechanics 109 6.3 The equivalence of mass and energy 111 6.4 Four-momentum identities 114 6.5 Relativistic billiards 115 6.6 The zero-momentum frame 117 6.7 Threshold energies 118 6.8 Light quanta and de Broglie waves 119 Contents xiii
6.9 The Compton effect 121 6.10 Four-force and three-force 123 Exercises 6 126
7 Four-tensors; Electromagnetism in vacuum 130 7.1 Tensors: Preliminary ideas and notations 130 7.2 Tensors: Definition and properties 132 7.3 Maxwell’s equations in tensor form 139 7.4 The four-potential 143 7.5 Transformation of e and b; The dual field 146 7.6 The field of a uniformly moving point charge 148 7.7 The field of an infinite straight current 150 7.8 The energy tensor of the electromagnetic field 151 7.9 From the mechanics of the field to the mechanics of material continua 154 Exercises 7 157 II General Relativity 163
8 Curved spaces and the basic ideas of general relativity 165 8.1 Curved surfaces 165 8.2 Curved spaces of higher dimensions 169 8.3 Riemannian spaces 172 8.4 A plan for general relativity 177 Exercises 8 180
9 Static and stationary spacetimes 183 9.1 The coordinate lattice 183 9.2 Synchronization of clocks 184 9.3 First standard form of the metric 186 9.4 Newtonian support for the geodesic law of motion 188 9.5 Symmetries and the geometric characterization of static and stationary spacetimes 191 9.6 Canonical metric and relativistic potentials 195 9.7 The uniformly rotating lattice in Minkowski space 198 Exercises 9 200
10 Geodesics, curvature tensor and vacuum field equations 203 10.1 Tensors for general relativity 203 10.2 Geodesics 204 10.3 Geodesic coordinates 208 10.4 Covariant and absolute differentiation 210 10.5 The Riemann curvature tensor 217 10.6 Einstein’s vacuum field equations 221 Exercises 10 224 xiv Contents
11 The Schwarzschild metric 228 11.1 Derivation of the metric 228 11.2 Properties of the metric 230 11.3 The geometry of the Schwarzschild lattice 231 11.4 Contributions of the spatial curvature to post-Newtonian effects 233 11.5 Coordinates and measurements 235 11.6 The gravitational frequency shift 236 11.7 Isotropic metric and Shapiro time delay 237 11.8 Particle orbits in Schwarzschild space 238 11.9 The precession of Mercury’s orbit 241 11.10 Photon orbits 245 11.11 Deflection of light by a spherical mass 248 11.12 Gravitational lenses 250 11.13 de Sitter precession via rotating coordinates 252 Exercises 11 254
12 Black holes and Kruskal space 258 12.1 Schwarzschild black holes 258 12.2 Potential energy; A general-relativistic ‘proof’ of E = mc2 263 12.3 The extendibility of Schwarzschild spacetime 265 12.4 The uniformly accelerated lattice 267 12.5 Kruskal space 272 12.6 Black-hole thermodynamics and related topics 279 Exercises 12 281
13 An exact plane gravitational wave 284 13.1 Introduction 284 13.2 The plane-wave metric 284 13.3 When wave meets dust 287 13.4 Inertial coordinates behind the wave 288 13.5 When wave meets light 290 13.6 The Penrose topology 291 13.7 Solving the field equation 293 Exercises 13 295
14 The full field equations; de Sitter space 296 14.1 The laws of physics in curved spacetime 296 14.2 At last, the full field equations 299 14.3 The cosmological constant 303 14.4 Modified Schwarzschild space 304 14.5 de Sitter space 306 14.6 Anti-de Sitter space 312 Exercises 14 314 Contents xv
15 Linearized general relativity 318 15.1 The basic equations 318 15.2 Gravitational waves; The TT gauge 323 15.3 Some physics of plane waves 325 15.4 Generation and detection of gravitational waves 330 15.5 The electromagnetic analogy in linearized GR 335 Exercises 15 341
III Cosmology 345
16 Cosmological spacetimes 347 16.1 The basic facts 347 16.2 Beginning to construct the model 358 16.3 Milne’s model 360 16.4 The Friedman–Robertson–Walker metric 363 16.5 Robertson and Walker’s theorem 368 Exercises 16 369
17 Light propagation in FRW universes 373 17.1 Representation of FRW universes by subuniverses 373 17.2 The cosmological frequency shift 374 17.3 Cosmological horizons 376 17.4 The apparent horizon 382 17.5 Observables 384 Exercises 17 388
18 Dynamics of FRW universes 391 18.1 Applying the field equations 391 18.2 What the field equations tell us 393 18.3 The Friedman models 397 18.4 Once again, comparison with observation 406 18.5 Inflation 411 18.6 The anthropic principle 415 Exercises 18 416
Appendix: Curvature tensor components for the diagonal metric 419
Index 423 This page intentionally left blank Introduction This page intentionally left blank 1 From absolute space and time to influenceable spacetime: an overview
1.1 Definition of relativity
At their core, Einstein’s relativity theories (both the special theory of 1905 and the general theory of 1915) are the modern physical theories of space and time, which have replaced Newton’s concepts of absolute space and absolute time by spacetime. We specifically call Einstein’s theories ‘physical’ because they claim to describe real structures in the real world and are open to experimental disproof. Since all (or, at least, all classical) physical processes play out on a background of space and time, the laws of physics must be compatible with the accepted theories of spaceandtime.Ifonechangesthebackground,onemustadaptthephysics.Thisprocess gave rise to ‘relativistic physics’, which from the outset made some startling predic- tions (like E = mc2) but which has nevertheless been amply confirmed by experiment. Originally, in physics, relativity meant the abolition of absolute space—a quest that had been recognized as desirable ever since Newton’s days. And this is indeed what Einstein’s two theories accomplished: special relativity (SR), the theory of flat spacetime, abolished absolute space in its Maxwellian role as the ‘ether’ that carried electromagnetic fields and, in particular, light waves, while general relativity (GR), the theory of curved spacetime, abolished absolute space also in its Newtonian role as the ubiquitous and uninfluenceable standard of rest or uniform motion. Surpris- ingly, and not by design but rather as an inevitable by-product, Einstein’s theory also abolished Newton’s concept of an absolute time. Since these ideas are fundamental, we devote the first chapter to a brief discussion centered on the three questions: What is absolute space? Why should it be abolished? How can it be abolished? A more modern and positive definition of relativity has evolved ex post facto from the actual relativity theories. According to this view, the relativity of any physical theory expresses itself in the group of transformations which leave the laws of the theory invariant and which therefore describe symmetries, for example of the space and time arenas of these theories. Thus, as we shall see, Newton’s mechanics pos- sesses the relativity of the so-called Galilean group, SR possesses the relativity of the Poincare´ (or ‘general’ Lorentz) group, GR possesses the relativity of the full group of smooth one-to-one space-time transformations, and the various cosmologies pos- sess the relativity of the various symmetries with which the large-scale universe is credited. Even a theory valid only in one absolute Euclidean space, provided that is 4 From absolute space and time to influenceable spacetime physically homogeneous and isotropic, would possess a relativity, namely the group of rotations and translations.
1.2 Newton’s laws and inertial frames
We recall Newton’s three laws of mechanics, of which the first (Galileo’s law of inertia) is really a special case of the second:
(i) Free particles move with constant vector-velocity (that is, with zero acceleration, or, in other words, with constant speed along straight lines). (ii) The vector-force on a particle equals the product of its mass into its vector- acceleration: f = ma. (iii) The forces of action and reaction are equal and opposite; for example, if a particle A exerts a force f on a particle B, then B exerts a force −f on A. (Newton’s absolute time is needed here: If the particles are at a distance and the forces vary, action today will not equal reaction tomorrow; they must be measured simultaneously, and simultaneity must be unambiguous.)
Physical laws are usually stated relative to some reference frame, which allows physical quantities like velocity, electric field, etc., to be defined. Preferred among reference frames are rigid frames, and preferred among these are the inertial frames. Newton’s laws apply in the latter. A classical rigid reference frame is an imagined extension of a rigid body. For example, the earth determines a rigid frame throughout all space, consisting of all those points which remain ‘rigidly’ at rest relative to the earth and to each other (like ‘geostationary’ satellites). We can associate an orthogonal Cartesian coordinate system with such a frame in many ways, by choosing three mutually orthogonal planes within it and measuring x,y,z as distances from these planes. Of course, this presupposes that the geometry in such a frame is Euclidean, which was taken for granted until 1915! Also, a time t must be defined throughout the frame, since this enters into many of the laws. In Newton’s theory there is no problem with that. Absolute time ‘ticks’ world-wide—its rate directly linked to Newton’s first law (free particles cover equal distances in equal times)—and any particular frame just picks up this ‘world-time’. Only the choice of units and the zero-setting remain free. Newton’s first law serves to single out inertial frames among rigid frames: a rigid frame is called inertial if free particles move without acceleration relative to it. And, as it turns out, Newton’s laws apply equally in all inertial frames. However, Newton postulated the existence of a quasi-substantial absolute space (AS) in which he thought the center of mass of the solar system was at rest, and which, to him, was the primary arena for his mechanics. That the laws were equally valid in all other reference frames moving uniformly relative to AS (the inertial frames) was to him a profoundly interesting theorem, but it was AS that bore, as it were, the responsibility for it all. The Galilean transformation 5
He called it the sensorium dei—God’s sensory organ—with which God ‘felt’ the world.
1.3 The Galilean transformation
Now consider any two rigid reference frames S and S in uniform relative motion with velocity v. Let identical units of length and time be used in both frames. And let their times t and t and their Cartesian coordinates x,y,z and x ,y ,z be adapted to their relative motion in the following way (cf. Fig. 1.1): The S origin moves with velocity v along the x-axis of S, the x -axis coincides with the x-axis, while the y- and y -axes remain parallel, as do the z- and z -axes; and all clocks are set to zero when the two origins meet. The coordinate systems S: {x,y,z,t} and S : {x ,y ,z ,t } are then said to be in standard configuration. Suppose an event (like the flashing of a light bulb, or the collision of two point- particles) has coordinates (x,y,z,t)relative to S and (x ,y ,z ,t ) relative to S . Then the classical (and ‘common sense’) relations between these two sets of coordinates are given by the standard Galilean transformation (GT):