Dinverno

INTRODUCING EINSTEIN'S RELATIVITY\n\nRAY D'INVERNO There is little doubt that Einstein's theory of relativity captures\nthe imagination. Not only has it radically altered the way we\nview the universe, but the theory has a considerable number\nof surprises in store. This is especially so in the three main\ntopics of current interest that this book reaches, namely: black\nholes, gravitational waves, and cosmology.\n\nThe main aim of this textbook is to provide students with a\nsound mathematical introduction coupled to an understanding\nof the physical insights needed to explore the subject. Indeed,\nthe book follows Einstein in that it introduces the theory very\nmuch from a physical point of view. After introducing the\nspecial theory of relativity, the basic field equations of\ngravitation are derived and discussed carefully as a prelude\nto first solving them in simple cases and then exploring the\nthree main areas of application.\n\nEinstein's theory of relativity is undoubtedly one of the\ngreatest achievements of the human mind. Yet, in this book,\nit author makes it possible for students with a wide range\nof abilities to deal confidently with the subject. Based on the\nauthor's fifteen years experience of teaching this subject, this\nis mainly achieved by breaking down the main arguments\ninto simple logical steps. The book includes numerous\nillustrative diagrams and exercises of varying degrees of\ndifficulty, and as a result this book makes an excellent course\nfor any student coming to the subject for the first time. Albert Einstein (1879–1955) Introducing\nEinstein's Relativity\n\nRay d'Inverno\nFaculty of Mathematical Studies, University of Southampton\n\nCLARENDON PRESS - OXFORD Oxford University Press, Great Clarendon Street, Oxford OX2 6DP\nOxford New York\n\nAthens Auckland Bangkok Bogota Bombay Buenos Aires Calcutta\nCape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul\nKarachi Kuala Lumpur Madrid Melbourne Mexico City Mumbai\nNairobi Paris S\u00e3o Paulo Singapore Taipei Tokyo Toronto Warsaw\nand associated companies in\nBerlin Ibadan\n\nOxford is a trade mark of Oxford University Press\n\nPublished in the United States\nby Oxford University Press Inc., New York\n\n\u00a9 Ray d'Inverno, 1992\nReprinted 1993, 1995 (with corrections), 1996, 1998\n\nAll rights reserved. No part of this publication may be reproduced, stored in\na retrieval system, or transmitted, in any form or by any means, without the prior\npermission in writing of Oxford University Press. Within the UK, exceptions are\nallowed in respect of any fair dealing for the purpose of research or private study, or\ncriticism or review, as permitted under the Copyright, Designs and Patents Act,\n1988, or in the case of reprographic reproduction in accordance with the terms of\na licences issued by the Copyright Licensing Agency. Enquiries concerning\nreproduction outside those terms and in other countries should be sent to the Rights\nDepartment, Oxford University Press, at the address above.\n\nThis book is sold subject to the condition that it shall not, by way\nof trade or otherwise, be lent, re-sold, hired out, or otherwise circulated\nwithout the publisher's prior consent in any form of binding or cover\nother than that in which it is published and without a similar condition\nincluding this condition being imposed on the subsequent purchaser.\n\nA catalogue record for this book is\navailable from the British Library\n\nLibrary of Congress Cataloging in Publication Data\n\n\\u00d8 Inverno, R. A.\nIntroducing Einstein's relativity/R. A. d'Inverno.\nIncludes bibliographical references and index.\n1. Relativity (Physics) 2. Black holes (Astronomy)\n3. Gravitation. 4. Cosmology. 5. Calculus of tensors. I. Title.\nQC173.51.158 1992 530, 71--dc20 91-24894\n\nISBN 0 19 859637 7 (Pbk)\nISBN 0 19 859868 3 (Pbk)\n\nPrinted in Malta by Interprint Limited Contents\n\nOverview 1\n1. The organization of the book 3\n1.1 Notes for the student 3\n1.2 Acknowledgements 4\n1.3 A brief survey of relativity theory 6\n1.4 Notes for the teacher 8\n1.5 A final note for the less able student 10\nExercises 11\n\nPart A. Special Relativity 13\n2. The k-calculus 15\n2.1 Model building 15\n2.2 Historical background 16\n2.3 Newtonian framework 16\n2.4 Galilean transformations 17\n2.5 The principle of special relativity 18\n2.6 The constancy of the velocity of light 19\n2.7 The k-factor 20\n2.8 Relative speed of two inertial observers 21\n2.9 Composition law for velocities 22\n2.10 The relativity of simultaneity 23\n2.11 The clock paradox 24\n2.12 The Lorentz transformations 25\n2.13 The four-dimensional world view 26\nExercises 28\n\n3. The key attributes of special relativity 29\n3.1 Standard derivation of the Lorentz transformations 29\n3.2 Mathematical properties of Lorentz transformations 31\n3.3 Length contraction 32\n3.4 Time dilation 33\n3.5 Transformation of velocities 34\n3.6 Relationship between space-time diagrams of inertial observers 35\n3.7 Acceleration in special relativity 36\n3.8 Uniform acceleration 37\n3.9 The twin paradox 38\n3.10 The Doppler effect 39\nExercises 40\n\n4. The elements of relativistic mechanics 42\n4.1 Newtonian theory 42\n4.2 Isolated systems of particles in Newtonian mechanics 44\n4.3 Relativistic mass 45\n4.4 Relativistic energy 47\n4.5 Photons 49\nExercises 51\n\nPart B. The Formalism of Tensors 53\n5. Tensor algebra 55\n5.1 Introduction 55\n5.2 Manifolds and coordinates 55\n5.3 Curves and surfaces 57\n5.4 Transformation of coordinates 58\n5.5 Contravariant tensors 60\n5.6 Covariant and mixed tensors 61\n5.7 Tensor fields 62\n5.8 Elementary operations with tensors 63\n5.9 Index-free interpretation of contravariant vector fields 64\nExercises 67 6. Tensor calculus 68\n6.1 Partial derivative of a tensor 68\n6.2 The Lie derivative 69\n6.3 The affine connection and covariant differentiation 72\n6.4 Affine geodesics 74\n6.5 The Riemann tensor 77\n6.6 Geodesic coordinates 77\n6.7 Affine flatness 78\n6.8 The metric 81\n6.9 Metric geodesics 82\n6.10 The metric connection 84\n6.11 Metric flatness 86\n6.12 The curvature tensor 86\n6.13 The Weyl tensor 87\nExercises 89\n\nPart C. General Relativity 105\n8. Special relativity revisited 107\n8.1 Minkowski space-time 107\n8.2 The null cone 108\n8.3 The Lorentz group 109\n8.4 Proper time 111\n8.5 An axiomatic formulation of special relativity 112\n8.6 A variational principle approach to classical mechanics 114\n8.7 A variational principle approach to relativistic mechanics 116\n\n 12. The energy-momentum tensor 155\n12.1 Preview 155\n12.2 Incoherent matter 155\n12.3 Perfect fluid 157\n12.4 Maxwell's equations 158\n12.5 Potential formulation of Maxwell's equations 160\n12.6 The Maxwell energy-momentum tensor 162\n12.7 Other energy-momentum tensors 163\n12.8 The dominant energy condition 164\n12.9 The Newtonian limit 165\n12.10 The coupling constant 167\nExercises 168\n\n13. The structure of the field equations 169\n13.1 Interpretation of the field equations 169\n13.2 Determinacy, non-linearity, and differentiability 170\n13.3 The cosmological term 171\n13.4 The conservation equations 173\n13.5 The Cauchy problem 174\n13.6 The hole problem 177\n13.7 The equivalence problem 178\nExercises 179\n\n14. The Schwarzschild solution 180\n14.1 Stationary solutions 180\n14.2 Hypersurface-orthogonal vector fields 181\n14.3 Static solutions 183\n14.4 Spherically symmetric solutions 184\n14.5 The Schwarzschild solution 186\n14.6 Properties of the Schwarzschild solution 188\n14.7 Isotropic coordinates 189\nExercises 190\n\n15. Experimental tests of general relativity 192\n15.1 Introduction 192\n15.2 Classical Kepler motion 192\n 18. Charged black holes 239\n18.1 The field of a charged mass point 239\n18.2 Intrinsic and coordinate singularities 241\n18.3 Space-time diagram of the Reissner–Nordström solution 242\n18.4 Neutral particles in Reissner–Nordström space-time 243\n18.5 Penrose diagrams of the maximal analytic extensions 244\nExercises 247\n\n19. Rotating black holes 248\n19.1 Null tetrads 248\n19.2 The Kerr solution from a complex transformation 250\n19.3 The three main forms of the Kerr solution 251\n19.4 Basic properties of the Kerr solution 252\n19.5 Singularities and horizons 254\n19.6 The principal null congruences 256\n19.7 Eddington–Finkelstein coordinates 258\n19.8 The stationary limit 259\n19.9 Maximal extension for the case a² < m² 260\n19.10 Maximal extension for the case a² > m² 261\n19.11 Rotating black holes 262\n19.12 The singularity theorems 265\n19.13 The Hawking effect 266\nExercises 268\n\nPart E. Gravitational Waves 269\n20. Plane gravitational waves 271\n20.1 The linearized field equations 271\n20.2 Gauge transformations 272\n20.3 Linearized plane gravitational waves 274\n20.4 Polarization states 278\n20.5 Exact plane gravitational waves 280\n20.6 Impulsive plane gravitational waves 282\n20.7 Colliding impulsive plane gravitational waves 283\n20.8 Colliding gravitational waves 284\n 23.4 The de Sitter model 337\n23.5 The first models 338\n23.6 The time-scale problem 339\n23.7 Later models 339\n23.8 The missing matter problem 341\n23.9 The standard models 342\n23.10 Early epochs of the universe 343\n23.11 Cosmological coincidences 343\n23.12 The steady-state theory 344\n23.13 The event horizon of the de Sitter universe 348\n23.14 Particle and event horizons 349\n23.15 Conformal structure of Robertson-Walker space-times 351\n\n23.16 Conformal structure of de Sitter space-time 352\n23.17 Inflation 354\n23.18 The anthropic principle 356\n23.19 Conclusion 358\nExercises 359\n\nAnswers to exercises 360\nFurther reading 370\nSelected bibliography 372\nIndex 375