This page intentionally left blank GENERAL RELATIVITY Startingwiththeideaofaneventandfinishingwithadescriptionof thestandardbig-bangmodeloftheUniverse,thistextbookprovides a clear, concise, and up-to-date introduction to the theory of gen- eral relativity, suitable for final-year undergraduate mathematics or physicsstudents.Throughout,theemphasisisonthegeometricstruc- ture of spacetime, rather than the traditional coordinate-dependent approach.Thisallowsthetheorytobepareddownandpresentedin itssimplestandmostelegantform.Topicscoveredincludeflatspace- time(specialrelativity),Maxwellfields,theenergy–momentumten- sor,spacetimecurvatureandgravity,SchwarzschildandKerrspace- times,blackholesandsingularities,andcosmology. In developing the theory, all physical assumptions are clearly spelledout,andthenecessarymathematicsisdevelopedalongwith the physics. Exercises are provided at the end of each chapter and keyideasinthetextareillustratedwithworkedexamples.Solutions and hints to selected problems are also provided at the end of the book. This textbook will enable the student to develop a sound under- standing of the theory of general relativity and all the necessary mathematicalmachinery. Dr.LudvigsenreceivedhisfirstPh.D.fromNewcastleUniversityand his second from the University of Pittsburgh. His research at the University of Botswana, Lesotho, and Swaziland led to an Andrew Mellon Fellowship in Pittsburgh, where he worked with the re- nownedrelativistTedNewmanonproblemsconnectedwithH-space and nonlinear gravitons. Dr. Ludvigsen is currently serving as both docentandlecturerattheUniversityofLinko¨pinginSweden. GENERAL RELATIVITY A GEOMETRIC APPROACH Malcolm Ludvigsen UniversityofLinko¨ping           The Pitt Building, Trumpington Street, Cambridge, United Kingdom    The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org ©Cambridge University Press 2004 First published in printed format 1999 ISBN 0-511-04006-7 eBook (netLibrary) ISBN 0-521-63019-3 hardback ISBN 0-521-63976-X paperback To Libby, John, and Elizabeth Contents Preface page xi PART ONE: THE CONCEPT OF SPACETIME 1 1 Introduction 3 EXERCISES, 11 2 Events and Spacetime 12 2.1 Events, 12 2.2 Inertial Particles, 13 2.3 Light and Null Cones, 15 EXERCISES, 17 PART TWO: FLAT SPACETIME AND SPECIAL RELATIVITY 19 3 Flat Spacetime 21 3.1 Distance, Time, and Angle, 21 3.2 Speed and the Doppler Effect, 23 EXERCISES, 26 4 The Geometry of Flat Spacetime 27 4.1 Spacetime Vectors, 27 4.2 The Spacetime Metric, 28 4.3 Volume and Particle Density, 35 EXERCISES, 38 5 Energy 40 5.1 Energy and Four-Momentum, 41 5.2 The Energy–Momentum Tensor, 43 5.3 General States of Matter, 44 5.4 Perfect Fluids, 47 5.5 Acceleration and the Maxwell Tensor, 48 EXERCISES, 50 6 Tensors 51 6.1 Tensors at a Point, 51 6.2 The Abstract Index Notation, 56 EXERCISES, 59 7 Tensor Fields 61 7.1 Congruences and Derivations, 62 vii viii CONTENTS 7.2 Lie Derivatives, 64 EXERCISES, 67 8 Field Equations 69 8.1 Conservation Laws, 69 8.2 Maxwell’s Equations, 70 8.3 Charge, Mass, and Angular Momentum, 74 EXERCISES, 78 PART THREE: CURVED SPACETIME AND GRAVITY 79 9 Curved Spacetime 81 9.1 Spacetime as a Manifold, 81 9.2 The Spacetime Metric, 85 9.3 The Covariant Derivative, 86 9.4 The Curvature Tensor, 89 9.5 Constant Curvature, 93 EXERCISES, 95 10 Curvature and Gravity 96 10.1 Geodesics, 96 10.2 Einstein’s Field Equation, 99 10.3 Gravity as an Attractive Force, 103 EXERCISES, 105 11 Null Congruences 106 11.1 Surface-Forming Null Congruences, 106 11.2 Twisting Null Congruences, 109 EXERCISES, 113 12 Asymptotic Flatness and Symmetries 115 12.1 Asymptotically Flat Spacetimes, 115 12.2 Killing Fields and Stationary Spacetimes, 122 12.3 Kerr Spacetime, 126 12.4 Energy and Intrinsic Angular Momentum, 131 EXERCISES, 133 13 Schwarzschild Geometries and Spacetimes 134 13.1 Schwarzschild Geometries, 135 13.2 Geodesics in a Schwarzschild Spacetime, 140 13.3 Three Classical Tests of General Relativity, 143 13.4 Schwarzschild Spacetimes, 146 EXERCISES, 150 14 Black Holes and Singularities 152 14.1 Spherical Gravitational Collapse, 152 14.2 Singularities, 155 14.3 Black Holes and Horizons, 158 14.4 Stationary Black Holes and Kerr Spacetime, 160 14.5 The Ergosphere and Energy Extraction, 167