This page intentionally left blank A First Course in General Relativity Second Edition Clarity,readability,andrigorcombineinthesecondeditionofthiswidelyusedtextbook to provide the first step into general relativity for undergraduate students with a minimal backgroundinmathematics. Topics within relativity that fascinate astrophysical researchers and students alike are coveredwithSchutz’scharacteristiceaseandauthority–fromblackholestogravitational lenses, from pulsars to the study of the Universe as a whole. This edition now contains recent discoveries by astronomers that require general relativity for their explanation; a revised chapter on relativistic stars, including new information on pulsars; an entirely rewritten chapter on cosmology; and an extended, comprehensive treatment of modern gravitationalwavedetectorsandexpectedsources. Over300exercises,manynewtothisedition,givestudentstheconfidencetoworkwith generalrelativityandthenecessarymathematics,whilsttheinformalwritingstylemakes thesubjectmattereasilyaccessible.Passwordprotectedsolutionsforinstructorsareavail- ableatwww.cambridge.org/Schutz. BernardSchutzisDirectoroftheMaxPlanckInstituteforGravitationalPhysics,aProfes- soratCardiffUniversity,UK,andanHonoraryProfessorattheUniversityofPotsdamand the University of Hannover, Germany. Heis alsoa Principal Investigator of theGEO600 detectorprojectandamemberoftheExecutiveCommitteeoftheLIGOScientificCollab- oration.ProfessorSchutzhasbeenawardedtheAmaldiGoldMedaloftheItalianSociety forGravitation. A First Course in General Relativity Second Edition Bernard F. Schutz MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute) and CardiffUniversity CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521887052 © B. Schutz 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13 978-0-511-53995-4 eBook (EBL) ISBN-13 978-0-521-88705-2 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. ToSiân Contents Prefacetothesecondedition pagexi Prefacetothefirstedition xiii 1 Specialrelativity 1 1.1 Fundamentalprinciplesofspecialrelativity(SR)theory 1 1.2 DefinitionofaninertialobserverinSR 3 1.3 Newunits 4 1.4 Spacetimediagrams 5 1.5 Constructionofthecoordinatesusedbyanotherobserver 6 1.6 Invarianceoftheinterval 9 1.7 Invarianthyperbolae 14 1.8 Particularlyimportantresults 17 1.9 TheLorentztransformation 21 1.10 Thevelocity-compositionlaw 22 1.11 Paradoxesandphysicalintuition 23 1.12 Furtherreading 24 1.13 Appendix:Thetwin‘paradox’dissected 25 1.14 Exercises 28 2 Vectoranalysisinspecialrelativity 33 2.1 Definitionofavector 33 2.2 Vectoralgebra 36 2.3 Thefour-velocity 41 2.4 Thefour-momentum 42 2.5 Scalarproduct 44 2.6 Applications 46 2.7 Photons 49 2.8 Furtherreading 50 2.9 Exercises 50 3 Tensoranalysisinspecialrelativity 56 3.1 Themetrictensor 56 3.2 Definitionoftensors 56 (cid:2) (cid:3) 3.3 The 0 tensors:one-forms 58 (cid:2)1(cid:3) 3.4 The 0 tensors 66 2 viii Contents (cid:2) 3.5 Metricas(cid:2)a(cid:3)mappingofvectorsintoone-forms 68 3.6 Finally: M tensors 72 N 3.7 Index‘raising’and‘lowering’ 74 3.8 Differentiationoftensors 76 3.9 Furtherreading 77 3.10 Exercises 77 4 Perfectfluidsinspecialrelativity 84 4.1 Fluids 84 (cid:3) 4.2 Dust:thenumber–fluxvectorN 85 4.3 One-formsandsurfaces 88 4.4 Dustagain:thestress–energytensor 91 4.5 Generalfluids 93 4.6 Perfectfluids 100 4.7 Importanceforgeneralrelativity 104 4.8 Gauss’law 105 4.9 Furtherreading 106 4.10 Exercises 107 5 Prefacetocurvature 111 5.1 Ontherelationofgravitationtocurvature 111 5.2 Tensoralgebrainpolarcoordinates 118 5.3 Tensorcalculusinpolarcoordinates 125 5.4 Christoffelsymbolsandthemetric 131 5.5 Noncoordinatebases 135 5.6 Lookingahead 138 5.7 Furtherreading 139 5.8 Exercises 139 6 Curvedmanifolds 142 6.1 Differentiablemanifoldsandtensors 142 6.2 Riemannianmanifolds 144 6.3 Covariantdifferentiation 150 6.4 Parallel-transport,geodesics,andcurvature 153 6.5 Thecurvaturetensor 157 6.6 Bianchiidentities:RicciandEinsteintensors 163 6.7 Curvatureinperspective 165 6.8 Furtherreading 166 6.9 Exercises 166 7 Physicsinacurvedspacetime 171 7.1 Thetransitionfromdifferentialgeometrytogravity 171 7.2 Physicsinslightlycurvedspacetimes 175 7.3 Curvedintuition 177