Relativity, Gravitation and Cosmology: A Basic Introduction

450 Pages·2010·2.948 MB·English

by  Ta-Pei Cheng

OXFORDMASTERSERIESINPARTICLEPHYSICS,ASTROPHYSICS,ANDCOSMOLOGY
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The Oxford Master Series is designed for final year undergraduate and beginning graduate students in physics and
related disciplines. It has been driven by a perceived gap in the literature today. While basic undergraduate physics
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CONDENSEDMATTERPHYSICS
1. M.T.Dove:Structureanddynamics:anatomicviewofmaterials
2. J.Singleton:Bandtheoryandelectronicpropertiesofsolids
3. A.M.Fox:Opticalpropertiesofsolids,secondedition
4. S.J.Blundell:Magnetismincondensedmatter
5. J.F.Annett:Superconductivity,superfluids,andcondensates
6. R.A.L.Jones:Softcondensedmatter
17. S.Tautz:Surfacesofcondensedmatter
18. H.Bruus:Theoreticalmicrofluidics
19. C.L.Dennis,J.F.Gregg:Theartofspintronics:anintroduction
ATOMIC,OPTICAL,ANDLASERPHYSICS
7. C.J.Foot:Atomicphysics
8. G.A.Brooker:Modernclassicaloptics
9. S.M.Hooker,C.E.Webb:Laserphysics
15. A.M.Fox:Quantumoptics:anintroduction
16. S.M.Barnett:Quantuminformation
PARTICLEPHYSICS,ASTROPHYSICS,ANDCOSMOLOGY
10. D.H.Perkins:Particleastrophysics,secondedition
11. Ta-PeiCheng:Relativity,gravitationandcosmology,secondedition
STATISTICAL,COMPUTATIONAL,ANDTHEORETICALPHYSICS
12. M.Maggiore:Amodernintroductiontoquantumfieldtheory
13. W.Krauth:Statisticalmechanics:algorithmsandcomputations
14. J.P.Sethna:Statisticalmechanics:entropy,orderparameters,andcomplexity
Relativity, Gravitation and
Cosmology
A Basic Introduction
Second Edition
TA-PEI CHENG
UniversityofMissouri–St.Louis
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ISBN 978–0–19–957363–9(Hbk.)
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1 3 5 7 9 10 8 6 4 2
Contents
Preface xi
PARTI PRELIMINARIES
1 Introductionandoverview 3
1.1 Relativityasacoordinatesymmetry 5
1.1.1 FromNewtonianrelativitytoether 5
1.1.2 Einsteinianrelativity 6
1.1.3 Coordinatesymmetrytransformations 7
1.1.4 Newkinematicsanddynamics 8
1.2 GRasagravitationalfieldtheory 8
1.2.1 Einstein’smotivationsforthegeneraltheory 8
1.2.2 Geometryasgravity 10
1.2.3 Mathematicallanguageofrelativity 11
1.2.4 ObservationalevidenceforGR 12
1.2.5 GRastheframeworkforcosmology 14
Reviewquestions 14
2 Specialrelativity:Thebasics 15
2.1 Coordinatesymmetries 15
2.1.1 NewtonianphysicsandGalileansymmetry 18
2.1.2 ElectrodynamicsandLorentzsymmetry 19
2.1.3 Velocityadditionruleamended 21
2.2 Thenewkinematics 22
2.2.1 Thebasicpostulatesofspecialrelativity 22
2.2.2 Relativityofequilocalityandsimultaneity 23
2.2.3 Timedilationandlengthcontraction 26
2.3 Lorentztransformation 28
2.3.1 Physicalmeaningofvarioustransformation
terms 30
2.3.2 Therelativisticinvariantinterval 31
2.3.3 Relativityistrulyrelative 32
2.3.4 TwoparadoxesasillustrativeSRexamples 32
Reviewquestions 35
Problems 36
vi Contents
PARTII RELATIVITY:METRICDESCRIPTIONOFSPACETIME
3 Specialrelativity:Thegeometricformulation 41
3.1 Minkowskispacetime 42
3.1.1 Basisvectors,themetricandscalarproduct 43
3.1.2 TheMinkowskimetricandLorentztransformation 45
3.2 Four-vectorsforparticledynamics 47
3.2.1 Thevelocity4-vector 47
3.2.2 Relativisticenergyandmomentum 48
3.3 Thespacetimediagram 51
3.3.1 Basicfeaturesandinvariantregions 52
3.3.2 Lorentztransformationinthespacetime
diagram 53
3.4 ThegeometricformulationofSR:Asummary 56
Reviewquestions 57
Problems 58
4 Theprincipleofequivalence 61
4.1 Newtoniangravitationpotential—areview 61
4.2 EPintroduced 63
4.2.1 Inertialmassvs.gravitationalmass 63
4.2.2 EPanditssignificance 66
4.3 ImplicationsofthestrongEP 67
4.3.1 Gravitationalredshiftandtimedilation 69
4.3.2 Lightraydeflectioncalculated 74
4.3.3 Energyconsiderationsofagravitatinglightpulse 77
4.3.4 Einstein’sinferenceofacurvedspacetime 78
Reviewquestions 79
Problems 79
5 Metricdescriptionofacurvedspace 81
5.1 Gaussiancoordinates 82
5.2 Metrictensor 83
5.2.1 Geodesicastheshortestcurve 86
5.2.2 LocalEuclideancoordinates 88
5.3 Curvature 90
5.3.1 Gaussiancurvature 91
5.3.2 CurvaturemeasuresthedeviationfromEuclidean
relations 92
5.3.3 Spaceswithconstantcurvature 94
Reviewquestions 98
Problems 98
6 GRasageometrictheoryofgravity–I 100
6.1 Geometryasgravity 100
6.1.1 EPphysicsandawarpedspacetime 102
6.1.2 Curvedspacetimeasagravitationalfield 103
Contents vii
6.2 GeodesicequationasGRequationofmotion 105
6.2.1 Thegeodesicequationrecalled 105
6.2.2 TheNewtonianlimit 107
6.3 Thecurvatureofspacetime 109
6.3.1 Tidalforceasthecurvatureofspacetime 110
6.3.2 TheGRfieldequationdescribed 113
Reviewquestions 115
Problems 116
7 Sphericallysymmetricspacetime–GRtests 117
7.1 DescriptionofSchwarzschildspacetime 118
7.1.1 Propertiesofasphericallysymmetricmetrictensor 118
7.1.2 TheSchwarzschildgeometryandtheembedding
diagram 121
7.2 Gravitationallensing 124
7.2.1 Lightraydeflection:GRvs.EP 124
7.2.2 Thelensequation 125
7.3 GeodesicsinSchwarzschildspacetime 129
7.3.1 PrecessionofMercury’sperihelion 130
7.3.2 TheShapirotimedelayofalightsignal 135
Reviewquestions 138
Problems 139
8 Blackholes 141
8.1 Nonrotatingblackholes 142
8.1.1 Timemeasurementsaroundablackhole 143
8.1.2 Eddington–Finkelsteincoordinates:Blackholes
andwhiteholes 145
8.1.3 Kruskalcoordinatesandthewormhole 151
8.2 Orbitsandaccretiondisksaroundablackhole 155
8.2.1 EffectivepotentialoftheSchwarzschildspacetime 156
8.2.2 Thebindingenergyofaparticlearoundablackhole 157
8.3 Physicalrealityofblackholes 158
8.3.1 Thelongroadtotheacceptanceoftheblackhole’s
reality 158
8.3.2 Observationalevidenceofblackholes 159
8.4 AppendixA:Rotatingsourceofgravity 161
8.4.1 Propertiesofanaxiallysymmetricmetrictensor 161
8.4.2 KerrgeometryandthePenroseprocess 164
8.4.3 BeyondtheSchwarzschildandKerrblackholes 170
8.5 AppendixB:Blackholesandquantumphysics 171
8.5.1 ThePlanckscale 171
8.5.2 Hawkingradiation 172
8.5.3 Blackholethermodynamics 173
8.5.4 Blackholesandquantumgravity 174
Reviewquestions 175
Problems 176
viii Contents
PARTIII COSMOLOGY
9 Thehomogeneousandisotropicuniverse 181
9.1 Thecosmosobserved 182
9.1.1 Matterdistributiononthecosmicdistancescale 182
9.1.2 Cosmologicalredshift:Hubble’slaw 184
9.1.3 Ageoftheuniverse 186
9.2 Massdensityoftheuniverse 188
9.2.1 Luminousmatterandthebaryonicdensity 189
9.2.2 Darkmatterandthetotalmassdensity 190
9.3 Thecosmologicalprinciple 194
9.4 TheRobertson–Walkerspacetime 195
9.4.1 Themetricinthecomovingcoordinatesystem 195
9.4.2 DistancesintheRWgeometry 197
Reviewquestions 202
Problems 203
10 Theexpandinguniverseandthermalrelics 205
10.1 Friedmannequations 206
10.1.1 TheGRfieldequationsforcosmology 206
10.1.2 Thequasi-Newtonianinterpretation 208
10.2 Timeevolutionofmodeluniverses 212
10.3 Bigbangcosmology 215
10.3.1 Scale-dependenceofradiation’stemperature 215
10.3.2 Differentthermalequilibriumstages 217
10.4 Primordialnucleosynthesis 220
10.5 PhotondecouplingandtheCMB 223
10.5.1 Theuniversebecametransparenttophotons 224
10.5.2 ThediscoveryofCMBradiation 225
10.5.3 Photons,neutrinosandtheradiation–matter
equalitytime 226
10.5.4 CMBtemperaturefluctuation 230
Reviewquestions 234
Problems 235
11 Inflationandtheacceleratinguniverse 237
11.1 Thecosmologicalconstant 238
11.1.1 Vacuumenergyassourceofgravitational
repulsion 240
11.1.2 Einstein’sstaticuniverse 241
11.2 Theinflationaryepoch 243
11.2.1 InitialconditionsfortheFLRWcosmology 244
11.2.2 Theinflationscenario 246
11.2.3 Inflationandtheconditionsitleftbehind 248
11.3 CMBanisotropyandevidenceforaflatuniverse 251
11.3.1 Threeregionsoftheangularpowerspectrum 251
11.3.2 Theprimarypeakandspatialgeometryof
theuniverse 254
Contents ix
11.4 Theacceleratinguniverseinthepresentepoch 256
11.4.1 Distantsupernovaeandthe1998discovery 257
11.4.2 Transitionfromdecelerationtoacceleration 261
11.4.3 Darkenergy:Furtherevidenceandthemystery
ofitsorigin 264
11.5 Theconcordantpicture 265
11.6 AppendixC:Falsevacuumandhiddensymmetry 268
11.7 AppendixD:Quantumvacuumenergyasthe
cosmologicalconstant 271
Reviewquestions 274
Problems 274
PARTIV RELATIVITY:FULLTENSORFORMULATION
12 Tensorsinspecialrelativity 279
12.1 Generalcoordinatesystems 280
12.1.1 Contravariantandcovariantcomponents 281
12.1.2 Coordinatetransformations 282
12.1.3 PositionanddeloperatorsinMinkowskispacetime 284
12.2 Manifestlycovariantformalismforelectromagnetism 286
12.2.1 Theelectromagneticfieldtensor 286
12.2.2 Electricchargeconservation 290
12.3 Energy–momentumtensors 291
Reviewquestions 296
Problems 296
13 Tensorsingeneralrelativity 298
13.1 Derivativesinacurvedspace 299
13.1.1 Generalcoordinatetransformations 299
13.1.2 Covariantdifferentiation 302
13.1.3 Christoffelsymbolsandthemetrictensor 304
13.2 Paralleltransport 307
13.2.1 Componentchangesunderparalleltransport 307
13.2.2 Thegeodesicasthestraightestpossiblecurve 308
13.3 Riemanniancurvaturetensor 309
13.3.1 Thecurvaturetensorinann-dimensionalspace 311
13.3.2 Symmetriesandcontractionsofthecurvaturetensor 313
Reviewquestions 315
Problems 316
14 GRasageometrictheoryofgravity–II 318
14.1 Theprincipleofgeneralcovariance 319
14.1.1 Theminimalsubstitutionrule 319
14.1.2 GeodesicequationfromSRequationofmotion 320
14.2 Einsteinfieldequation 321
14.2.1 Findingtherelativisticgravitationalfieldequation 321
14.2.2 NewtonianlimitoftheEinsteinequation 323