The General Theory of Relativity - A Mathematical Approach

427 Pages·2021·13.272 MB·English

by  Farook Rahaman

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The General Theory of Relativity
Thegeneraltheoryofrelativity,Einstein’stheoryofgravitation,hasbeenincludedasacompulsory
subject in undergraduate and graduate courses in Physics and Applied Mathematics all over the
world.However,thephysics-firstapproachthatistakenbymanytextbooksisnotuniversallyused,as
theapproachoftendependsontheinstructors’orstudents’background.Conceivedfromthelecture
notesmadebytheauthoroverateachingcareerspanning18years,thisbookintroducesthegeneral
theoryofrelativityforadvancedstudentswithastrongmathematicalbackground.
The proposed book takes a ‘math-first approach’, for which the mathematical formalism
comes first and is then applied to physics. It presents a concise yet comprehensive and structured
understandingofthegeneraltheoryofrelativity.Thebookdiscussesthemathematicalfoundation
ofthegeneraltheoryofrelativityandfocusesheavilyontopicssuchastensorcalculus,geodesics,
Einsteinfieldequations,linearizedgravity,Liederivativesandtheirapplications,thecausalstructure
ofspacetime,rotatingblackholes,andbasicknowledgeofcosmologyandastrophysics.Allofthese
areexplainedthroughalargenumberofworkedexamplesandexercises.
FarookRahamanisaProfessorofMathematicsatJadavpurUniversity,Kolkata.Besideswriting
a book, The Special Theory of Relativity, he has published numerous research papers on galactic
dark matter, wormhole geometry, charged fluid model, topological defects in the early universe,
gravastars,blackholephysics,starmodeling,andthecosmologicalmodeloftheuniverse.
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The General Theory of
Relativity
AMathematicalApproach
FarookRahaman
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Names:Nandakumaran,A.K.,author.|Datti,P.S.,author.
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NUamniveesr:siRtyaPhraemss,a2n0,2F0.a|rIonocklu,daeustbhibolrio.graphicalreferencesandindex.
ITdietnltei:fieTrhs:eLCgeCnNer2a0l20th00e1o1r4y3o(pfrrienlta)t|iLvCitCyN:2a0m20a0t0h1e1m44a(teibcoaolka)p|pIrSoBaNc h/FarookRahaman.
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Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversityPress,2021.
Subjects:LCSH:Differentialequations,Partial–Textbooks.
Includesbibliographicalreferencesandindex.
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To
myparents
MajedaRahamanandLateObaidurRahaman
and
mysonandwife
MdRahilMirajandPakizahYasmin
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Contents
ListofFigures xiii
ListofTables xvii
Preface xix
Acknowledgments xxi
Chapter1 TensorCalculus—ABrief Overview 1
1.1 Introduction 1
1.2 TransformationofCoordinates 1
1.3 CovariantandContravariantVectorandTensor 2
1.4 OperationsonTensors 6
1.5 GeneralizedKroneckerDelta 9
1.6 TheLineElement 11
1.7 Levi-CivitaTensororAlternatingTensor 18
1.8 ChristoffelSymbols 20
1.9 AffineConnection 22
1.10 CovariantDerivative 24
1.11 CurvatureTensor 27
1.12 RicciTensor 29
1.13 RicciScalar 30
1.14 SpaceofConstantCurvature 32
1.15 TheAffineConnectioninRiemannianGeometry 36
1.16 GeodesicCoordinate 37
1.17 BianchiIdentity 38
1.18 EinsteinTensor 39
1.19 WeylTensor 41
Chapter2 Geodesics 45
2.1 GeodesicsEquation 45
2.2 DerivationofEuler–LagrangeEquation 46
2.3 GeodesicEquationinCurvedSpacetime 47
2.4 GeodesicDeviation 49
2.5 GeodesicsAreAutoParallel 49
2.6 RaychaudhuriEquation 50
vii
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viii Contents
Chapter3 Einstein FieldEquations 61
3.1 Introduction 61
3.2 ThreeTypesofMass 62
3.3 EinsteinTensor 62
3.4 SomeUsefulVariations 63
3.5 ActionIntegralfortheGravitationalField 63
3.6 Einstein’sEquationfromVariationalPrinciple 64
3.7 SomeModifiedTheoriesofGravity 76
Chapter4 LinearizedGravity 85
4.1 NewtonianGravity 85
4.2 Newtonian Limit of Einstein Field Equations or Weak Field Approximation
ofEinsteinEquations 88
4.3 PoissonEquationasanApproximationofEinsteinFieldEquations 90
4.4 GravitationalWave 92
Chapter5 Lie DerivativesandKilling’sEquation 95
5.1 Introduction 95
5.2 LieDerivativeofaScalar 96
5.3 LieDerivativeofContravariantVector 97
5.4 LieDerivativeofCovariantVector 97
5.5 LieDerivativeofCovariantandContravariantTensorsofOrderTwo 98
5.6 KillingEquation 101
5.7 StationaryandStaticSpacetimes 108
5.8 SphericallySymmetricSpacetime 109
5.9 CylindricallySymmetricSpacetime(AxiallySymmetry) 110
Chapter6 Spacetimes of Spherically Symmetric Distribution of
MatterandBlackHoles 115
6.1 SphericallySymmetricLineElement 115
6.2 SchwarzschildSolutionorExteriorSolution 117
6.3 VacuumSolutionorExteriorSolutionwithCosmologicalConstant 122
6.4 Birkhoff’sTheorem 123
6.5 SchwarzschildInteriorSolution 126
6.6 TheTolman–Oppenheimer–VolkoffEquation 127
6.7 TheStructureofNewtonianStar 129
6.8 IsotropicCoordinates 138
6.9 InteractionbetweenGravitationalandElectromagneticFields 146
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Contents ix
Chapter7 ParticleandPhotonOrbitsinthe SchwarzschildSpacetime 159
7.1 MotionofTestParticle 159
7.2 ExperimentalTestforGeneralRelativity 161
7.3 GravitationalRedshift 171
7.4 StableCircularOrbitsintheSchwarzschildSpacetime 173
Chapter8 CausalStructureofSpacetime 187
8.1 Introduction 187
8.2 Causality 187
8.3 CausalRelation 196
8.4 CausalFunction 210
Chapter9 Exact Solutions of Einstein Equations and Their Causal
Structures 219
9.1 MinkowskiSpacetime 219
9.2 deSitterSpacetime 225
9.3 Anti-deSitterSpace 230
9.4 Robertson–WalkerSpaces 233
9.5 PenroseDiagramsofRobertson–WalkerSpacetimefortheDustCase 235
9.6 SpatiallyHomogeneousCosmologicalModels 237
9.7 SchwarzschildSolutions 240
9.8 NullCurvesinSchwarzschildSpacetime 241
9.9 Time-likeGeodesicsinSchwarzschildSpacetime 242
9.10 TortoiseCoordinates 245
9.11 Eddington–FinkelsteinCoordinates 245
9.12 Kruskal–SzekeresCoordinates 248
9.13 Reissner–NordströmSolution 253
Chapter10 RotatingBlackHoles 261
10.1 NullTetrad 261
10.2 NullTetradofSomeBlackHoles 266
10.3 TheKerrSolution 271
10.4 TheKerrSolutionfromtheSchwarzschildSolution 272
10.5 TheKerr–NewmannSolutionfromtheReissner–NordströmSolution 274
10.6 TheHigherDimensionalRotatingBlackHoleSolution 276
10.7 DifferentFormsofKerrSolution 279
10.8 SomeElementaryPropertiesoftheKerrSolution 283
10.9 SingularitiesandHorizons 284
10.10 StaticLimitandErgosphere 286
10.11 ZeroAngularMomentumObserversintheKerrSpacetime 288
10.12 StationaryObserverintheKerrSpacetime 288
10.13 NullGeodesicsinKerrSpacetime 290