Table Of ContentUNITEXT for Physics
Philippe Jetzer
Applications
of General
Relativity
With Problems
UNITEXT for Physics
SeriesEditors
MicheleCini,UniversityofRomeTorVergata,Roma,Italy
AttilioFerrari,UniversityofTurin,Turin,Italy
StefanoForte,UniversityofMilan,Milan,Italy
GuidoMontagna,UniversityofPavia,Pavia,Italy
OresteNicrosini,UniversityofPavia,Pavia,Italy
LucaPeliti,UniversityofNapoli,Naples,Italy
AlbertoRotondi,Pavia,Italy
PaoloBiscari,PolitecnicodiMilano,Milan,Italy
NicolaManini,UniversityofMilan,Milan,Italy
MortenHjorth-Jensen,UniversityofOslo,Oslo,Norway
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Philippe Jetzer
Applications of General
Relativity
With Problems
PhilippeJetzer
DepartmentofPhysics
UniversityofZurich
Zürich,Switzerland
ISSN2198-7882 ISSN2198-7890 (electronic)
UNITEXTforPhysics
ISBN978-3-030-95717-9 ISBN978-3-030-95718-6 (eBook)
https://doi.org/10.1007/978-3-030-95718-6
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ForFiorenza
Preface
Theaimofthisbookistopresentinacomprehensivewayseveraladvancedtopics
related to general relativity. In Chap. 1, I review the basic knowledge on general
relativity needed for the understanding of the next topics. The following chapter
is dedicated to some applications of general relativity, among which it is useful
to mention gravitational lensing, relativistic stellar structure, time delay as well as
geodeticprecession.Following,Idiscussindetailgravitationalwavesandthepost-
Newtonian approximation. Chapter 4 is devoted to several topics related to black
holes, including the treatment of spinors in curved spacetime and the four laws of
blackholedynamics.Finally,thelastchaptercoverstheequivalenceprincipleand
ways to test it. The selection of the topics included in this book has been made
according to ongoing important research activities, both theoretically and experi-
mentally,inordertoprovideadvancedundergraduateandgraduatestudentswitha
modernpedagogicaltextdesignedtobroadentheknowledgeacquiredduringacore
courseongeneralrelativity.Idevelopedthisbookfromaseriesoflecturesgivenona
yearlybasiswiththeintentionofofferingagentleintroductiontothesubject,which
helpsreadingthemorespecializedliteratureandthatcanbeusedasafirstreading
togetquicklyintothefieldwhenstartingresearch.Thebookincludesproblemsto
practice understanding of key concepts as well as a list of selected references to
deepenthematerial.
Zürich,Switzerland PhilippeJetzer
December2021
vii
Acknowledgments
ThisbookhasgrownupfromlecturesIgaveinthelastyearsforstudentsofboththe
UniversityofZürichandETHasacontinuationofmylectureongeneralrelativity.
Thenotes,originallyintendedasinternallecturenotes,havebeentypedbyseveral
studentsattendingthecoursesaswellasbysomeofmyPh.D.students.Ithankthem
allfortheirinvaluablecontributionandthenumeroussuggestions,improvementsand
corrections,inalphabeticorder:RaymondAngélil,TobiasBaldauf,SimoneBavera,
YannickBoetzel,AdrianBoitier,FelixHähl,D.MoosbruggerandespeciallyMichael
Ebersold,whotookcareoftheproblems.Iwishtothankseveralcollaboratorsinthe
teamofSpringer,whocontactedandencouragedmetowritethisbook,inparticular
Lisa Scalone. After some initial reluctance in doing it, they convinced and helped
meallalongthedifferenteditorialprocesses.Withouttheirpatiencethebookwould
nothavebeenwritten.
ix
Contents
1 ElementsofGeneralRelativity .................................. 1
1.1 Einstein’sFieldEquations ................................... 1
1.1.1 TheCosmologicalConstant ........................... 5
1.2 StaticIsotropicMetric ...................................... 6
1.2.1 FormoftheMetric ................................... 6
1.2.2 Christoffel Symbols and Ricci Tensor
fortheSphericallySymmetricCase ..................... 6
1.2.3 RobertsonExpansion ................................. 7
1.2.4 SchwarzschildMetric ................................. 8
1.3 GeneralEquationsofMotion ................................. 9
References ..................................................... 12
2 SomeApplicationsofGeneralRelativity .......................... 13
2.1 GravitationalLensing ....................................... 13
2.1.1 HistoricalIntroduction ................................ 13
2.1.2 Point-LikeLens ...................................... 14
2.1.3 ThinLensApproximation ............................. 17
2.1.4 LensEquation ....................................... 18
2.1.5 SchwarzschildLens .................................. 20
2.2 GeneralRelativisticStellarStructureEquations ................. 22
2.2.1 Introduction ......................................... 22
2.2.2 GeneralRelativisticStellarStructureEquations .......... 23
2.2.3 InterpretationoftheGravitationalMass M ............... 26
2.2.4 TheInteriorofNeutronStars .......................... 27
2.3 TimeDelayofRadarEchoes ................................. 28
2.4 Precession ................................................. 31
2.4.1 GeodeticPrecession .................................. 33
2.5 LinearizedFieldEquations ................................... 38
2.5.1 TheEnergy-MomentumTensoroftheGravitational
Field ............................................... 38
2.5.2 LinearizedFieldEquations ............................ 40
xi
xii Contents
2.6 Lense–ThirringEffect ....................................... 42
2.6.1 MetricoftheRotatingEarth ........................... 42
2.6.2 GravitomagneticForces ............................... 47
2.7 Summary .................................................. 48
2.8 Problems .................................................. 49
References ..................................................... 52
3 GravitationalWavesandPost-NewtonianApproximation .......... 55
3.1 Introduction ............................................... 55
3.1.1 ElectromagneticWaves ............................... 56
3.2 GravitationalWaves ........................................ 57
3.2.1 HelicityofGravitationalWaves ........................ 59
3.2.2 ParticlesintheFieldofaGravitationalWave ............. 60
3.2.3 EnergyandMomentumofaGravitationalWave .......... 62
3.2.4 QuadrupoleRadiation ................................ 65
3.3 SourcesofGravitationalWaves ............................... 71
3.3.1 RigidRotator ........................................ 71
3.3.2 Example:TheBinaryStarSystem ...................... 73
3.4 BinariesonEllipticOrbits ................................... 75
3.4.1 EllipticKeplerianOrbits .............................. 75
3.4.2 RadiatedPower ...................................... 77
3.4.3 ParabolicOrbits ..................................... 80
3.4.4 FrequencySpectrumofGravitationalWaves ............. 81
3.4.5 EvolutionoftheOrbitDuetoGravitationalWaves
Emission ........................................... 84
3.4.6 ThePSR1913+16System ........................... 85
3.5 Waveform ................................................. 87
3.5.1 QuadrupoleRadiationfromaMassinCircularOrbit ...... 91
3.6 ThePost-NewtonianApproximation .......................... 94
3.6.1 The1PNApproximation .............................. 96
3.6.2 EquationsofMotionofTestParticlesinthe1PNMetric ... 98
3.6.3 TwoBodyProbleminthe1PNApproximation ........... 101
3.7 Summary .................................................. 106
3.8 Problems .................................................. 106
References ..................................................... 112
4 BlackHoles .................................................... 113
4.1 TheKerrSolution .......................................... 113
4.1.1 InterpretationoftheParametersa andm ................ 114
4.1.2 Kerr–NewmanSolution ............................... 115
4.1.3 EquationsofMotionforTestParticles .................. 116
4.2 TetradFormalism ........................................... 117
4.2.1 Introduction ......................................... 117
4.2.2 SomeDifferentialGeometry ........................... 120
4.2.3 TheSchwarzschildMetricRevisited .................... 122
4.3 SpinorsinCurvedSpacetime ................................. 126
