Table Of ContentLecture Notes in Physics
Peter A. Hogan
Dirk Puetzfeld
Frontiers
in General
Relativity
Lecture Notes in Physics
Volume 984
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Peter A. Hogan (cid:129) Dirk Puetzfeld
Frontiers in General
Relativity
PeterA.Hogan DirkPuetzfeld
SchoolofPhysics CenterofAppliedSpaceTechnology
UniversityCollegeDublin andMicrogravity
Dublin,Ireland UniversityofBremen
Bremen,Germany
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-030-69369-5 ISBN978-3-030-69370-1 (eBook)
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Preface
Inourexperience,whilestudentsmayinitiallybeattractedtogeneralrelativityby
thefactthatitistheareaoftheoreticalphysicsthatpredictstheexistenceofblack
holes,gravitationalwavesandthe big-bangbeginningofthe universe,andthatall
three of these phenomenaare currentlybeing confirmedby observations,students
areattractedtothetheoryofthesephenomenabecausegeneralrelativityprovidesa
frameworkformodellingthemusingabstractmathematics.Whensuchstudentsare
beginningresearchandlookingaroundforwaysto“getinvolved”,weenvisagethat
thisbookmightpossiblyprovidewhattheyarelookingfor.Ourapproachisstrongly
geometrical. Even the speculative explicit models which we describe in order to
stimulate further ideas and research are geometrically motivated. These include a
modelofthegravitationalfieldofa Kerrblackholeincorporatingmatterescaping
atthespeedoflightanddiminishingthemassandangularmomentum;alight-like
chargeinelectromagnetictheorywhoseMaxwellfieldisalight-likeanalogueofthe
Liénard-Wiechertfield;amagneticblackhole;theanalogueingeneralrelativityof
amagneticmonopole,movinginexternalgravitationalandelectromagneticfields;
run-awaymotionofReissner-Nordströmparticlesintheabsenceofexternalfields;
and a model of colliding gravitational waves leading to a de Sitter or an anti-de
Sitteruniverse.Noneoftheseexamplescouldbeclaimedtobe“fullyunderstood”
andtherebyleaveroomfordevelopment.
We assume a strong mathematical background in differential geometry (and
thus in tensor calculus); although we only occasionally refer to it, knowledge of
the powerfulCartan calculus is very useful for carrying out some of the involved
explicit calculations referred to in the text. Photons and material particles play
important roles in general relativity, and this fact is strongly represented in the
topicsdescribedinthisbook.Suchobjectsareinvolvedinanalysingandmeasuring
gravitationalfieldsandinconstructingmathematicalmodelsofgravitationalfields
ofvarioustypes.Thismeansthatfromthespace-timegeometrypointofview,the
studyofcongruencesofworldlines,bothtime-likeandlight-like,areofparamount
importance in general relativity. For this reason, the book begins with a standard
description of both types of congruence. A key role in measuring a gravitational
fieldisplayedbythetime-likecongruencesand,inparticular,theuseofdeviation
equations in this context. These are therefore discussed at the beginning of the
book making use of Synge’s world function in the process. With an emphasis
v
vi Preface
on tensor calculus, we make extensive use of the theory of bivectors (or skew-
symmetric tensors) in general relativity and thus an early chapter is devoted to
them.An applicationofthe time-likecongruencesis providedbythe construction
of models of Bateman electromagnetic and gravitational waves in the linear
approximation using the gauge invariant and covariant formalism demonstrating
that the gravitational waves exist owing to the acquisition of shear or distortion
bythet-lines.Therefollows,interspersedwiththeprovocativeexamplesmentioned
above,studiesofgravitationalradiationinthecontextofdeSitter(oranti-deSitter)
cosmology and the gravitational compass or clock compass (general relativistic
gradiometers)forprovidinganoperationalwayofmeasuringagravitationalfield.
Finallyweshouldpointoutthatamongtherelativistswithwhomoneorotherof
ushascollaboratedareJ.L.Synge,I.Robinson,A.Trautman,G.F.R.Ellis,W.Israel,
Y.N.ObukhovandF.W.Hehl,andthereareidentifiablethreadsrunningthroughthe
textwhichreflecttheirinfluenceuponus.
Dublin,Ireland PeterA.Hogan
Bremen,Germany DirkPuetzfeld1
October2020
1D.P.acknowledgesthesupportbytheDeutscheForschungsgemeinschaft(DFG)throughthegrant
PU461/1-2projectnumber369402949.
Contents
1 CongruencesofWorldLines................................................ 1
1.1 Time-LikeConguences................................................. 1
1.2 NullGeodesicCongruences ........................................... 8
1.3 TheWorldFunctionandDeviationEquations........................ 11
References..................................................................... 17
2 BivectorFormalisminGeneralRelativity ................................ 19
2.1 BivectorsandElectromagneticFields................................. 19
2.2 ElectromagneticRadiation............................................. 27
2.3 BivectorsandGravitationalFields..................................... 30
2.4 TheKerrSpace-Time................................................... 38
2.5 PassagetoChargedKerrSpace-Time ................................. 44
2.6 UsingtheBianchiIdentities............................................ 46
References..................................................................... 48
3 HypotheticalObjectsinElectromagnetismandGravity................ 51
3.1 PartI:ALight-LikeCharge............................................ 51
3.2 GeometryBasedonaNon-geodesicNullWorldLine................ 53
3.3 MaxwellFieldofaChargewithNon-geodesicWorldLine.......... 57
3.4 PartII:AKerrBlackHoleandLight-LikeMatter.................... 61
3.5 AxialSymmetry........................................................ 62
3.6 Energy-Momentum-StressTensor..................................... 64
3.7 TwoConservationLaws................................................ 66
References..................................................................... 67
4 BatemanWaves .............................................................. 69
4.1 ElectromagneticRadiation............................................. 69
4.2 BatemanElectromagneticWaves...................................... 77
4.3 Some‘Spherical’ElectromagneticWaves ............................ 82
4.4 GravitationalRadiation ................................................ 85
4.5 BatemanGravitationalWaves ......................................... 89
4.6 Some‘Spherical’GravitationalWaves................................ 93
References..................................................................... 97
vii
viii Contents
5 Gravitational(Clock)Compass ............................................ 99
5.1 DeterminationoftheGravitationalFieldbyMeansofTest
Bodies................................................................... 99
5.2 GravitationalCompass................................................. 100
5.3 DeterminationoftheGravitationalFieldbyMeansofClocks....... 106
5.4 GravitationalClockCompass.......................................... 113
References..................................................................... 124
6 deSitterCosmology.......................................................... 127
6.1 NullHyperplanesinSpace-TimesofConstantCurvature............ 127
6.2 IntersectingNullHyperplanes......................................... 135
6.3 GeneralizedKerr–SchildSpace-TimesandGravitationalWaves.... 137
6.4 deSitterSpace-TimeRevisited........................................ 140
6.5 CollisionofGravitationalWavesand(cid:2)(cid:2)=0.......................... 146
6.6 PostCollisionPhysicalProperties..................................... 151
References..................................................................... 152
7 SmallMagneticBlackHole................................................. 153
7.1 MagneticPoles ......................................................... 153
7.2 BackgroundSpace-Time/ExternalFields ............................. 157
7.3 MagneticBlackHolePerturbationofBackground................... 162
7.4 EquationsofMotioninFirstApproximation ......................... 164
7.5 EquationsofMotioninSecondApproximation ...................... 171
7.6 ReviewofApproximations ............................................ 184
References..................................................................... 186
8 Run-AwayReissner–NordströmParticle.................................. 187
8.1 Robinson–TrautmanSolutionsoftheEinstein–Maxwell
Equations ............................................................... 187
8.2 AcceleratingReissner–NordströmParticle............................ 190
8.3 PerturbationsofaReissner–NordströmParticle...................... 193
8.4 EquationsofMotionofaReissner–NordströmParticle.............. 197
8.5 EnergyRadiationRateandRun-AwayMotion ....................... 203
8.6 Run-AwayMagneticReissner–NordströmParticle................... 207
References..................................................................... 208
A CongruencesofWorldLines................................................ 209
A.1 Propertiesofη ...................................................... 209
abcd
B BatemanWavesintheLinearApproximation............................ 211
B.1 CovariantTreatmentonGeneralSpace-Time......................... 211
B.2 RicciIdentitiesandBianchiIdentities................................. 215
References..................................................................... 217
C Gravitational(Clock)Compass ............................................ 219
C.1 CoordinatesforMinkowskianSpace-Time ........................... 219
C.2 PlaneGravitationalWavesI............................................ 227
Contents ix
C.3 PlaneGravitationalWavesII........................................... 229
C.4 WavesMovingRadiallyRelativetor =0............................ 232
References..................................................................... 235
D deSitterCosmology.......................................................... 237
D.1 PropertiesofGeneralisedKerr–SchildMetrics....................... 237
Reference...................................................................... 239
E SmallMagneticBlackHole................................................. 241
E.1 TheElectromagneticField............................................. 241
Index............................................................................... 245
