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Lecture Notes in Physics Peter A. Hogan Dirk Puetzfeld Frontiers in General Relativity Lecture Notes in Physics Volume 984 FoundingEditors WolfBeiglböck,Heidelberg,Germany JürgenEhlers,Potsdam,Germany KlausHepp,Zürich,Switzerland Hans-ArwedWeidenmüller,Heidelberg,Germany SeriesEditors MatthiasBartelmann,Heidelberg,Germany RobertaCitro,Salerno,Italy PeterHänggi,Augsburg,Germany MortenHjorth-Jensen,Oslo,Norway MaciejLewenstein,Barcelona,Spain AngelRubio,Hamburg,Germany WolfgangSchleich,Ulm,Germany StefanTheisen,Potsdam,Germany JamesD.Wells,AnnArbor,MI,USA GaryP.Zank,Huntsville,AL,USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new developmentsin physicsresearch and teaching-quicklyand informally,but with a highqualityandtheexplicitaimtosummarizeandcommunicatecurrentknowledge in an accessible way. 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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) https://doi.org/10.1007/978-3-030-69370-1 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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