史瓦西时空中的准正模式分析

PHYSICALREVIEWDVOLUME2,NUMHER1015NOVEMHER1970GravitationalFieldofaParticleFallinginaSchwarzschildGeometryAnalyzedinTensorHarrIIonics*tFRANKJ.ZERILLItJosephHenryLaboratories,Princeton,¹arJersey08540(Received5January1970;revisedmanuscriptreceived4August1970)Weareconcernedwiththepulseofgravitationalradiationgiveno6'whenastarfallsintoablackholenearthecenterofourgalaxy.WelookattheproblemofasmallparticlefallinginaSchwarzschildback-ground(blackhole)andexamineitsspectruminthehigh-frequencylimit.Informulatingtheproblemitisessentialtoposethecorrectboundarycondition:gravitationalradiationnotonlyescapingtoin6nitybutalsodisappearingdownthehole.Wehaveexaminedtheproblemintheapproximationoflinearpertur-bationsfromaSchwarzschildbackgroundgeometry,utilizingthedecompositionintothetensorsphericalharmonicsgivenbyReggeandWheeler(1957)andbyMathews(1962).Thefallingparticlecontributesa8-functionsourceterm(geodesicmotioninthebackgroundSchwarzschildgeometry)whichisalsodecom-posedintotensorharmonics,eachofwhichdrivesthecorrespondingperturbationharmonic.Thepowerspectrumradiatedinin6nityisgiveninthehigh-frequencyapproximationintermsofthetracelesstrans-versetensorharmonicscalledelectricandmagneticbyMathews.I.INTRODUCTION''TwaspointedoutbyDyson'thatapulseofgravita-~-tionalradiationwillresultfromthecaptureofastarbyablackhole(thatis,acollapsedstar).WeconsidertheproblemofaparticleofmassmofallingalongageodesicofaSchwarzschildgeometryproducedbyalargermassm.TheparticleemitsgravitationalradiationasitfallsuntilitisabsorbedthroughtheSchwarzschildsurfaceat2'.Thequestionofboundaryconditionsisinterestinghere.InaEuclideantopologywewouldrequireoutgoingwavesatinfinityandregularityattheorigin.IntheSchwarzschildcasethereisnoorigin.How-ever,theSchwarzschildsurfaceat2mhasthepropertythatfuturetimelikeornulltrajectoriespassthroughitonlytowardtheinteriorregion.HenceanaturalboundaryconditiontoreplaceregularityattheoriginistorequirethatthereareonlyingoingwavesattheSchwarzschildsurface,thatis,nothingcomingoutoftheblackhole.Zel'dovichandNovikov'haveconsideredtheproblemoftheradiationofgravitationalwavesbybodiesmovinginthefieldofacollapsingstar.Theybasetheircalcula-tionsontheformula,givenbyLandauandLifshitz,'forthegravitationalpowerradiatedintermsofthethirdtimederivativeofthequadrupolemomentofthesystem.Unfortunately,suchconsiderationscanonlybevalidforbodieswhichmoveatdistanceslargecomparedtotheSchwarzschildradiusofthecentralbody.Butasubstantialpartoftheradiationcomesfromtheregionrnear2m.Itisforthisreasonthatweconsiderthefieldproducedbythefallingparticleasaperturbationonthe*Basedontheauthor'sPh.D.thesis,PrincetonUniversity,1969(unpublished).fWorksupportedinpartbyNSFGrantNo.GP7769andNSFGraduateFellowship.tPresentaddress:PhysicsDepartment,UniversityofNorthCarolina,ChapelHill,N.C.27514.~F.Dyson(privy.tecommunication).'Ya.Zel'dovichandI.Novikov,Dokl.Akad.NaukSSSRI155,1033(1964))SovietPhys.Doklady9,246(1964)j.'L.LandauandE.Lifshitz,TheClassicalTheoryofFields(Addison-Wesley,Reading,Mass.,1962).2backgroundSchwarzschildgeometrysothatwearenotrestrictedtolargedistancesfrom2m.ThesourcetermT„„isgivenbyanintegralovertheworldlineoftheparticle,theintegrandcontainingafour-dimensionalinvariant6function.Thesourcetermisthenguaranteedtobedivergence-freeiftheworldlineisageodesicinthebackgroundgeometry.BecauseofthesphericalsymmetryoftheSchwarz-schildfield,thefieldequationsfortheperturbationh„„areintheformofarotationallyinvariantdifferentialoperatoronh„,setequaltothesourcetermT„,.Weusethisrotationalinvariancetoseparatetheangularvariablesintheheldequations.TheusefulnessofscalarharmonicsII,~inseparating,forexample,Laplace'sequationliesinthefactthattheytransformunderaparticularirreduciblerepresentationoftherotationgroup.ThusarotationallyinvariantoperatoronI'gMgivesaquantitywhichtransformsunderthesameir-reduciblerepresentationandhenceisalinearcombina-tionofI'I,~ofthesameorderI..Whendealingwithtensorfields,weusetensorharmonicswhichtransformunderaparticularrepresentationoftherotationgroup.InAppendix6wediscussthesolutionsoftheI.=OandI.=1equations.ThereisnoL=0odd-parity-type(magnetic)harmonic.Bysuitablegaugetransforma-tions,itispossibletosolveexplicitlythepartialdiBer-entialequationsinrandtfortheL,=0andI.=1even-parity-type(electric)harmonicsandtheI.=1magneticharmonics.TheL=Oelectricequationsgivetheex-pectedresult.LetR(f),O~(t),andC(t)denotetheSchwarzschildcoordinatesofthefallingparticleatSchwarzschildtimet.Theninsidethespherer(R(t)theperturbationfromthebackgroundiszero,whileoutsideitsimplyrepresentsanaugmentationoftheSchwarzschildmassbymgro,themass-energyofthefallingparticle.TheI)=1magneticequationsgiveasasolutionzeroperturbationforr(R(t),butgiveanhastermoutsideofR(t)whichgoesas(sin'0/r)andwhich,accordingtothecriteriongiveninLandauandLifshitz,'representsametricwithangularmomentumboa,where21412142FRANKJ.ZERILLImouistheconservedangularmomentumofthefallingparticle.TheL=j.electricequationsalsogivezeroinsideE(t)andanonzeroh„„outsideE(t)whichcanberemovedbyagaugetransformationwhichisinterpret-ablebyadist