<%BANNER%>

Radiation Reaction in Curved Spacetime

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FirstandforemostIwouldliketothankmyresearchadvisorProfessorStevenDetweiler,forhisconstantencouragementandguidancethroughouttheentirecourseofmyresearchwork.IwouldalsoliketothankProfessorBernardWhitingandProfessorRichardWoodardforvaluablediscussions.IamhonoredandgratefultoProfessorJamesFry,ProfessorDavidReitze,andProfessorAtaSarajediniforservingonmysupervisorycommittee. iv

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page ACKNOWLEDGMENTS ............................. iv FIGURE ...................................... viii ABSTRACT .................................... ix CHAPTER 1INTRODUCTION .............................. 1 2GENERALFORMALSCHEMESFORRADIATIONREACTION ... 5 2.1Dirac:RadiatingElectronsinFlatSpacetime ............ 5 2.1.1TheFieldsAssociatedwithanElectron ............ 5 2.1.2TheEquationsofMotionofanElectron ............ 8 2.2DewittandBrehme:ElectromagneticRadiationDampinginCurvedSpacetime ................................ 12 2.2.1Bi-tensors ............................ 12 2.2.2Green'sFunctionsinCurvedSpacetime ............ 18 2.2.3ElectrodynamicsinCurvedSpacetime ............. 21 2.2.4DerivationofEquationsofMotionforanElectricChargeviaWorld-tubeMethod .................... 25 3CALCULATIONSOFSELF-FORCE:REVIEWOFGENERALSCHEMESANDANALYTICALAPPROACHES ................... 33 3.1GeneralFormalSchemesRevisited .................. 34 3.1.1Dirac:RadiatingElectronsinFlatSpacetime ........ 34 3.1.2DewittandBrehme:ElectromagneticRadiationDampinginCurvedSpacetime ........................ 35 3.1.3Quinn:RadiationReactionofScalarParticlesinCurvedSpace-time ............................... 36 3.1.4Mino,Sasaki,andTanaka;QuinnandWald:GravitationalRadiationReactionofParticlesinCurvedSpacetime .... 37 3.2AnalyticalCalculationsofSelf-force .................. 38 3.2.1DewittandDewitt:FallingCharges ............. 38 3.2.2PfenningandPoisson:Scalar,Electromagnetic,andGravi-tationalSelf-forcesinWeaklyCurvedSpacetimes ...... 40 v

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............................ 43 4.1SplittingtheRetardedField ...................... 45 4.1.1ConventionalMethodofSplittingtheRetardedField .... 45 4.1.2NewMethodofSplittingtheRetardedField ......... 49 4.2Mode-sumDecompositionandRegularizationParameters ...... 51 4.3DescriptionofSingularFieldandTHZCoordinates ......... 56 4.3.1IntroductionofTHZCoordinates ............... 56 4.3.2ApproximationfortheSingularFieldinTHZCoordinates 60 4.3.3TheDeterminationofTHZCoordinates ............ 65 4.3.4DeterminationoftheSingularField .............. 80 4.4DeterminationofRegularizationParameters ............. 82 4.4.1Aa-terms ............................. 87 4.4.2Ba-terms ............................. 93 4.4.3Ca-terms ............................. 101 4.4.4Da-terms ............................. 104 4.5AnExample:Self-forceonCircularOrbitsaboutaSchwarzschildBlackHole ................................ 110 5PRACTICALSCHEMESFORCALCULATIONSOFSELF-FORCE(B):EFFECTSOFGRAVITATIONALSELF-FORCE .......... 113 5.1MiSaTaQuWaGravitationalSelf-forceandGaugeIssues ...... 113 5.2FirstOrderPerturbationAnalysis ................... 114 5.3DecompositionofthePerturbationFieldhab 117 5.3.1SingularFieldhSab 118 5.3.2RegularFieldhRab 121 5.4AnExample:Self-forceEectsonCircularOrbitsintheSchwarzschildGeometry ................................ 121 5.4.1GaugeInvariantQuantities ................... 122 5.4.2Mode-sumRegularization .................... 123 5.4.3RegularizationParameters ................... 125 6CONCLUSION ................................ 144 APPENDIX AHYPERGEOMETRICFUNCTIONSANDREPRESENTATIONSOFREGULARIZATIONPARAMETERS ................... 146 BTHEGEOMETRYOFFERMINORMALCOORDINATES ....... 149 CTHETRANSFORMATIONBETWEENFERMINORMALANDTHZNORMALGEOMETRIES .......................... 154 REFERENCES ................................... 158 vi

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............................ 160 vii

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Figure page 4{1Self-forceofascalareldintheSchwarzschildspacetime ......... 112 viii

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Abinaryinspiralofasmallblackholeofsolarmassandasupermassiveblackholeof105to107solarmass,calledanextrememass-ratiosystem,isoneofthepossibletargetsourcesofgravitationalwavesforLISA(LaserInterferometerSpaceAntenna)detection.Anaccuratedescriptionoftheorbitalmotionofthesmallblackhole,includingtheeectsofradiationreactionandtheself-forceisessentialtodesigningthetheoreticalwaveformfromthisbinarysystem. Onecancalculatetheeectsofradiationreactionandtheself-forceforthetwomodelsofsuchsystems:thecaseofascalarparticleorbitingaSchwarzschildblackholeandthecaseofapointmassorbitingaSchwarzschildblackhole.Asfortheformer,theinteractionofascalarpointchargewithitsowneldresultsintheself-forceontheparticle,whichincludesbutismoregeneralthantheradiationreactionforce.Inthevicinityoftheparticleincurvedspacetime,onemayfollowDiracandsplittheretardedeldoftheparticleintotwoparts:(1)thesingularsourceeldwhichresemblestheCoulombpotentialneartheparticle,and(2)theregularremaindereld.Thesingularsourceeldexertsnoforceontheparticle,andtheself-forceisentirelycausedbytheregularremainder.Asforthelatter,a ix

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Inthisdissertationwedescribesystematicmethodsforndingmultipoledecompositionsofthesingularsourceeldsforbothcases.Thisimportantstepleadstothecalculationoftheself-forceonascalar-chargedparticleorapointmassorbitingaSchwarzschildblackhole. x

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Einstein'sGeneralTheoryofRelativityisafundamentaltheoryofgravitationandspacetime.Ithasdescribedwithgreataccuracyandprecisionmanyphenom-enainourphysicaluniversethatclassicalphysicshasnotbeenabletoexplainsuccessfully,suchastheperihelionmotionofplanetsandthebendingofstarlightbytheSun.Ithasalsomademanysignicantpredictionssuchastheexistenceofgravitationalwaves,blackholesandtheexpansionoftheuniverse. Amongotherpredictions,gravitationalwavesmightbethemostexcitingproblemthesedays,sincethepossibledetectionofthemcouldhelprevealinfor-mationabouttheverystructureoftheiroriginsandaboutthenatureofgravity,thuswouldopenupanewwindowforourunderstandingoftheuniversebothfromphysicsandfromastronomy.Gravitationalwavescanbedescribedasripplesinthefabricofspacetimecausedbyviolentastrophysicaleventsinthedistantuni-verse,forexamplethecoalescenceofbinaryblackholesortheinspiralofcompactobjectsintothesupermassiveblackholes.Althoughthedetectionofgravita-tionalwaveshasbeenknowntobetechnicallychallenging,scientistsareeagertoimplementexperimentswhichproposetodetectgravitationalwaves.Currently,severalground-baseddetectorsareinoperationorunderconstruction,includingLIGO(USA),VIRGO(Italy/France),GEO(Germany/GreatBritain)andTAMA(Japan),andthespace-basedobservatoryLISAisscheduledtolaunchin2011. Sincetherearesomanysourcesatagiventime,inordertodetectgravita-tionalwaves,itisnecessarytomodelthegravitationalwaveformwhichisbaseduponadetailedtheoreticalstudyofthetargetsources.Thenthetheoreticalmodels 1

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ofgravitationalwaveswouldhelpscientiststosortoutwhattolookforfromaseeminglyhugemessofobservationaldata. AsanexampleofthepossiblesourcesofgravitationalwavesforLISAdetec-tion[ 1 ],abinaryinspiralofasmallblackholeofsolarmassandasupermassiveblackholeof105to107solarmass,whatwecallanextrememass-ratiosystem,canbetaken.Suchblackholesarenowbelievedtoresideinthecoresofmanygalaxies,includingourown. Designingthetheoreticalwaveformfromthisbinarysystemwouldrequireanaccuratedescriptionoftheorbitalevolutionofthesmallblackhole.Theorbitalmotioncanbemodeledbyconsideringapointliketestparticlemovinginthegravitationaleldwhichresultsfromcombiningtheeldofthelargeblackholewiththemuchsmallereldofthesmallblackholeusingperturbationtechniques.Theresultingmotionthenincludestheeectsofradiationreactionandtheself-force. Thisdissertationpresentsspecicmethodsforcalculatingtheeectsofradiationreactionandtheself-forcefortheextrememass-ratiosystems.Weexploretwomodelsofsuchsystemsinthemainbodyofthedissertation.ThecaseofascalarparticleorbitingaSchwarzschildblackholeisinvestigatedrst,andthecaseofapointmassorbitingaSchwarzschildblackholefollows.Thestudyoftheformeritselfmightnotprovidephysicalinterpretationsasdirectlyapplicabletoourgravitationalwavephysics,butitprovidesvaluablecomputationaltoolswithwhichwecanapproachthelatter.Theentiredissertationcanbeoutlinedasfollows. InChapter 2 weintroducegeneralformalschemesonradiationreaction.TwomainarticlesonthissubjectbyDirac[ 2 ]andbyDewittandBrehme[ 3 ]arereviewed. InChapter 3 werevisitthegeneralformalschemesandreviewbrieythestructureoftheequationsofmotionfortheself-forceforeachcasefromDiracto

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Mino,Sasaki,andTanaka,andQuinnandWald[ 2 3 4 5 6 ].Then,weprovidetwoexamplesofthepurelyanalyticattemptstotheself-forcecalculationsbyDewittandDewitt[ 7 ]andPfenningandPoisson[ 8 ]. InChapter 4 weintroduceahybridofbothanalyticalandnumericalmethods,knownasthe\mode-sum"methoddevisedbyBarackandOri[ 9 ],inordertohandlemoregeneralproblemsthanthepurelyanalyticalapproachescan.WethenworkonthecaseofascalarparticleorbitingaSchwarzschildblackholeviathismethod.Theself-forcecalculationsforthiscaseinvolveanalyticalworkfordeterminingRegularizationParameters,whichrefertothemode-decomposedmultipolemomentsofthesingularpartofthescalareld.Thecomputationsoftheregularizationparametersarefacilitatedviaalocalanalysisofspacetime,andanelaborateperturbationanalysisofthelocalgeometryisdevelopedforthispurpose.Theregularizationparametersarecalculatedtosucientlyhighorderssothattheiruseinthemodesumsfortheself-forcecalculationwillresultinmorerapidconvergenceandmoreaccuratenalresults.Theseanalyticalresultsarethencombinedwiththenumericalcomputationsoftheretardedeldtoprovidetheself-forceultimately. InChapter 5 weprovideamethodtodeterminetheeectsofthegravitationalself-forceonapointmassorbitingaSchwarzschildblackhole.First,weaddressthegaugeissuesinrelationtoMiSaTaQuWaGravitationalSelf-force[ 4 5 ].ThenwefollowarecentanalysisbyDetweiler[ 10 ]todescribethegravitationaleld,whichistheperturbationcreatedbythepointmassfromthebackgroundspacetime.Toavoidthegaugeproblem,ratherthancalculatingtheself-forcedirectly,wefocusongaugeinvariantquantitiesanddeterminetheirchangesduetotheself-forceeects.Techniquesinvolvedincalculatingtheregularizationparametersforthegravitationaleldcasearemorecomplicatedthanforthescalareldcase.We

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followanalysesbyDetweilerandWhiting[ 11 ]tondthemethodsforcalculatingtheregularizationparameters.

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Historically,Diracgavetherstformalanalysisoftheradiationreactioneectfortheelectromagneticeldofaparticlemovinginatspacetimein1938[ 2 ].Intheequationofmotionforamovingelectron,hewasabletoobtaintheadditionalforceterm,namedthe\Abraham-Lorentz-Dirac(ALD)dampingterm,"apartfromtheLorentzforceduetotheexternalelectromagneticeld.ButthisALDdampingtermeventuallyturnsouttovanishinfreefall,leavingtheparticle'smotioningeodesic,andnoradiationdampingor\self-force"eectoccursinatspacetime. However,Dirac'spioneeringideawassucceededandgeneralizedtocurvedspacetimeinsimilarlyformalapproachesbythefollowinggenerations.DewittandBrehme[ 3 ]extendedDirac'sanalysistocurvedspacetime.Mino,Sasaki,andTanaka[ 4 ]developedasimilaranalysisforthegravitationaltensoreld.QuinnandWald[ 5 ]andQuinn[ 6 ]workedoutsimilarschemesfortheradiationreactionofthegravitational,electromagnetic,andscalareldsbytakingaxiomaticapproaches.Allthesegeneralizedversionsoftheradiationreactionproblemshowtheobviousexistenceofnon-vanishingdampingtermsinadditiontotheALDdampingterm,whichwouldeventuallycauseradiationreactionincurvedspacetime. InthisChapterwereviewthetwomainarticlesonthissubject,onebyDirac[ 2 ]andtheotherbyDewittandBrehme[ 3 ]. 2.1.1TheFieldsAssociatedwithanElectron 5

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Letusdescribetheworld-lineoftheelectroninspacetimebytheequation whereza(s)isafunctionoftheproper-times,anddz0=ds>0.TheelectromagneticpotentialatthepointxasatisestheMaxwell'sequations whereJaisthecharge-currentdensityvector.Withourpresentmodeloftheelectron,Javanisheseverywhereexceptontheworld-lineoftheelectron,whereitisinnite foranelectronofchargee.TheelectromagneticeldtensorFabcanbederivedfromthepotentialAa Eqs( 2-2 )and( 2-3 )havemanysolutionsandthusdonotxtheelduniquely.Onemayuseasolutionprovidedbythewell-knownretardedpotentialsofLienardandWiechert.WecalltheeldderivedfromthesepotentialsFabret.OnecanobtainothersolutionsbyaddingtothisoneanysolutionofEq.( 2-2 )and representingaeldofradiation.Then,theactualeldFabactforourone-electronproblemwillbethesuperpositionoftheeldfromtheretardedpotentialsandtheeldfromthesolutionsofEq.( 2-6 )thatrepresenttheincomingelectromagneticwavesincidentonourelectron

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AlsowehavetheeldFabadvderivedfromanothersolutionofEqs.( 2-2 )and( 2-3 ),whichisprovidedbytheadvancedpotentials.FabadvisexpectedtoplayasymmetricalroletoFabretinallquestionsofgeneraltheory.Thus,correspondingtoEq.( 2-7 )onemayput whereaneweldFaboutisexpectedtoplayasymmetricalroleingeneraltheorytoFabin,andshouldbeinterpretableastheeldofoutgoingradiationleavingtheneighborhoodoftheelectron.Thedierence wouldthenbetheeldofradiationproducedbytheelectron.Alternatively,fromEqs.( 2-7 )and( 2-8 ),thisdierencemaybeexpressedas whichshowsthatFabradiscompletelydeterminedbytheworld-lineoftheelectron.Throughsomecalculations,itisfoundtobe neartheworld-line,andisfreefromsingularity. Withtheattainedsymmetrybetweentheuseofretardedandadvancedelds,onedenesaeld 2Fabret+Fabadv;(2-12) whichwillbeusedtodescribethemotionoftheelectron.ThiseldisderivablefrompotentialssatisfyingEq.( 2-6 )andisfreefromsingularityontheworld-lineoftheelectron.FromEqs.( 2-7 )and( 2-8 ),itisinfactjustthemeanoftheincoming

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andoutgoingeldsofradiation, 2Fabin+Fabout:(2-13) 4Tac=FabFcb+1 4gacFbdFbd:(2-14) Bytheconservationlaws,thetotalowofenergy(ormomentum)outfromthesurfaceofanynitelengthofworld-tubemustbeequaltothedierenceintheenergy(ormomentum)residingwithinthetubeatthetwoendsofthislength:dependingonlyonconditionsatthetwoendsofthislength,therateofowofenergy(ormomentum)outfromthesurfaceofthetubemustbeaperfectdierential. Theinformationobtainedinthismannerisindependentofshapeandsizeoftheworld-tubeprovidedthatitismuchsmallerthantherealmoftheTaylorexpansionsusedinthecalculations.Ifwetaketwoworld-tubessurroundingthesingularworld-line,thedivergenceofthestresstensor@Tac=@xcwillvanisheverywhereintheregionofspacetimebetweenthem,sincetherearenosingularities

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inthisregionandEq.( 2-6 )issatisedthroughoutit.Theintegral overtheregionofspacetimebetweenthetwoworld-tubesofacertainlengthcanbeexpressedasasurfaceintegraloverthethree-dimensionalsurfaceofthisregion.Thenthedierenceintheowsofenergy(ormomentum)acrossthesurfacesofthetwotubesshoulddependonlyonconditionsatthetwoendsofthelengthconsidered.Thustheinformationprovidedbytheconservationlawsiswelldened. Foreasiercalculations,thesimplestcongurationoftheworld-tubeischosen,withasphericalsurfaceandofaconstantradiusforeachinstantofthepropertimeinthatLorentzframeofreferenceinwhichtheelectronisatrest.Also,wenotethefollowingelementaryequationsforlateruse wherevadza=dsanddotsdenotedierentiationswithrespecttos.AfterratherlengthycalculationswiththeintegralofthestresstensorTacovertheworld-tube,onecanshowthattheowofenergyandmomentumoutfromthesurfaceofanynitelengthoftubeisgivenas 2e21_vaevbfabds;(2-19) wheretermsthatvanishwithareneglected.Sincethisintegralmustdependonlyonconditionsatthetwoendsofthelengthoftube,theintegrandmustbeaperfectdierential,i.e., 1 2e21_vaevbfab=_Ba:(2-20)

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Thisisallonecangetfromthelawsofconservationofenergyandmomentum.Todevelopthisfurtherintotheequationofmotionfortheelectron,oneneedstoxthevectorBabymakingsomeassumptions.TakingadotproductofthebothsidesofEq.( 2-20 )withva,wehave 2e21va_vaevavbfab=0;(2-21) byEq.( 2-17 )andfromtheantisymmetryofthetensorfab.ThenwemayassumethatBacouldbeanyvectorfunctionofvaanditsderivatives.ThesimplestchoicethatsatisesEq.( 2-21 )wouldbe wherekisaconstant. SubstitutingEq.( 2-22 )intotherighthandsideofEq.( 2-20 ),oneseesthattheconstantkmustbeoftheform 2e21m;(2-23) wheremisanotherconstantindependentof,inorderthatourequationsmayhaveadenitelimitingformwhentendstozero.Thenonegets astheequationsofmotionfortheelectron.Thisistheusualformoftheequationofmotionofanelectroninanexternalelectromagneticeld,withmbeingtherest-massoftheelectronandfab=Fabact1 2Fabret+Fabadv,beingtheexternaleld.

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Inpracticalproblems,however,wearegivennotfabbuttheincidenteldFabin.ThesetwoeldsareconnectedviaEqs.( 2-12 ),( 2-7 )and( 2-10 ), 2Fabrad=Fabin+2 3evavbvbva withthehelpofEq.( 2-11 ).SubstitutingthisintoEq.( 2-24 )andusingEqs.( 2-16 )and( 2-18 ),oneobtains 3e2va+_v2va=evbFabin;(2-26) where_v2_va_va.Eq.( 2-26 )wouldbeequaltotheequationofmotionderivedfromtheLorentztheoryoftheextendedelectronbyequatingthetotalforceontheelectrontozero,ifoneneglectstermsinvolvinghigherderivativesofvathethesecond. TodiscussthephysicalinterpretationsofEq.( 2-26 ),oneneedstoexaminetheequationfora=0component,describingtheenergybalance.Therighthandsidegivestherateatwhichtheincidentelddoesworkontheelectron,andisequatedtothesumofthethreetermsm_v0,2 3e2v0and2 3e2_v2v0.Thersttwoofthesearetheperfectdierentialsofthequantitiesmv0and2 3e2_v0,respectively,andmaybeconsideredasintrinsicenergiesoftheelectron:theformeristheusualexpressionforaparticleofrest-massmandthelatterthe\accelerationenergy"oftheelectron[ 12 ].Changesintheaccelerationenergycorrespondtoareversibleformofemissionorabsorptionoftheeldenergyneartheelectron.However,thethirdterm2 3e2_v2v0correspondstoirreversibleemissionofradiationandgivestheeectofradiationdampingonthemotionoftheelectron.AccordingtoEq.( 2-17 ),thistermmustbepositivesince_vaisorthogonaltothetime-likevectorvaandisthusaspace-likevector,andhenceitssquareisnegative(inthesignatureconvention(+1;1;1;1)).

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Later,wewillcompareEq.( 2-26 )withtheequationsofmotionforaparticlemovinginelectromagnetic[ 3 ],scalar[ 6 ]andgravitationalelds[ 4 5 ]incurvedspacetime.Then,itwouldbemoreconvenienttowriteEq.( 2-26 )inthealternativesignatureconvention(1;+1;+1;+1)tobeconsistentwiththeotherequationsofmotioninsign,namely 3e2va_v2va:(2-27) 2.2.1Bi-tensors 2.1 wasdevelopedunderLorentzinvariancethroughout,DewittandBrehme'scurved-spacetimegeneralizationofDirac'siscarriedoutundergeneralcovariancethroughout.Thiscovariantgeneralizationinvolvesnon-localityquestions,anditisessentialtointroducebi-tensors,whichareageneralizationofordinarytensors.Abi-tensorisasetoffunctionsoftwospacetimepoints,eachmemberofwhichtransformsunderacoordinatetransformationlikeanordinarylocaltensor,withthedierencethatthetransformationindicesdonotallrefertothesamepoint,butrathertothetwoseparatepoints.Thesimplestexampleofabi-tensoristheproductoftwolocalvectors,Aa(x)andBb0(z),takenatdierentspacetimepoints,xandzwiththeindicesaandb0runningfrom0to3: Heretheconventionisthattheusual,non-primedindicesarealwaystobeassoci-atedwiththepointx,whiletheprimedindicesarealwaystobeassociatedwiththepointz.Thenthecoordinatesofthepointsthemselvesareexpressedasxaandzb0.

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Thecoordinatetransformationlawforthisbi-tensorisgivenby Inaddition,theusualoperationssuchascontractionandcovariantdierentiationsmaybeimmediatelyextendedtobi-tensorswiththeprecautions:(i)contractionmaybeperformedonlyovertheindicesreferringtothesamepoint,(ii)intakingcovariantderivativesallindicesexceptthosereferringtothevariableinquestionshouldbeignored.Onemaytakecovariantderivativeswithrespecttoeithervariable, wherethesemicolondenotescovariantdierentiationandthecommadenotesordinarydierentiation.Indicesassociatedwithcovariantdierentiationatdierentpointscommute,whiletheusualcommutationlawsholdforindicesreferringtothesamepoint. Onemaydeneabi-scalar,whichisaninvariantbi-tensorbearingnoindices.Onemayalsointroduceabi-density,anditsmostelementaryexampleisthefour-dimensionaldeltafunction Ingeneral,thedeltafunctionmayberegardedasadensityofweightwatthepointxandweight1watthepointz,wherewisarbitrary.Onemaychoosew=1=2forthesakeofsymmetry,andthetransformationlawforthedeltafunctionmaybegiveintheform @~x1=2@z @~z1=2(4)(x;z):(2-33)

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Onemayintroduceabi-scalarofgeodeticinterval,whichisoffundamentalimportanceinthestudyofthenon-localpropertiesofspacetime.Itisdenedasthemagnitudeoftheinvariantdistancebetweenxandzasmeasuredalongthegeodesicjoiningthem.Denotingitbys(x;z),onemayexpressitsbasicpropertiesintheequations limx!zs=0; wherethesignatureofthemetricistakenas(1;+1;+1;+1)(comparethiswithDirac'sconventioninSection 2.1 ).Theintervalbetweenxandzissaidtobespacelikewhenthesignis+andtimelikewhenthesignisinEq.( 2-34 ).However,thebi-scalaritselfistakennon-negative.Whens=0,thelocusofpointsxdenethelightconethroughz. Geodesicsjoiningxandzmaynotnecessarilybeunique,andthebi-scalarofgeodeticintervalcanbemultiple-valued.However,therewillbearegioninwhichthegeodeticintervalissinglevalued,andourattentionisconnedtothisregionindevelopingourargument:thegeodeticintervalinthissingle-valuedregioncanserveasthestructuralelementofcovariantexpansiontechniqueslater.Andinordertoavoid\branchpoint"problems,insteadofs,itwillbemoreconvenienttoworkwiththequantity,whichisknownasSynge'sworldfunction[ 13 ], 2s2;(2-36) whichsatises 1 2gab;a;b=1 2ga0b0;a0;b0=; limx!z=0; wheretheintervalissaidtobespacelikewith+signandtimelikewithsign.

PAGE 25

Using,abi-tensorTa0b0,whoseindicesallrefertothesamepointz,canbeexpandedaboutzinthecovariantform 2Aa0b0c0d0;c0;d0+O(s3);(2-39) wheretheexpansioncoecientsAa0b0,Aa0b0c0,Aa0b0c0d0,etc.areordinarylocaltensorsatz.ThesecoecientscanbedeterminedintermsofthecovariantderivativesofTa0b0: Aparticularexampleofsuchexpansionstonoteis 3Ra0c0b0d0;c0;d0+O(s3):(2-43) Onecandeveloptheexpansionstohigherordersandobtainfurther 3Ra0c0b0d0+Ra0d0b0c0;d0+O(s2); 3(Ra0c0b0d0+Ra0d0b0c0)+O(s): Forexpandingabi-tensorwhoseindicesdonotallrefertothesamepoint,forexampleTab0,oneintroducesadevicecalledthebi-vectorofgeodeticparal-leldisplacementanddenotesitbygab0(x;z).Thisbi-vectorhasthesignicantgeometricalinterpretationinthedeningequations gab0;cgcd;d=0; gab0;c0gc0d0;d0=0; limx!zgab0=gab0orlimx!zgab0=ab0:(2-48)

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FromEqs.( 2-46 )and( 2-47 )itisinferredthatitscovariantderivativesvanishinthedirectionstangenttothegeodesicjoiningxandz,whileEq.( 2-48 )statesthatitreducestotheordinarymetric(orKroneckerdelta)inthecoincidencelimit.Also,thisbi-vectorhassymmetricreciprocity gab0(x;z)=gb0a(z;x):(2-49) Theroleofthebi-vectorgab0isto\homogenize"theindices.Forinstance,alocalvectorAb0atthepointztransformsintothelocalvectorAaatthepointxbyparalleldisplacement.Theapplicationcanalsobeextendedtolocaltensorsofarbitraryorder.Inparticular,onehas gab0gcd0gb0d0=gac; gab0gcd0gac=gb0d0; gab0;b0=;a; gab0;a=;b0; gab0gcb0=ac; gab0gad0=b0d0: Tensordensitiesarealsosubjectedtoageodesicparalleldisplacementbymeansofthebi-vectorgab0.Onecanintroduceitsdeterminant =gab0:(2-56) Thisdeterminantisabi-scalardensity,havingweight1atthepointxandweight1atthepointz.Itsatisestheequations ;agab;b=0; ;a0ga0b0;b0=0; limx!z=1:

PAGE 27

Eqs.( 2-57 )-( 2-59 )havetheuniquesolution (x;z)=g1=2(x)g1=2(z)=1(z;x);(2-60) where AlocalvectordensityAb0ofweightwtransformsintothelocalvectorAaalongthegeodesicfromztoxbyparalleldisplacementinthemanner Aa=wgab0Ab0:(2-62) Thetransformationbyparalleldisplacementcanbeextendedtothegeneralcase. Abi-scalaroffundamentalimportanceinthetheoryofgeodesicsistheVanVleckdeterminant,givenby =g1j;ab0j;(2-63) where g=jgab0j;(2-64) withtheproperty g(x;z)=g1=2(x)g1=2(z)=g(z;x):(2-65) DierentiatingEq.( 2-37 )repeatedlyandusingEq.( 2-63 ),onecanshowthat 1(;a);a=4:(2-66) Alsoimportantistheexpansionofthisdeterminant,knowntobe =11 6Ra0b0;a0;b0+O(s3):(2-67)

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onefollowsHadamard[ 14 ],accordingtowhichanelementarysolutioncanbewrittenintheform (2)2uaa0 wherethefunctionsuaa0,vaa0,waa0arebi-vectors.IfEq.( 2-69 )issubstitutedintoEq.( 2-68 ),therstfunctionisuniquelydetermined,usingtheboundaryconditionatx!z, whiletheothertwoaremosteasilyobtainedbyexpandingthefunctionsinapowerseries andobtainingtherecurrenceformulaeforthecoecients.UsingEq.( 2-67 )forEq.( 2-70 ),oneobtains 12Rb0c0;b0;c0+O(s3)gaa0:(2-73) Byrepeatedlydierentiatinggaa0,however,onends gaa0;bc=1 2gda0Rbcad0+O(s):(2-74) Then,dierentiatingEq.( 2-73 )repeatedlyandusingEq.( 2-74 ),onealsonds 6gaa0R+O(s):(2-75)

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Also,insertingEqs.( 2-71 )and( 2-72 )intotheequation andmakinguseofEq.( 2-75 ),onearrivesat limx!zvaa0=1 2gab0Ra0b01 6ga0b0R:(2-77) OneintroducestheFeynmanpropagator (2)21=2gaa0 whichcanbeseparatedintorealandimaginaryparts, Usingtheidentities (+i0)1=P1i(); ln(+i0)=lnjj+i(); wherePdenotestheprincipalvalueand onendsforthe\symmetric"Green'sfunction,Gaa0, Gaa0=(8)11=2gaa0()vaa0():(2-83)

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ThevariousGreen'sfunctionsarenowdened, where(x)isanarbitraryspacelikehypersurfacecontainingx,and[(x);z]=1[z;(x)]isequalto1whenzliestothepastof(x)andvanisheswhenzliestothefuture.TheseGreen'sfunctionssatisfytheequations Gaa0=1 2Gretaa0+Gadvaa0;(2-87) Also,theyhavethesymmetryproperties Gaa0(x;z)=Ga0a(z;x); Finally,onecannotethatthesubstitutionofEq.( 2-69 )intoEq.( 2-76 )viaEq.( 2-72 )leavesw0aa0arbitraryinthesolutionforwaa0,whichcorrespondstoaddingtoG(1)anysingularity-freesolutionofthewaveequation.However,thisarbitrarinessdisappearsinthesolutionforthesymmetricGreen'sfunctionsasitisevidentfromEq.( 2-83 ).

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. TheLagrangiandensityforapointparticleofchargeeandbaremassm0,interactingwithanelectromagneticeldFabinaspacetimewithmetricgab,canbewrittenas where Here,theworld-lineoftheparticleisdescribedbyasetoffunctionsza0(),withrepresentinganarbitraryparameter,andthedotoverzdenotesdierentiationwithrespectto.Multipledotswillbeusedtodenoterepeatedabsolutecovariantdierentiationwithrespectto, _za0=dza0=d; za0=d_za0=d+a0b0c0_zb0_zc0; ...za0=dza0=d+a0b0c0zb0_zc0; ... Theactionforthesystemisgivenby

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wheretheintegrationisperformedovertheregionbetweenanytwospacelikehypersurfaces.Withvariationstakeninthedynamicalvariablesza0andAawhichvanishonthesehypersurfaces,theactionsuersthevariation providedthatistakentobethepropertimeoftheparticle(andwillhenceforthbeassumed)suchthat Applicationofthisactionprincipleyieldsthedynamicalequations Bythefactthat 2RbaacFcb+RbabcFac=0;(2-108) onecanshowviaEq.( 2-107 )thatthecurrentdensityisconserved. Conservationofthestress-energytensor Thestress-energytensorofthesystemisgivenby

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where 4gabFcdFcd: Thedivergencesofthesetensorsarefoundtobe 2gab(Fcd;b+Fdb;c+Fbc;d)Fcd=FabJb: Combiningtheseresults,oneobtainstheconservationlaw Vectorpotentialsandelectromagneticelds IntheLorenzgauge theelectromagneticeldequation( 2-107 )mayberewrittenasaninhomogeneousvectorwaveequation Particularsolutionsofthisequationaregivenby

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bywhichtheadvancedandretardedeldsoftheparticlearewritten Thetotaleldmaybeexpressedintheforms Alternatively,onemayexpressthetotaleldintheform where Fab=1 2Fretab+Fadvab; 2Finab+Foutab=Finab+1 2Fradab=Foutab1 2Fradab; SimilarlytoEqs.( 2-119 )and( 2-120 ),theeldsFabandFradabmaybeexpressedintermsofpotentialsAaandArada,whicharedenedbytheintegralexpressionsoftheform( 2-117 )and( 2-118 ),involvingthefunctionsGaa0andGradaa0,respectively.Thevariouseldsdenedthussatisfytheequations

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SubstitutingEqs.( 2-83 ),( 2-84 ),( 2-85 )and( 2-97 )intoEqs.( 2-117 )and( 2-118 ),oneobtains whereinthesecondlineisthevalueofthepropertimeattheintersectionoftheworld-lineoftheparticlewithanarbitraryspacelikehypersurface(x)containingx,andinthethirdlineadv=retdenotestheadvancedorretardedpropertimeoftheparticlerelativetothepointx.ThesepotentialsarethecovariantLienard-Wiechertpotentials.Correspondingtothese,theeldstrengthtensorsisexpressedas wherethelasttermisgenerallynamed\tail"term,whichinvolvesintegrationsovertheentirepastorfuturehistoryofparticle. Inordertodeterminetheeectofradiationreactionontheparticleonemustkeeparecordoftheenergy-momentumbalancebetweentheparticleandtheeld.

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Thiseectisexaminedviatheequationsofmotionoftheparticlewhichdescribeitslocalbehavior,andtheycanbeobtainedonlyifonekeepsaninstantaneousrecordintheimmediateneighborhoodoftheparticle.Forthispurposeoneconstructsathree-dimensionalhypersurfacearoundtheworld-lineoftheparticle,ortheworld-tube,whichisgeneratedbyasmallspheresurroundingtheparticleastimevaries.IntermsofSynge'sworldfunction,thegeneratingsphereofradius,astimevaries,producesahyperspheredenedby 22; wherena0(i)(I=1;2;3)denotesspatialbasisvectorswhichareorthogonaltoeachotherandspanthehypersurfaceorthogonaltotheworld-lineoftheparticle, andirepresentsasetofdirectioncosineswhichsatisfy ii=1:(2-135) Intermsofionecanspecifythedirectionrelativetona0(i)ofanarbitraryunitvectorwhichisperpendiculartotheworld-lineatz.Then,startinginthedirectionofthisarbitraryvector,oneconstructsageodesicemanatingfromzextendingouttoaxeddistancetoapointx.Thecoordinatesofxwilldependonthedirectioncosinesiandonthepropertimewhichistheparameterforthepointz.

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Avariationiinthedirectioncosinesproducesavariationinthepointx,whichisviaEq.( 2-131 )givenby Apairofindependentvariations1iand2iinthedirectioncosinesdeneanelementdofsolidanglebytherelation id=ijk1j2k;(2-137) whereijkisthethree-dimensionalLevi-Civita.Thissolidangledenesanelementoftwo-dimensionalareaonthesurfaceofthesphere,enclosedbytheparallelogramformedfromthecorrespondingdisplacements1xaand2xa.However,oneisratherinterestedinathree-dimensionalsurfaceelementoftheworld-tubegeneratedbythesphereaspropertimevaries.Thenoneshallconstructageneraldisplacementofthepointxontheworld-tube,whichisproducedbyindependentvariationsofandi,withalinearcombinationof1xa,2xaandthethirddisplacement3xaorthogonaltotherstandsecond,formingaparallelepiped: Later,integralsovertheworld-tubewillbeevaluatedtocomputetheenergy-momentumow,andforthispurposeonedenesthedirectedsurfaceelementda,whichisavectordensity,formedfromindependentdisplacements1xa,2xaand3xa Intermsoftheradiusoftube,variationofsolidangledandvariationofpropertimed,thesurfaceelementatxisexpressedas 62Rb0c0b0c0dd+O(5);(2-140)

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where a0na0(i)i;(2-141) Theequationsofmotion Theconservationlawofenergyandmomentum,whosedierentialformwasgivenbyEq.( 2-114 ),canbeexpressedinintegralformusingthebi-vectorofthegeodeticparalleldisplacement,inwhichthecontributionstotheintegralatthevariablepointxisreferredbacktosomexedpointz.Thisintegralisalocalcovariantvectoratz,andGauss'stheoremcanbeemployed.Then,onemaywrite 0=ZVgaa0Tab;bd4x=Z+Z1+Z2gaa0TabdbZVgaa0;bTabd4x; where1and2arethehypersurfacesor\caps"atthepropertimes1and2,respectively,andrepresentsthesurfaceoftheworld-tubebetween1and2,andVisthevolumeofthetube,enclosedby1,2and.Nowbytakingthelimit!0,theintegralsover1,2andVwillretaincontributionsonlyfromtheparticlestress-energytensor.Furthermore,takingthexedpointztolieontheparticle'sworld-lineatapropertime,whichis1<<2,willgive 0=lim!0Z21Z4gaa0Tabdb+m0hgb00a0(z(00);z())_zb00(00)i00=200=1m0Z21gb00a0;c00(z(00);z())_zb00(00)_zc00(00)d00; wherethereplacementhasbeenmade,

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suchthattheintegraloverthesurfacecanbecomputedexplicitlyintermsofanintegraloverpropertimeandanintegraloversolidangle.Byletting1and2bothapproach,Eq.( 2-144 )becomes 0=m0za0d+lim!0Z4gaa0Tabdb:(2-146) Oneshallfocusontheevaluationofthesecondtermofthisequationtoderivetheequationsofmotionoftheelectriccharge. First,theretardedandadvancedeldstrengthtensorsofEq.( 2-129 )mustbeexpressedintheformofexpansions.Afteraverytediousalgebrainvolvinganumberofperturbationsonends 213za0_zb0+1 85_za0b0z21 23...za0b02 34...za0_zb0+1 121_za0b0R1 61_za0Rb0c0c0+1 21a0Rb0c0_zc0+1 121_za0b0Rc0d0c0d0+1 21Ra0c0b0d0_zc0d01 123_za0b0Rc0d0_zc0_zd0+1 63_za0Rb0c0d0e0_zc0_zd0e01 32_za0Rb0c0_zc02eZadv=retr[bGadv=reta]c0_zc0()d+O(); wherethetailtermhasbeenwrittenintermsoftheGreen'sfunctionGadv=retaa0(x;z())ratherthantheHadamardexpansiontermvaa0(x;z())forlaterconvenience. FromEq.( 2-147 )itfollowsthattheeldFradabiseverywherenite.Atthelocationoftheparticle,itisdescribedas 3e_za0...zb0_zb0...za0+4 3e2_z[a0Rb0]c_zc+2eZretr[b0Greta0]c00_zc00(00)d00Z1advr[b0Gadva0]c00_zc00(00)d00:

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Ontheotherhand,forthemeanoftheretardedandadvancedeldsonehastheexpression Fab=1 2Fretab+Fadvab=e(gaa0gbb0gba0gab0)21_za0b0+1 213za0_zb0+1 85_za0b0z21 23...za0b0+termslinearandcubicinthe'sinvolvingtheRiemanntensor+eZretr[bGreta]c0_zc0()d+Z1advr[bGadva]c0_zc0()d+O(): BysplittingthetotalelectromagneticeldasinEq.( 2-122 ),onecannowcomputethestress-energytensorviaEq.( 2-149 ).Takingadvantageofthefactthatfabissingularity-free,onemaywrite gaa0Tabdb=(4)1g1=2gaa0FacFbc+facFbc+Facfbcdb1 4FcdFcd+1 2fcdFcdgaa0da+O(): UsingEqs.( 2-103 ),( 2-104 ),( 2-105 ),( 2-140 )andtheexpansion onecomputestherighthandsideofEq.( 2-150 )andnds gaa0Tabdb=(4)1e21 22a0+1 21za03 4za0zb0b0+1 2a0z2+termsofodddegreeinthe'sinvolvingtheRiemanntensor_zb0Zretr[b0Greta0]c00_zc00(00)d00+Z1advr[b0Gadva0]c00_zc00(00)d00dd(4)1efa0b0_zb0dd+O():

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Carryingouttheintegration,oneeliminatesallthetermscontainingodddegreeinthedirectioncosinesandobtains ThedivergentterminEq.( 2-153 )hasthesamekinematicalstructureasthemassterminEq.( 2-146 ).Therefore,ithastheeectofanunobservablemassrenormalization,andbyintroducingtheobservedmass 2e21;(2-154) onemaynowrewriteEq.( 2-146 )as Then,substitutingEqs.( 2-124 )and( 2-148 )togetherwith( 2-151 )intoEq.( 2-155 )andusingEqs.( 2-103 )and( 2-105 ),onenallyobtainstheequationsofmotionfortheelectriccharge 15 ]andisslightlydif-ferentfromtheoriginal,Eq.(5.26)inDewittandBrehme[ 3 ].ThisisduetothecorrectionsmadetoEqs.(5.12)and(5.14)inDewittandBrehme,whosemodiedformsarenowEqs.( 2-147 )and( 2-148 ),respectively.

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3e2(...zaz2_za)+1 3e2(Rab_zb+_zaRbc_zb_zc)+e2_zbZretr[bGreta]c0(z();z(0))_zc0(0)d0: TheintegralterminvolvingthisGreen'sfunctioninEq.( 2-156 ),oftenreferredtoasthe\tail"term,givesanimplicationthatthemotionoftheparticleisaectedbytheentirehistoryoftheparticleitself.This,togetherwiththethirdtermontherighthandsidewillmaketheparticledeviatefromitsoriginalworld-linetotheorderofe2,evenintheabsenceofanexternalincidenteldFabin,whichmeansthatradiationdampingisexpectedtooccurevenforaparticleinfreefallincurvedspacetime.

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InChapter 2 westudiedgeneralformalschemesofradiationreactioninavari-etyofcontexts,fromDirac'sradiatingelectronsinatspacetimetoMino,Sasaki,andTanakaandalsoQuinnandWald'sgravitationalradiationreactionincurvedspacetime[ 2 3 4 5 6 ].Theseformalschemesaretheoreticallywelldevelopedandprovideagoodfoundationforradiationreactionincurvedspacetime.However,thepractical,quantitativecalculationsofradiationreactionremainachallenge.Thedicultyliesinthe\tail"integraltermsappearingintheequationsofmotion:itisextremelydiculttodeterminepreciselytheretardedGreen'sfunctionsintheintegralsforgeneralgeometryandforgeneralgeodesicofparticle'smotion.Someattemptsweremadetoevaluatetheself-forcebycomputingthose\tail"integraltermsdirectly,buttheirapplicationshadtobelimitedtotheproblemshavingcertainsymmetriesandconditionsthatwouldsimplifytheGreenfunctionsintheintegrals[ 7 8 ].Hence,formorerealisticphysicalproblems,inwhichspecialconditionsandrestrictionsmightnotbealwaysexpected,dierentschemesofcalculationswouldbedemandedtocomputethe\tail"integralterms,thencetheself-force. InSection 3.1 werevisitthegeneralformalschemesandreviewbrieythestructureoftheequationsofmotionfortheself-forceforeachcasefromDiractoMino,Sasaki,andTanaka,andQuinnandWald[ 2 3 4 5 6 ].Then,Section 3.2 presentstwoexamplesofthepurelyanalyticattemptstotheself-forcecalculations,inwhichthetailintegraltermsaredirectlycalculatedastheretardedGreen'sfunctionsaresimpliedbysomespecialconditions.DewittandDewitt[ 7 ]and 33

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PfenningandPoisson[ 8 ]areprovidedastheexamples.Analternativeschemefortheself-forcecalculations,whichhasbeendevisedtoworkformoregeneralproblems,isahybridofbothanalyticalandnumericalmethods.Thiswillbethemainapproachthatthisdissertationisgoingtotake,andweleaveitsfulldiscussionforthenexttwoChapters. 3.1.1Dirac:RadiatingElectronsinFlatSpacetime 2 ]derivedthefollowingequationofmotionusingtheconservationofthestress-energytensorinsideanarrowworld-tubesurroundingtheparticle'sworld-line, 3e2...zaz2_za;(3-1) whereFabin=@aAb@bAarepresentstheincidentelectromagneticeldandthesecondtermontherighthandside,knownastheAbraham-Lorentz-Dirac(ALD)force,resultsfromtheradiationeldproducedbythemovingelectron.Inthisanalysis,theretardedelectromagneticeldisdecomposedintotwoparts: 2Fabret+Fabadv(i)+1 2FabretFabadv(ii):(3-2) Therstterm(i)ontherighthandsideofEq.( 3-2 )isthesolutionoftheinhomo-geneousequation withthecharge-currentdensity andcorrespondstotheeldresemblingtheCoulombq=rpieceofthescalarpotentialneartheparticle,whichdoesnotcontributetotheforceontheparticle

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itself.Andthethesecondterm(ii),denedastheradiationeld,comesfromthehomogeneoussolutionoftheequation andiscompletelyresponsiblefortheALDforce. ItturnsoutthatintheabsenceoftheincidenteldFabin,theonlyphysicalsolutionofEq.( 3-1 )is_va=0,i.e.geodesicmotion,hencethereisnoself-forceontheparticle. 3 ]generalizedDirac'sapproach[ 2 ]tothegeneralcurvedspacetime.UsingtheHadamardexpansiontechniquesforthevectoreldincurvedspacetime,theequationofmotionforthechargedparticleturnsouttobe 3e2(...zaz2_za)+1 3e2(Rab_zb+_zaRbc_zb_zc)+lim"!0e2_zbZ"r[bGreta]c0(z();z(0))_zc0(0)d0; whereGretaa0(z();z(0))isabi-vectorretardedGreen'sfunctionforthevectorwaveequationincurvedspacetime withthecurrentdensity (gaa0(x;z):bi-vectorofgeodesicparalleldisplacement). TheintegralterminvolvingthisGreen'sfunctioninEq.( 3-6 )isoftencalledthe\tail"part,givinganimplicationthattheparticle'smotionisaectedbytheentirehistoryofthesource.Thistailterm,togetherwiththethirdtermonthe

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righthandsideofEq.( 3-6 )resultsfromtheparticle'smotionandthecurvatureofspacetime,andwillmaketheparticledeviatefromitsoriginalworld-linetotheorderofe2,evenintheabsenceofanexternalincidenteldFabin.Hence,radiationdampingisexpectedtooccurevenforaparticleinfreefall. 6 ]wasabletoderivetheequationofmotionforascalarpointparticlemovingincurvedspacetimeas 3q2(...zaz2_za)+1 6q2(Rab_zb+_zaRbc_zb_zc)1 12q2R_za+lim"!0q2Z"raGret(z();z(0))d0; whereGret(z();z(0))isabi-scalarretardedGreen'sfunctionforthescalarwaveequationincurvedspacetime withthescalarchargedensity Again,wehavea\tail"terminvolvingtheGreen'sfunctioninEq.( 3-9 ),whichgivesthesameimplicationasthatofDewittandBrehme's.Similarlyasinthecaseofelectromagneticvectoreld,thelastthreetermsincludingthistailtermontherighthandsideofEq.( 3-9 )resultfromtheparticle'smotionandthecurvatureofspacetime,andwillberesponsibleforradiationreactionofthescalarparticleinfreefall.

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4 ]andQuinnandWald[ 5 ]obtainedthefollowingequationofmotion 2rahbcinrbhacin1 2_za_zdrdhbcin_zb_zc+lim"!0m2_zb_zcZ"1 2raGretbca0b0(z();z(0))rbGretcaa0b0(z();z(0))1 2_za_zdrdGretbca0b0(z();z(0))_za0(0)_zb0(0)d0; whereGretaba0b0(z();z(0))isabi-tensorretardedGreen'sfunctionforthetensorwaveequationincurvedspacetime withthetrace-reversedelddenedby habhab1 2(hcc)gab(3-14) andthestress-energytensorgivenby andwiththeharmonicgaugecondition hab;b=0:(3-16) ItshouldbenotedthattheAbraham-Lorentz-Dirac(ALD)term(...zaz2_za)isabsentfromtheequationofmotion( 3-12 )unliketheothercases:ithasbeeneectivelydroppedobythereductionoforderprocedure.The\tail"term

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ispresent,givingthesameimplicationasintheothercases,andistheonlycontributiontotheself-forceforthegravitationaleld. 3.2.1DewittandDewitt:FallingCharges 7 ]computedtheself-forceontheelectricchargefallingfreelyinthenon-relativisticlimitofsmallvelocitiesinastaticweakgravitationaleldwhichischaracterizedby(inharmoniccoordinates) rab;(3-17) whereGisthegravitationalconstantandMisthetotalmasscontainedinthespacetime.Bysimplifyingthebi-vectorretardedGreen'sfunctioninthislimit,theywereabletoevaluatethe\tailintegral"inEq.( 3-3 )directly.Theresultofcalculationshowsthat,inthislimit,theforceseparatesnaturallyintothefollowingtwoparts(lookingintothespatialcomponentsoftheforce): 3e2_r~r~rGM r; whereFCisaconservativeforcewhicharisesfromthefactthatthemassoftheparticleisnotconcentratedatapointbutispartlydistributedaselectriceldenergyinthespacesurroundingtheparticle,andFNCisanon-conservativeforcewhichgivesrisetoradiationdamping,havingalineardependenceonboththevelocityofthechargeandthecurvatureofthebackgroundgeometry. C=e2GM

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Thisisshowntomaketheretrogradecontributiontotheperihelionprecession, whereaisthesemi-majoraxisoftheorbit,isitseccentricity,andre=e2=m,theclassicalradiusoftheparticle. FromEq.( 3-17 ) r;(3-21) andthisleadsto 2h00;ij=@2 r:(3-22) Thenwiththis,FNCcanberewrittenintheform 3e2Ri0j0_xj;(3-23) whichshowsdirectlythatthedampingeectcomesfromthecurvatureofspace-time.FNCmaybewritteninanotherformbymakinguseoftheundampedequationofmotion r=~rGM r(3-24) astherstapproximation.Then,onegets 3e2...r;(3-25) whichisinagreementwiththeatspacetimetheory.Fromthis,anintegrationbypartsgives Eorbit=2 3e2Zorbitr2dt;(3-26) whichexpresseseitherthetotalenergylossforanunboundorbitorthelossinoneperiodforaboundorbit.ThiswouldbeidenticalwiththeeectfromthetraditionaldampingtermofEq.( 3-6 )whichisusedforaccelerationscausedby

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nongravitationalforces.Whengravitationalforcesarepresentalone,itisimportanttonotethatthephenomenonofpreaccelerationdoesnotoccurasitwouldbearguedbyEq.( 3-25 ),sinceEq.( 3-23 )showsthatthenonconservativeforcedependsonthevelocityoftheparticleratherthanits...r. FromEq.( 3-18 ),theeectofradiationdampingrepresentedbyFNCisnegligiblysmallinmagnitudecomparedtotheconservativeforceFC,owingtothedependenceofFNConthevelocityoftheparticle.Hence,itsexperimentaldetectionwouldbevirtuallyimpossible,andallthediscussionsaboveonradiationdampingwouldbeofconceptualinterestonly. 8 ]calculatedtheself-forceexperiencedbyapointscalarchargeq,apointelectricchargee,andapointmassmmovinginaweaklycurvedspacetimecharacterizedbyatime-independentNewtonianpotential.AsitwasinDewittandDewitt[ 7 ],thematterdistributionresponsibleforthispotentialisassumedtobebounded,so M r(3-27) atlargedistancesrfromthematter,whosetotalmassisM(withtheconventionG=c=1).Theprocedureofcalculatingtheself-forceissimilartoDewittandDewitt[ 7 ],i.e.rstcomputingtheretardedGreen'sfunctionsforscalar,electromagnetic,andgravitationaleldsintheweaklyspacetime,andthenforeachcaseofeldevaluatingthe\tailintegral"overtheparticle'spastworld-line. Forthescalarcharge,theresultis r3^r+1

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whereisadimensionlessconstantmeasuringthecouplingofthescalareldtothespacetimecurvature,and^risaunitvectorpointingintheradialdirection,andg=~ristheNewtoniangravitationaleld.Here,isintroducedtoimplythattheconservativetermdisappearswhentheeldisminimallycoupled. Fortheelectriccharge,thesameresulttoDewittandDewitt[ 7 ]isreproduced, r3^r+2 3q2dg Forthepointmassparticle,theconservativeforcevanishesandonlythenon-conservative(radiation-reaction)forceispresent, 3m2dg where()signimpliesradiation\antidamping".However,thisresultforthegrav-itationalself-forcehassomeproblemsofinterpretation:(i)Aradiation-reactionforceshouldnotappearintheequationofmotionatthislevelofapproxima-tion,whichcorrespondsto1.5post-Newtonianorder.(ii)Itshouldnotgiverisetoradiationantidamping.Theseproblemscanberesolvedbyincorporatinga\matter-mediatedforce"intotheequationofmotion:thematter-mediatedforceoriginatesfromadisturbedspacetimewhichhasbecomelocallynon-vacuumduetothechangesinitsmassdistributioninducedbythepresenceofaparticleintheregion.Itisobtainedas 3m2dg wherethersttermrepresentsthechangeintheparticle'sNewtoniangravitationaleldassociatedwithitsmotionaroundthexedcentralmass,thesecondtermisapost-NewtoniancorrectiontotheNewtonianforcemg,andthethirdtermisaradiationdampingterm.WhenthetwoforcesfromEqs.( 3-30 )and( 3-31 )are

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combined,theradiationdampingforcecancelsouttheantidampingforce,sotheequationofmotionisconservative,anditagreeswiththeappropriatelimitofthestandardpost-Newtonianequationofmotion.

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InChapter 3 wereviewedtheanalyticapproachestotheself-forcecalculationsbyDewittandDewitt[ 7 ]andPfenningandPoisson[ 8 ].Fromthereviews,itwasmeaningfultoseethattheircalculationsshowdesiredcorrespondencelimittotheatspacetimetheoryortheagreementwiththelow-orderpost-Newtonianapproximations.Intheircalculationsoftheself-force,however,specialconditionssuchasnon-relativisticvelocitiesofparticlesandaweakgravitationaleldhadtobeimposedtoenabletheentirecalculationstobetreatedfullyanalytically.Morerealisticself-forceproblemshavingmoregeneralconditionswouldrequirecompletelydierentapproachesforthecalculations,inwhichwecombinebothanalyticalandnumericalmethods. Here,weintroduceanalternativeapproachtotheself-forcecalculations,knownasthe\mode-sum"method,whichwasoriginallydevisedbyBarackandOri[ 9 ].Employingbothanalyticalandnumericaltechniques,thismethoddoesnotlimittheparticle'svelocitiesandtheeld'sstrength,andshouldpracticallyworkforallkindsofeldsunderconsideration,whetheritisscalar,electromagnetic,orgravitational.Itisparticularlypowerfulfortheproblemsinasphericallysymmetricspacetime,suchasSchwarzschild.Inprinciple,wetakeadvantageofthesphericalsymmetryofthebackgroundgeometrytodecomposetheretardedGreen'sfunctioninthe\tail"termintospherical-harmonicmodeswhichcanbecomputedindividually.Then,fromthemode-decompositionoftheretardedGreen'sfunctionweobtainamode-decompositionoftheretardedeld,andfromthissubtractamode-decompositionofthesingulareld,whichislocallywelldescribed.The 43

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InthisChapterwedealwiththeself-forceofascalarchargeorbitingaSchwarzschildblackhole.Section 4.1 introducesarecentmethodtosplittheretardedscalareldincurvedspacetimesuggestedbyDetweilerandWhiting[ 16 ].ThisfollowsDirac'sideainhisatspacetimeproblem[ 2 ],andgivesgoodinterpretationsnotonlyinthesingularbehavioroftheretardedeld,butalsoinitsdierentiability.InSection 4.2 wegivetheoverviewofthemode-summethodoriginatedbyBarackandOri[ 9 ],andpresenttheanalyticresultsforthesingulareld,i.e.theregularizationparameters,whichwereobtainedbyKim[ 17 ].Theregularizationparametersarelocallywelldenedandshoulddescribethesingularbehaviorandthedierentiabilityoftheeldprecisely.Higherorderexpansionsofthesingulareldwillgeneratehigherorderregularizationparameters,andtheiruseinthemodesumsfortheself-forcecalculationwillresultinmorerapidconvergenceandmoreaccuratenalresults.Tofacilitatethecomputationsoftheregularizationparameters,anin-depthanalysisofthelocalspacetimewouldbedemanded,andinSection 4.3 wedevelopanelaborateperturbationanalysisofthelocalgeometryforthispurpose.Then,Section 4.4 isdevotedtothecalculationsoftheregularizationparameters.Theseresultsarethencombinedwiththenumericalcomputationsoftheretardedeldtoprovidetheself-forceultimately.ThisnaltaskisdoneintheSection 4.5

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4.1.1ConventionalMethodofSplittingtheRetardedField 2 ]rstgavetheanalysisoftheself-forcefortheelectromag-neticeldofaparticleinatspacetime.Hewasabletoapproachtheprobleminaperturbativeschemebyallowingtheparticle'ssizetoremainniteandinvokingtheconservationofthestress-energytensorinsideanarrowworld-tubesurroundingtheparticle'sworld-line.Inhisanalysis,theretardedeldisdecomposedintotwoparts:(i)Therstpartisthe\meanoftheadvancedandretardedelds"whichisasolutionoftheinhomogeneouseldequationresemblingtheCoulombq=rpieceofthescalarpotentialneartheparticle.(ii)Thesecondpartisa\radiation"eldwhichisahomogeneoussolutionofMaxwell'sequations.Diracdescribestheself-forceastheinteractionoftheparticlewiththeradiationeld,awell-denedvacuumeldsolution. Intheanalysesoftheself-forceincurvedspacetime,rstbyDewittandBrehme[ 3 ],andsubsequentlybyMino,Sasaki,andTanaka[ 4 ],byQuinnandWald[ 5 ]andbyQuinn[ 6 ],theHadamardformoftheGreen'sfunctionisemployedtodescribetheretardedeldoftheparticle.Traditionally,takingthescalareldcaseforexample,theretardedGreen'sfunctionGret(p;p0)isdividedinto"direct"and"tail"parts:(i)Therstparthassupportonlyonthepastnullconeoftheeldpointp.(ii)Thesecondparthassupportinsidethepastnullconeduetothepresenceofthecurvatureofspacetime.Accordingly,theself-forceontheparticleconsistsoftwopieces:(i)Therstpiececomesfromthedirectpartoftheeldandtheaccelerationoftheworld-lineinthebackgroundgeometry;thiscorrespondstoAbraham-Lorentz-Dirac(ALD)forceinatspacetime.(ii)Thesecondpiececomesfromthetailpartoftheeldandispresentincurvedspacetime.Thus,thedescriptionoftheself-forceincurvedspacetimereducestoDirac'sresultintheatspacetimelimit.

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Inthisapproach,theself-forceisconsideredtoresultvia fromtheinteractionofthechargewiththeeld Nowselfisinvestigatedinthefollowingmanner.Wehavethescalareldequation where isthesourcefunctionforascalarchargeqmovingalongaworld-line,describedbyp0(),withrepresentingthepropertimealongtheworld-line.ThiseldequationissolvedintermsofaGreen'sfunction, Thescalareldofthischargeisthen DewittandBrehmeanalyzethescalareldincurvedspacetimeusingtheHadamardexpansionsoftheGreen'sfunctionnear.Abi-scalarquantity(p;p0),termedSynge's\worldfunction"[ 13 ]isdenedashalfofthesquareofthedistancemeasuredalongageodesicfromptop0,and<0foratimelikegeodesic,=0onthepastandfuturenullconesofp,and>0foraspacelikegeodesic.TheusualsymmetricscalareldGreen'sfunctionisderivedfromtheHadamardformtobe

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8[u(p;p0)()v(p;p0)()];(4-7) whereu(p;p0)andv(p;p0)arebi-scalarsdescribedbyDewittandBrehme,andtheirexpansionsareknowntobeconvergentwithinaniteneighborhoodofifthegeometryisanalytic.Inthevicinityof,DewittandBrehmeshowthat 12Rabrarb+O(3=R3);(4-8) andthat 12R(p0)+O(=R3);(4-9) whereistheproperdistancefromptomeasuredalongthespatialgeodesicwhichisorthogonalto,andRrepresentsalengthscaleofthebackgroundgeometry(thethesmallestoftheradiusofcurvature,thescaleofinhomogeneitiesandtimescaleforchangesincurvaturealong).The()guaranteesthatonlywhenpandp0aretimelike-relatedisthereacontributionfromv(p;p0).InanyGreen'sfunctionthetermscontaininguandvarefrequentlyreferredtoasthe\direct"and\tail"parts,respectively.Also,theretardedandadvancedGreen'sfunctionsareexpressedintermsofGsym(p;p0)as, respectively,where[(p);p0]=1[p0;(p)]equals1ifp0isinthepastofaspacelikehypersurface(p)thatintersectsp,andis0otherwise.AsDiracdecomposedtheretardedelectromagneticeldFretintotwopartsasinEq.( 3-2 ),wemaytrytodecomposeourscalareldretinto

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wheresymdirect1 2retdirect+advdirectandraddirect1 2retdirectadvdirectsuchthatretdirect=symdirect+raddirect.Weseparatesymdirectfromtherestontherighthandsideoftheaboveequationsincethistermissingularandexertsnoforceontheparticle.Then,wesingleoutraddirectandrettailfromEq.( 4-11 ),whicharetheonlycontributionstotheforceontheparticle,andmaywritedown WiththehelpofEqs.( 4-6 ),( 4-7 )and( 4-10 )togetherwiththedenitionofraddirectabove,onecanexpressEq.( 4-12 )as 2_advretqZretv[p;p0()]d;(4-13) whichgivesourself-forceviaEq.( 4-1 ). Althoughthistraditionalapproachprovidesadequatemethodstocomputetheself-force,itdoesnotsharethephysicalsimplicityofDirac'sanalysiswheretheforceisdescribedentirelyintermsofanidentiable,vacuumsolutionoftheeldequations[ 16 ]:unlikeDirac'sradiationeld,theselfinEq.( 4-13 )isnotasolutionofthevacuumeldequationr2=0. Inaddition,theselfisnotfullydierentiableontheworld-line.TherstterminEq.( 4-13 )isniteanddierentiableinthecoincidencelimit,p!.Thisterm,infact,providesthecurvedspacetimegeneralizationoftheALDforce,andiseventuallyexpressedintermsoftheaccelerationofandcomponentsoftheRiemanntensorvialocalexpansionsofu(p;p0)and_(p;p0)asinRefs.[ 3 4 5 6 ].TheintegralterminEq.( 4-13 )comesfromthetailpartoftheGreen'sfunction.

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Takingitsderivativewithrespecttoxa,thecoordinatesforp,oneobtains[ 6 ] 12(xax0a)+O(=R3);p!; whereEq.( 4-9 )wasusedforv[p;p0(ret)]near.ThespatialpartoftherighthandsideofEq.( 4-14 )isnotdenedwhenpison,thusthedierentiabilityisnotguaranteedingeneralontheworld-lineiftheRicciscalarofthebackgroundisnotzero|similarly,theelectromagneticpotentialAtailaandthegravitationalmetricperturbationhtailabarenotdierentiableatthepointoftheparticleunlessRab1 6gabRubandRcadbucud,respectively,arezerointhebackground[ 16 ].Therefore,inordertoobtainawelldenedcontributiontotheself-forceoutofthetailpart,onerstaveragesraselfoverasmall,spatialtwo-spheresurroundingtheparticle,thusremovingthespatialpartofEq.( 4-14 ),thentakesthelimitofthisaverageastheradiusofthetwo-spheretendstozero[ 3 4 5 6 ]. 16 ],anewsymmetricGreen'sfunctioncanbeconstructedbyaddingtotherstinEq.( 4-7 )anybi-scalarwhichisahomogeneoussolutionofEq.( 4-5 ).DewittandBrehme[ 3 ]showthatthesymmetricbi-scalarv(p;p0)isasolutionofthehomogeneouswaveequation, Then,usingthiswegenerateanewsymmetricGreen'sfunction 8v(p;p0)=1 8[u(p;p0)()+v(p;p0)()]: ThisnewsymmetricGreen'sfunctionhassupportonthenullconeofp,justasGsymdoes,andhassupportoutsidethenullcone,butnotwithinthenullcone,

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unlikeGsym.WeconsiderGS(p;p0)onlyinalocalneighborhoodoftheparticle,thustheuseofGS(p;p0)isnotcomplicatedbytheneedforknowledgeoftheentirepasthistoryofthesourceandisamenabletolocalanalysis.ByEqs.( 4-6 )and( 4-16 ),thecorrespondingeldis 2j_jret+qu[p;p0()] 2j_jadv+q whichisaninhomogeneoussolutionofEq.( 4-3 )justasretis,andisanalogoustoDirac'ssingulareld1 2Fabret+Fabadv.FollowingDirac'spioneeringidea,onecandene ItisremarkablethatlikeGret(p;p0),GR(p;p0)hasnosupportinsidethefuturenullcone.CorrespondingtoGR(p;p0),weconstruct 2_advretqZret+1 2Zadvretv[p;p0()]d; whichisanalogoustoDirac'sradiationeld1 2FabretFabadv. AsbothretandSareinhomogeneoussolutionsofthesamedierentialequation,Eq.( 4-3 ),consequently,R,asdenedbytherstlineofEq.( 4-19 )isahomogeneoussolutionandthereforeexpectedtobedierentiableon.TherelationbetweenRandselfis Hereisobservedaresultofgreatsignicance:Rcanreplaceselfforanex-plicitcomputationoftheself-force,sincetheintegralterminEq.( 4-20 )givesnocontributiontoaself-force.Foraeldpointpnear,viaEq.( 4-9 )and

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4-20 )is 12qR(p)+O(2=R3);p!:(4-21) Takingthederivativeoftherighthandsideofthisequationgives 1 12qR(p)ra+O(=R3)=qR(p) 12(xax0a)+O(=R3);p!:(4-22) WhenthisresultiscombinedwithEq.( 4-14 )viaEq.( 4-20 ),thetroublesomepartofraselfinEq.( 4-14 )iscanceledbyitsnegativecounterpartinEq.( 4-22 ),andwesimplyendupwith wheretheremaindertermO(=R3)vanishesinthelimitthatpapproachesandgivesnocontributiontotheself-force. FortherestofthisChapter,Rreplacesselfforanexplicitcomputationoftheself-force,andthealternativesplitofretisadopted,namely whereSistermedtheSingularSourceeld,andRtheRegularRemaindereld.WedetermineananalyticalapproximationofSviaamultipoleexpansion,thensubtractthisfromthenumericalsolutionofretfortheultimatecalculationoftheself-force. 9 ]suggestedamethodtoanalyzesuchproblemswhenthebackgroundspacetimeissphericallysymmetric

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bycombiningbothanalyticalandnumericalcomputations.Intheiranalysis,theself-forcemaybeconsideredtobecalculatedfrom wherep0istheeventonwheretheself-forceistobedeterminedandpisaneventintheneighborhoodofp0,andFa(p)isrelatedto(p)viaEq.( 4-1 ).Foruseofthisequation,bothFreta(p)andFdira(p)wouldbeexpandedintomultipole`-modes,i.eP`Fret`a(p)andP`Fdir`a(p),respectively,whereFret`a(p)isdeterminednumericallyandFdir`a(p)determinedanalytically.InordertodetermineFret`a(p),wesolveEq.( 4-3 )usingsphericalharmonicexpansions.Thesource%intheequationisexpandedintosphericalharmonics,andsimilarlytheeldretisexpanded whereret`m(r;t)isfoundnumerically.Theindividualcomponentsret`m(r;t)inthisexpansionareniteatthelocationoftheparticleeventhoughtheirsum,therighthandsideofEq.( 4-26 )issingular.Thenwehave whichisalsonite.TheremainingpartFdir`a(p)isdeterminedbyalocalanalysisoftheGreen'sfunctionintheneighborhoodoftheparticle'sworld-line.Ref.[ 9 ]provides limp!p0Fdir`a(p)=`+1 2Aa+Ba+Ca 2+O(`2);(4-28) whereAa,BaandCaareconstantsandaregenericallyreferredtoasRegularizationParameters.Theremainderisdenedas 2AaBaCa 2;(4-29)

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53 whichisshowntovanish[ 9 ].Thentheself-forceisultimatelycalculatedas F self a = 1 X ` =0 lim p p 0 F ret `a ( p ) ` + 1 2 A a B a C a ` + 1 2 D 0 a (4-30) OurapproachcloselyfollowsBarackandOri[ 9 ],butthereissomedierencein theregularizationschemeduetoourdierentsplitof ret asdescribedin Eq.( 4-24 ).Fromourperspective,viaEq.( 4-1 )theself-forcecanbeexplicitly evaluatedfrom F self a =lim p p 0 F ret a ( p ) F S a ( p ) = F R a ( p 0 ) = q lim p p 0 r a ( ret S )= q r a R ; (4-31) wheresimilarlytotheabove,weexpandboth F ret a ( p )and F S a ( p )intomultipole ` modes P ` F ret `a ( p )and P ` F S `a ( p ),respectively,with F ret `a ( p )determinednumerically and F S `a ( p )determinedanalytically.Thisimpliesthatourself-forceis F self a = X ` lim p p 0 F ret `a ( p ) F S `a ( p ) = X ` F R `a ( p 0 ) = q X ` lim p p 0 r a X m ( ret `m S `m ) Y `m = q X ` lim p p 0 r a X m R `m Y `m ; (4-32) evaluatedatthelocationoftheparticle.Heretheindividual `m components ret `m and S `m areniteatthelocationoftheparticleeventhoughtheirsums arebothsingular.The ` -modederivatives F ret `a = q r a P m ( ret `m Y `m )and F S `a = q r a P m ( S `m Y `m )arealsoniteatthepointoftheparticle,andwetakethe dierencebetweenthetwo,whichis F R `a = q r a P m R `m Y `m ,thentakethesum ofthisquantityover ` ,whichproducesaconvergentvaluefortheself-force. OurcomputationoftheretardedeldpartisidenticaltothatofBarackand Ori,butour ` mode-decompositionofthesingulareldpart,i.e. F S `a isslightly dierentfromtheir F dir `a .Wedescribe F S `a inthecoincidencelimit p p 0 viathe regularizationparameters

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limp!p0FS`a=`+1 2Aa+Ba+Ca2p wherethersttwotermslookjustidenticaltothoseinEq.( 4-28 ),butthethirdtermforCalooksdierentfromitscounterpart.OurregularizationparametersareclassiedintermsofsingularityanddierentiabilityofFS`ainthelimitp!p0,namelyinto2-order,1-order,0-order,1-orderterms,etc.(seeSection 4.4 ),andallthe`-dependencesassociatedwiththem,asseeninEq.( 4-33 ),naturallyre-sultfromthemultipoledecompositionofFS`aviaLegendrepolynomialexpansions.WewillseelaterinSection 4.4 thatour0-ordertermhasnocleardependenceon`.ButinBarackandOri[ 9 ]L=`+1 2isintroducedasaperturbationfactorandlimp!p0Fdir`aisexpandedasapowerseriesinL,inwhichthethirdtermgainsLinitsdenominatorasshowninEq.( 4-28 ).Thisdiscrepancybetweenthetwoapproaches,however,isresolvedbythefactthatCavanishesalways.WewillprovethisinSection 4.4 .Also,oneshouldnotethatourlastparameterDaisdeneddif-ferentlyfromD0ainEq.( 4-29 )(notethedierenceinnotation).OurDaoriginatesfromthenon-singularbutnon-dierentiablebehaviorsoftheeldintheneighbor-hoodoftheworld-lineoftheparticle.Again,itscoecient2p InSection 4.4 wepresentindetailthederivationsofalltheseregularizationparameters.Theresultsaresummarizedasbelow: ;(4-34) f;(4-35)

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23=2;(4-39)

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whererro,andEut=(12M=ro)(dt=d)o(:propertime)andJu=r2o(d=d)oaretheconservedenergyandangularmomentum,respectively,and_rur=(dr=d)o,f(12M=ro),(1+J2=r2o).Thesubscriptodenotesevaluationatthelocationoftheparticle.Also,shorthandforthehypergeometricfunctionisFp2F1p;1 2;1;J2=(r2o+J2)(seeAppendix A formoredetailsaboutthehypergeometricfunctionsandtherepresentationsoftheregularizationparametersintermsofthem). 4.3.1IntroductionofTHZCoordinates 2 ].Infact,thisintuitioncanbesupportedviasomelocalanalysisofS.IfSresemblestheCoulombq=rpieceofthescalarpotentialneartheparticlewithscalarchargeq,whereristhedistancebetweenasourcepointp0andanearbyeldpointp,wecanthinkofSastheeldmeasuredbyalocalobserversittingontheparticle,towhomthebackgroundgeometryinthevicinityofhislocationlooksat.ThedescriptionofSwillthenbeadvantageouslysimpleinthisobserver'sframeofreference,andwearemotivatedtousesome

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19 ].Normalcoordinatesforageodesic,however,arenotuniqueandhaveanambiguityatO(3),whereistheproperdistancefromptomeasuredalongthespatialgeodesicwhichisorthogonalto.Forexample,dierencesofO(3)distinguishRiemannnormalfromFerminormalcoordinates[ 19 ].ForourpurposesanormalcoordinatesystemintroducedbyThorneandHartle[ 20 ]andlaterextendedbyZhang[ 21 ](henceforth,referredtoasTHZnormalcoordinatesystem)isparticularlyadvantageous.ItwillbeshownlaterinSubsection 4.3.2 thatinthiscoordinatesystemthescalarwaveequationtakesasimpleformandthatasaresultweobtain whereRrepresentsalengthscaleofthebackgroundgeometry.TheapproximationinEq.( 4-45 )isaccurateenoughforself-forceregularizationbecause andtheO(2=R4)remaindervanishesinthecoincidencelimitp!p0. TheTHZcoordinatesXA=(T;X;Y;Z)associatedwithagivengeodesichavethefollowingfeatures[ 21 ]: (i) LocallyinertialandCartesian;morespecically,on,gAB=ABand@CgAB=0.AndTmeasuresthepropertimealongthegeodesic;andX=Y=Z=0on.Also,themetricisexpandableaboutinpowersofp X2+Y2+Z2inaparticularformlike

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withp0andp+q2. (ii) ThecoordinatessatisfythedeDondergaugecondition wheregABp ggAB. ThemetricperturbationinTHZcoordinatesisdescribedas[ 18 ] with 3KPQBQIXPXIdTdXK20 21_EIJXIXJXK2 52_EIKXIdTdXK+5 21XIJPQ_BQKXPXK1 52PQI_BJQXPdXIdXJ and 3EIJKXIXJXK(dT2+KLdXKdXL)+2 3KPQBQIJXPXIXJdTdXK+O(4=R4)IJdXIdXJ; whereABistheatMinkowskimetric,IJKistheatspaceLevi-Civitatensor,=(X2+Y2+Z2)1=2,andtheindicesI,J,K,L,PandQarespatialandraisedandloweredwiththethreedimensionalatspacemetricIJwhiletheindex0denotesthetimecomponent.Theexternalmultipolemomentsarespatial,symmetric,tracefreetensorsandaredenedintermsoftheRiemanntensor

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evaluatedonas 2IPQRPQJ0;(4-53) 8IPQrKRPQJ0STF;(4-55) whereSTFmeanstotakethesymmetric,tracefreepartwithrespecttothespatialindicesI,J,:::andthedotdenotesdierentiationofthemultipolemomentwithrespecttoTalong.Dimensionally,EIJBIJO(1=R2)andEIJKBIJK_EIJ_BIJO(1=R3).ThefactthatalloftheexternalmultipolemomentsaretracefreecomesfromtheassumptionthatthebackgroundgeometryisavacuumsolutionoftheEinsteinequations. TheTHZcoordinatesareaspecialkindofharmonic(ordeDonder)coordi-nates.Wemayexpresstheperturbedeldbydening HABABgAB;(4-56) wheregABp ggAB.Acoordinatesystemisharmonicifandonlyif Zhang[ 21 ]providesanexpansionofgABforanarbitrarysolutionofthevacuumEinsteinequationsinTHZcoordinates,inhisequation(3.26).Intheleadinglowerorderterms,themetricperturbationHABinthisexpansionisdescribedas[ 18 ] HAB=2HAB+3HAB+O(4=R4);(4-58)

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where 3KPQBQIXPXI+10 21_EIJXIXJXK2 5_EIKXI22HIJ=5 21X(IJ)PQ_BQKXPXK1 5PQ(I_BJ)QXP2 and 3EIJKXIXJXK3H0K=1 3KPQBQIJXPXIXJ3HIJ=O(4=R4): ThemetricperturbationHABisthetracereversedversionofHABatlinearorder, 2gABHCC;(4-61) andtheexpansionshowninEqs.( 4-49 )-( 4-51 )preciselycorrespondstoZhang's[ 21 ],therstleadingtermsofwhichareexpressedinEqs.( 4-58 )-( 4-60 ). 4-45 ).TheresultinthisequationcanbederivedviaEq.( 4-17 ).WedeveloplocalexpansionsintheTHZcoordinatesfortheelementsu(p;p0),_andv(p;p0)ontherighthandsideoftheequation,andcombinethemtogiveanapproximateexpressionforS. First,foravacuumspacetime(RAB=0)whichisnearlyat,accordingtoThorneandKovacs[ 22 ]wehave

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Whenthesourcepointp0ison,Synge'sworldfunction(p;p0)isparticularlyeasytoevaluateinTHZcoordinatesforpclosetop0.TheworldfunctionisshownbyThorneandKovacs[ 22 ]tobe 2XAXBAB+ZCHABd+O(6=R4);(4-63) whereXAistheTHZcoordinaterepresentationoftheeldpointpwhilethesourcepointp0isrepresentedby(T0;0;0;0)[ 18 ].Theintegrationofthecoordinatealongastraightpathisgivenby whererunsfrom0to1. Workingthroughonlythelowerorderexpansionsoftheperturbedeld,namelyHAB=2HAB+3HAB+O(4=R4),theintegralofHABalongthestraightpathCisevaluatedwiththehelpofEq.( 4-64 )tobe[ 18 ] 3EKLMKLMd+O(4=R4)=1 3EKLXKXL1 12EKLMXKXLXM+O(4=R4); 3IKPBPLKL10 21_EKLKLI+4 21_EKIKLL+1 3IKPBPLMKLMd+O(4=R4)=2 9IKPBPLXKXL5 42_EKLXKXLXI+1 21_EKIXK2+1 12IKPBPLMXKXLXM+O(4=R4) (4-66)

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and 3IJEKLMKLM+5 21IKP_BPLKLJ1 21IKP_BPJKLLd+O(4=R4)=1 3IJEKLXKXL1 12IJEKLMXKXLXM+5 84IKP_BPLXKXLXJ1 84IKP_BPJXK2+O(4=R4): IntermsofHAB,Synge'sworldfunctionisexpressedas[ 18 ] 2XAXBAB+1 2(TT0)2H00+(TT0)XIHI0+1 2XIXJHIJ+O(6=R4)=1 2(1H00)(T0T+XIHI0)2XIXJ(IJ+HIJ)=(1H00)+O(6=R4): Withthesourcepointp0on,thesecondterminthesquarebracketsabovecanbemodiedwiththehelpofEqs.( 4-65 )and( 4-67 ), where SubstitutingEq.( 4-69 )intoEq.( 4-68 )andfactorizing,weobtain[ 18 ] 2(1H00)T0T+XIHI0(1+H00)T0T+XIHI0+(1+H00)+O(6=R4):

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Attheretardedtime,p0isonthepastnullconeemanatingfromp,where(p;p0)=0,andtherstfactorinthesquarebracketsinEq.( 4-71 )is duetothefactthatT0T<0andjT0Tj,andthesecondmustbe tocancelthetermO(6=R4)inEq.( 4-71 )suchthat(p;p0)=0precisely.Then,thedierentiationofEq.( 4-71 )withrespecttoT0,evaluatedattheretardedtimeisdominatedbythedierentiationofthesecondfactor, 2(1H00)T0T+XIHI0(1+H00)ret+O(6=R5)=1 2(1H00)2(1+H00)+O(5=R4)=1+O(4=R4); wherethesecondequalityfollowsfromthefactthatT0Tfortheretardedtime,andthethirdequalityfromEq.( 4-65 )[ 18 ]. Similarly,attheadvancedtime,p0liesonthefuturenullconeofp,where(p;p0)=0,andtherstandsecondfactorsinthesquarebracketsinEq.( 4-71 )nowreversetheirroles, and duetothefactsthatT0T>0andjT0Tjandthat(p;p0)=0.Then,thedierentiationofEq.( 4-71 )withrespecttoT0,evaluatedattheadvancedtimeis

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nowdominatedbythedierentiationoftherstfactor, 2(1H00)T0T+XIHI0+(1+H00)adv+O(6=R5)=1 2(1H00)2(1+H00)+O(5=R4)=1+O(4=R4); wherethesecondequalityresultsfromthefactthatT0Tfortheadvancedtime. DewittandBrehmeshowthatingeneral 12R(p0)+O(=R3);p!:(4-78) However,invacuumspacetime,whereR=0, Whenintegratedoverthepropertime,thedominantcontributionfromthistermisO(3=R4)inthecoincidencelimitp!. ThensubstitutingalltheresultsinEqs.( 4-62 ),( 4-74 ),( 4-77 )and( 4-78 )intoEq.( 4-17 ),weeventuallyobtain[ 18 ] +O(3=R4):Q:E:D:(4-80) FromEqs.( 4-49 )-( 4-51 ),onenotesthatourexpressionsofTHZcoordinatesarewelldeneduptotheadditionofatermO(5=R4),whichcorrespondstoatermO(4=R4)inthemetricperturbation.ThechangeinduetotheadditionofsuchatermisO(3=R4),whichwouldbeconsistentwiththeorderterminEq.( 4-80 ).Thedierentiabilityoftheordertermisofinterest,andatermofO(3=R4)isC2inthelimit!0.FromthefactthatR=retS,whereRisahomogeneoussolutionofthescalarwaveequation,wendthatEq.( 4-80 )clariestherelationshipbetweentheaccuracyofanapproximationforSandthe

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dierentiabilityofthesubsequentapproximationforR,andtheself-force@aR:iftheapproximationforSisinerrorbyaCnfunction,thentheapproximationforRisnomoredierentiablethanCnandtheapproximationfor@aRisnomoredierentiablethanCn1.Thisconcernofdierentiabilityisassociatedwithourlastregularizationparameters,Da-terms.AccordingtoEq.( 4-33 ),Da-termsaredeterminedbythe1-ordertermsof@aS,andcorrespondtotheaccuracy2=R3inS.ThisallowserrorsinSbyO(3=R4),whichisC2inthelimit!0.Then,ourself-force@aRisnomoredierentiablethanC1. ItwouldbeinstructivetogiveanintuitiveinterpretationofEq.( 4-80 )usingthefeaturesofTHZcoordinates.ThescalarwaveoperatorinTHZcoordinatesis[ 18 ] grArA=@AAB@B@AHAB@B=AB@A@BHIJ@I@J2HI0@(I@0)H00@0@0; wherethesecondequalityfollowsfromthedeDondergaugecondition,Eq.( 4-57 ).Whenisreplacedbyq=inEq.( 4-81 )[ 18 ], grArA(q=)=4q(3)(~X)+O(=R4);=R!0;(4-82) forwhichweusedEqs.( 4-58 )-( 4-60 )andthefactthatisindependentofT.AC2correctiontoq=,ofO(3=R4),wouldremovetheordertermontherighthandsideofEq.( 4-82 )andweareledtotheconclusionthatS=q=+O(3=R4)isaninhomogeneoussolutionofthescalareldwaveequation[ 18 ].TheerrorintheapproximationofSbyq=isC2. 4-80 ).Inourself-forceproblem,thisSdependsonthegeodesicofthe

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particleaswellasonthegeometryofthebackgroundspacetime.Thus,inordertoderivetheregularizationparametersfromthemultipolecomponentsofraS,whereadenotesthecoordinatesofthebackgroundgeometry,onerequiresthatinEq.( 4-80 )beexpressedintermsofthecoordinatesofthebackgroundgeometry.Forthispurpose,giventhedenition2=X2+Y2+Z2intheTHZcoordinates,weneedclarifytherelationshipbetweenthebackgroundcoordinates(t;r;;)andtheTHZcoordinates(T;X;Y;Z)associatedwithaneventp0ontheworld-line.InthisSubsectionweprovidetheexpressionsoftheTHZcoordinatesinthevicinityofthesourcep0movingonageneralorbit,intermsofexpansionsoftheSchwarzschildbackgroundcoordinatesuptothequarticorder.Theproceduretocompletethistaskcanbesummarizedinthefollowingsteps: (i) Findinitialinertialcoordinates^XA(A=0;1;2;3)intheneighborhoodoftheeventp0onintermsofTaylorexpansionsofthebackgroundcoordi-natesxa=(t;r;;)aboutxao,wherexaorepresentsp0inthebackgroundcoordinates(henceforth,thesubscriptodenotestheevaluationatp0).Thiscoordinateframeisstaticinthesensethattheeventp0isnotinmotionalongyet. (ii) ConstructFerminormalcoordinates,whichhavevanishingChristoelsymbolsalong.Withthemetriccomponentsbeingexpandableaboutinpowersofproperdistancetoforalltime,Ferminormalcoordinatesprovideastandardizedwayinwhichfreelyfallingobservercanreportobservationsandlocalexperiments[ 23 ]. (iii) DeterminethetransformationfromFerminormalcoordinatestoTHZcoordinatesandnallycombinethiswiththeresultsofSteps(i)and(ii). Step(i) Webuildinitialinertialcoordinates^XAviatheexpansionsoftheSchwarzschildcoordinatesxa=(t;r;;)aboutxao.Weinberg's[ 24 ]Eq.(3.2.12)showsthat

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^XA=^XAo+MAa(xaxao)+1 2MAaabcjo(xbxbo)(xcxco);(4-83) wherewemaychoose^XAo=0and forconvenienceasthischoicere-centersandre-scalestheSchwarzschildcoordinatesto^T=(12M=ro)1=2(tto),^X=(12M=ro)1=2(rro),^Y=rosino(o),^Z=ro(o).Takingadvantageofthesphericalsymmetryofthebackground,wemaytakeonlytheequatorialplaneo==2forthistransformation.Then,forEq.( 4-83 )thenon-zeroChristoelsymbolsevaluatedatxoare ttro=M fr2o;rttjo=fM r2o;rrrjo=M fr2o;rjo=fro;ro=fro;ro=ro=1 wheref12M ro.Thesecoordinatesarestaticandwehavenoinformationabouttheparticle'smotion. FollowingRef.[ 24 ],themetricexpansionsinthesecoordinatescanbedeter-minedvia ^gAB=gab@xa wherethebackgroundmetricgabcanbeexpandedaboutxaointheTaylorseries 2gab;cdjo(xcxco)(xdxdo)+1 6gab;cdejo(xcxco)(xdxdo)(xexeo)+O[(xxo)4]:

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TocarryoutthecalculationswithEqs.( 4-86 )and( 4-87 ),weneedinvert^XA(xa)inEq.( 4-83 )byiterationto 2apqoNpANqB^XA^XB+1 2apqoprsjoNqANrBNsC^XA^XB^XC1 84apqoprsjosuvjoNqANrBNuCNvD+apqoprsjoquvjoNrANsBNuCNvD^XA^XB^XC^XD+O(^X5); whereNaAistheinverseofMAasuchthatMAaNaB=ABandMAaNbA=ba.Then,usingEqs.( 4-86 )-( 4-88 )togetherwith( 4-84 )and( 4-85 ),weobtain ^gAB=AB+^HABCD=AB+^HABCD^XC^XD+^HABCDE^XC^XD^XE+O(^X4=R4) (4-89) orinthecontravariantform ^gAB=AB^HAB^gAB=AB^HABCD^XC^XD^HABCDE^XC^XD^XE+O(^X4=R4); wherethenon-zerocomponents^HABCD=^H(AB)(CD)and^HABCDE=^H(AB)(CDE)turnouttobe ^H0000=M2 r3o3M2 r3oM2 r2o;^H1212=^H1313=1 21 r3o;^H2222=f r2o;^H2233=1 r2o;

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and ^H00001=1 3f3=22M2 r4o6M2 3f1=22M r4o3M2 3f3=24M2 6f1=2M r4o3M2 3f3=22M2 r4o6M2 3f1=24 r4o+15M2 6f1=2M r4oM2 6f1=24 r4o+9M2 r4o;^H12233=f1=2 r4o;^H13003=1 6f1=2M r4oM2 6f1=24 r4o+9M2 r4o;^H13333=f1=2 r4o;^H22122=^H33133=2f1=2 r4o;^H22133=2f1=2 r4o: Thecomponents^HABCDand^HABCDEserveasbuildingblockstoevaluatethequantities^ABC;D,^RABCD,^ABC;DEand^RABCD;Eatthelocationoftheparticle ^ABC;Do=^HABCD+^HACBD^HBCAD; ^RABCDo=^HBCAD^HACBD^HBDAC+^HADBC; ^ABC;DEo=3^HABCDE+^HACBDE^HBCADE; ^RABCD;Eo=3^HBCADE^HACBDE^HBDACE+^HADBCE;

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andthesequantitiesareessentialfortransformingtheinitialcoordinates^XAintoFerminormalcoordinatesandthennallyintoTHZcoordinatesthroughSteps(ii)and(iii). Step(ii) ToconstructFerminormalcoordinatesoutoftheinitialcoordinates^XA,rstweevolvetheparticle'smotionalongfromtheinitialpoint^XAo=0,whichcorrespondstoxaointheSchwarzschildcoordinates.Sinceisageodesicoftheparticle'smotion,itstangentvector^uA,thefour-velocityoftheparticleistransportedparalleltoitselfalong ^uBrB^uA=0:(4-97) Wecallthetimelikegeodesic,andthetime-axisofanobserver'sframethatisco-movingwiththeparticleistangenttothisgeodesic.Whileanobserveristravelingwiththeparticlealong,hisspacetriadremainsorthogonalto|paralleltransportpreservesorthogonalityto[ 25 ],i.e. ^uBrB^nA(I)=0;(4-98) where^nA(I)(I=1;2;3)arebasisvectorsforthespacetriad,spanningthehyper-surfaceorthogonalto.Alongeachdirectionof^nA(I),thephysicalmeasurementmadebytheobservershouldnotbeaectedbywhereitismade,thuseachof^nA(I)shouldbetransportedparalleltoitself.Atthesametime,eachof^nA(I)shouldalwaysremainorthogonaltotheothers^nA(J)(J6=I).Then,altogether ^nB(J)rB^nA(I)=0;(4-99) whichgivesthreespacelikegeodesics,(I)(I=1;2;3). Thesetofvectorsn^uA;^nA(1);^nA(2);^nA(3)oaboveformanorthonormalbasisfortheco-movingobserver'sframe.Nowhavingthisbasiswemayconstructafamily

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71 ofgeodesics ^ X A ( s; 1 ; 2 ; 3 ),whichwillbeinvertedlatertogiveFerminormal coordinates:here s isaparameterforatemporalmeasurealongandbecomes thetimecoordinate T FN viatheinversion,and I ( I =1 ; 2 ; 3)areparameters forspatialmeasuresalong ( I ) andbecomethespatialcoordinates X I FN viathe inversion.Bycombiningtheintegralsolutionsofthegeodesicequations( 4-97 )( 4-99 )weobtain ^ X A ( s; 1 ; 2 ; 3 )= Z ds Z ds @ ^ u A @s ~ =0 + 3 X I =1 Z ( I ) d I Z ds @ ^ n A ( I ) @s ~ =0 # + 3 X I =1 Z ( I ) d I 3 X J =1 Z ( J ) d J @ ^ n A ( I ) @ J s # ; (4-100) where ~ ( 1 ; 2 ; 3 ),andthesubscriptsoutside()meanthatthesevariables areheldxedwhilethepartialdierentiationsareperformedwithrespecttothe others. Toevaluatetheaboveintegral,oneneedndproperexpansionsofeach integrandintermsof s and ~ .FromEqs.( 4-97 )-( 4-99 )wehave @ ^ u A @s ~ =0 = ^ A BC ^ u B ^ u C ~ =0 = ^ A BC;D o ^ u B ^ u C ^ X D + 1 2 ^ A BC;DE o ^ u B ^ u C ^ X D ^ X E ~ =0 + O ( ^ X 3 = R 4 ) ; (4-101) @ ^ n A ( I ) @s ~ =0 = ^ A BC ^ n B ( I ) ^ u C ~ =0 = ^ A BC;D o ^ n B ( I ) ^ u C ^ X D + 1 2 ^ A BC;DE o ^ n B ( I ) ^ u C ^ X D ^ X E ~ =0 + O ( ^ X 3 = R 4 ) ; (4-102)

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2^ABC;DEo^nB(I)^nC(J)^XD^XEs=0+O(^X3=R4); where^ABCisexpandedaround^XAo=0,anditsexpansioncoecients^ABC;Doand^ABC;DEoarecomputedviaEqs.( 4-93 )and( 4-95 )alongwith( 4-91 )and( 4-92 ).Therstorderapproximationfor^XA(s;1;2;3)neartheinitialpoint^XAo=0is ^XA(s;1;2;3)=@^XA wheren^uAo;^nAo(1);^nAo(2);^nAo(3)oaretheorthonormalbasisvectorsevaluatedatthelocationoftheparticle,andthesummationisassumedovertherepeatedindexI=1;2;3(hereafter,weomitthesummationsignandassumethesummationconventionfortheup-and-downrepeatedspatialindicesI;J;K;L;:::=1;2;3).SubstitutingthisintoEqs.( 4-101 )-( 4-103 ),weobtainthequadraticapproximationsfortheintegrands 2^ABC;DEo^uBo^uCo^uDo^uEos2+O(s3=R4); 2^ABC;DEo^nBo(I)^uCo^uDo^uEos2+O(s3=R4); 2^ABC;DEo^nBo(I)^nCo(J)^uDos+^nDo(K)K^uEos+^nEo(L)L+O[(s;~)3=R4]:

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OuraimhereistondthequarticorderexpansionsforFerminormalcoordinates,whicharederivedfromtheinversetransformationsofthequarticorderexpansionsof^XAinsand~.^XAaredeterminedtothequarticorderinsand~whenweevaluatetheintegralinEq.( 4-100 )alongwithEqs.( 4-105 )-( 4-107 ).ThecontributionsfromtheordertermsO(s3=R4)orO[(s;~)3=R4]canbedisregardedsincethroughtheintegrationsviaEq.( 4-100 )thesebecomehigherthanquartic. SubstitutingEqs.( 4-105 )-( 4-107 )intoEq.( 4-100 )andperformingtheintegral,weobtain ^XA(s;1;2;3)=^uAos+^nAo(K)K1 6^ABC;Do^uBo^uCo^uDos3+3^uCo^uDos2^nBo(K)K+3^uDos^nBo(K)K^nCo(L)L+^nBo(K)K^nCo(L)L^nDo(M)M1 24^ABC;DEo^uBo^uCo^uDo^uEos4+4^uCo^uDo^uEos3^nBo(K)K+6^uDo^uEos2^nBo(K)K^nCo(L)L+4^uDos^nBo(K)K^nCo(L)L^nEo(M)M+^nBo(K)K^nCo(L)L^nDo(M)M^nEo(N)N+O[(s;~)5=R4]: TheparametersforatemporalmeasurealongbecomesthetimecoordinateTFNandtheparametersIforspatialmeasuresalong(I)becomethespatialcoordinatesXIFN(I=1;2;3)whenEq.( 4-108 )isinvertedandsolvedforsandI.

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Tosolvefors,takeaninnerproductofbothsidesofEq.( 4-108 )with^uoA,exploitingthefactthat^uAo^uoA=1andthat^nAo(K)^uoA=0.Then 6^ABC;Do^uBo^uCo^uDos3+3^uCo^uDos2^nBo(K)K+3^uDos^nBo(K)K^nCo(L)L+^nBo(K)K^nCo(L)L^nDo(M)M+1 24^ABC;DEo^uBo^uCo^uDo^uEos4+4^uCo^uDo^uEos3^nBo(K)K+6^uDo^uEos2^nBo(K)K^nCo(L)L+4^uDos^nBo(K)K^nCo(L)L^nDo(M)M+^nBo(K)K^nCo(L)L^nDo(M)M^nEo(N)N+O[(s;~)5=R4]: TosolveforI,takeaninnerproductofbothsidesofEq.( 4-108 )with^n(I)oA,exploitingthefactthat^uAo^n(I)oA=0andthat^n(I)oA^nAo(K)=IK.Then 6^ABC;Do^uBo^uCo^uDos3+3^uCo^uDos2^nBo(K)K+3^uDos^nBo(K)K^nCo(L)L+^nBo(K)K^nCo(L)L^nDo(M)M+1 24^ABC;DEo^uBo^uCo^uDo^uEos4+4^uCo^uDo^uEos3^nBo(K)K+6^uDo^uEos2^nBo(K)K^nCo(L)L+4^uDos^nBo(K)K^nCo(L)L^nDo(M)M+^nBo(K)K^nCo(L)L^nDo(M)M^nEo(N)N+O[(s;~)5=R4]: Similarly,wemayinvertEq.( 4-104 )tosolveforsandIbycontractingbothitssideswith^uoAand^n(I)oA

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SubstitutingtheselinearapproximationsforsandIontherighthandsidesofEqs.( 4-109 )and( 4-110 ),wenallyobtaintheexpressionsofFerminormalcoordinates,writtenintermsoftheinitialcoordinates^XA,preciselyuptothequarticorder where 6^APQ;RoPBQCRD+3QBRChPD+3RBhPChQD+hPBhQChRD; 24^APQ;RSoPBQCRDSE+4QBRCSDhPE+6RBSChPDhQE+4RBhPChQDhRE+hPBhQChRDhSE; withABbeingthetime-projectiontensorandhABbeingthespace-projectiontensor,whicharedenedas respectively. 1 4-118 )isobviousfromthelocaltetrad,orlo-calvierbein:A(P)nuA;nA(1);nA(2);nA(3)o,whereP2(0;1;2;3)isthelabelfor

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ThetransformationviaEqs.( 4-113 )and( 4-114 )reproducesthedesiredgeometryofFerminormalcoordinates.WeexaminethisinAppendix B Step(iii) OutofanyLocallyInertialCartesiancoordinates,wecandevelopthethreegeodesicequationsasdescribedbyEqs.( 4-97 )-( 4-99 ).Thusthecoordinates^XAinStep(ii)maybereplacedbyanotherlocallyinertialcoordinatesXAtomakeinitialcoordinates.Thenbycombiningtheintegralsolutionsofthegeodesicequations( 4-97 )-( 4-99 )wenowconstructanewfamilyofgeodesicsXA(s;1;2;3) wheresbecomesthetimecoordinateTFNandI(I=1;2;3)becomethespatialcoordinatesXIFNwhentheequationisinverted. eachvectorofthetetrad.Thesevectorsconstitutealocalorthonormalframeateachpointalongthetimelikeworld-lineofaparticle,inwhichthetimelikevec-toruA,thefour-velocityoftheparticleistangenttoandgivesthedirectionforthetime-axis,andthespacelikevectorsnA(I)(I=1;2;3)serveasthebasisvec-torsforthespacetriad.Thenitfollowsthat(i)gABA(P)B(Q)=PQandthat(ii)PQA(P)B(Q)=gAB.SplittingthetetradA(P)intouAandnA(I),therelation(ii)canberewrittenasuAuB+P3I=1nA(I)n(I)B=gAB,whichprovestheidentityinEq.( 4-118 ).

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FollowingthesameprocedureasinEqs.( 4-101 )-( 4-108 ),Eq.( 4-119 )developsinto 6ABC;DouBouCouDoT3FN+3nBo(K)uCouDoT2FNXKFN+3nBo(K)nCo(L)uDoTFNXKFNXLFN+nBo(K)nCo(L)nDo(M)XKFNXLFNXMFN1 24ABC;DEouBouCouDouEoT4FN+4nBo(K)uCouDouEoT3FNXKFN+6nBo(K)nCo(L)uDouEoT2FNXKFNXLFN+4nBo(K)nCo(L)uDonEo(M)TFNXKFNXLFNXMFN+nBo(K)nCo(L)nDo(M)nEo(N)XKFNXLFNXMFNXNFN+O(X5FN=R4); wherethesubscriptodenotestheevaluationatthelocationoftheparticle,andthequantitiesuAo,nAo(I),ABC;DoandABC;DEoareevaluatedinthecoordinatesXA.IfweidentifyXAwithTHZnormalcoordinates,thenthisequationexactlytellsushowFerminormalcoordinatestransformintoTHZnormalcoordinates. Thelinearpartoftheabovetransformationimpliestheinverse-Lorentzboostandisresponsiblefortherelationshipbetweenthemetricsofthetwogeometriesatthelocationoftheparticle.ThetwometricsareinfactbothMinkowskianthere, AccordingtoRef.[ 24 ],thisrelationshipmustbesatisedvia

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Thentheinverse-Lorentzboostmustbetheidentitytransformation, andthewholetransformationinEq.( 4-120 )ischaracterizedbycubicorhigherordercorrectionsbetweenthetwocoordinateexpressions.Also,wecanspecify ABC;Do=HABCD+HACBDHBCAD; ABC;DEo=3HABCDE+HACBDEHBCADE usingthecomponentsofthemetricperturbationstakenfromEqs.( 4-50 )and( 4-51 ).Then,withuAo,nAo(I),ABC;DoandABC;DEospecied,thetransformationinEq.( 4-120 )iscompletelydeterminedtobe 168_EKLXKFNXLFN2FN+O(X5FN=R4); 6EIKXKFN2FN+1 3EKLXKFNXLFNXIFN1 6_EIKXKFN2FNTFN+1 3_EKLXKFNXLFNXIFNTFN1 24EIKLXKFNXLFN2FN+1 12EKLMXKFNXLFNXMFNXIFN2 63IMK_BMLXKFNXLFN2FN+O(X5FN=R4); whereTTTHZ,XIXITHZand2FN=X2FN+Y2FN+Z2FN.InAppendix C itisveriedthatthecoordinatetransformationviaEqs.( 4-127 )and( 4-128 )properlyconvertsthemetricofFerminormalgeometryintothatofTHZnormalgeometrywiththehelpofsomepropertiesontheRiemanntensorsforvacuumspacetime. Finally,inordertoexpresstheTHZcoordinatesXAintermsoftheback-groundcoordinatesxa=(t;r;;),wecombineEqs.( 4-113 )and( 4-114 )with( 4-127 )and( 4-128 )alongwith( 4-83 ).Thenalresultis

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with ^XA=MAa(xaxao)+1 2MAaabcjo(xbxbo)(xcxco);(4-131) 6^APQ;RoPBQCRD+3QBRChPD+3RBhPChQD+hPBhQChRD; 24^APQ;RSoPBQCRDSE+4QBRCSDhPE6RBSChPDhQE+4SBhPChQDhRE+hPBhQChRDhSE5 168^RAPQR;SoQShPBhRChDE; 6^APQ;RoPBQCRD+3QBRChPD+3RBhPChQD+hPBhQChRD^REPQRo1 6AEPQhRBhCD+1 3ABQEhPChRD; 24^APQ;RSoPBQCRDSE+4QBRCSDhPE6RBSChPDhQE+4SBhPChQDhRE+hPBhQChRDhSE^RFPQR;So1 6AFPQSBhRChDE+1 3ABQFSChPDhRE+1 24AFPQhRBhSChDE+1 24ABQFhPChRDhSE+2 63AFRShPBhQChDE;

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whereA;;EandP;;S=0;1;2;3,andI=1;2;3.FortheSchwarzschildspacetimeasthebackground,wemaytaketheequatorialplaneo==2anddescribethesourcepointinEq.( 4-131 )as Wealsohave togetherwithallnon-zeroabcjoforthebackgroundtakenfromEq.( 4-85 ).Describ-ingthefour-velocities, ^uAo=f1=2E;f1=2_r;J ro;0; ^n(1)Ao=f1=2_r;1+_r2 ro(E+f1=2);0; ^n(2)Ao=J ro;J_r ro(E+f1=2);1+J2 ^n(3)Ao=(0;0;0;1); wheref=12M ro,andEut=(12M=ro)(dt=d)o(:propertime)andJu=r2o(d=d)oaretheconservedenergyandangularmomentuminthebackground,respectively,and_rur=(dr=d)o.ThesearealsousedtocomputeABandhABviaEqs.( 4-117 )and( 4-118 ).Also,onecanevaluate^ABC;Do,^RABCDo,^ABC;DEoand^RABCD;EousingEqs.( 4-93 )-( 4-96 )alongwith( 4-91 )and( 4-92 ). 4-45 ),isdenedastheproperdistancefromptomeasuredalongthespatialgeodesicwhichisorthogonalto.IntheTHZ

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coordinates,itssquareisexpressedas SubstitutingEqs.( 4-130 )alongwith( 4-132 )-( 4-135 )intothisequationandsimplifyingthealgebra,weobtain 3hAE^ABC;Do^XB^XC^XD^XE1 3AC^RABCDo^XB^XD^XE^XE+1 12hAF^ABC;DEo^XB^XC^XD^XE^XF1 12AC^RABCD;Eo^XB^XD^XE^XF^XF1 4ACEF^RABCD;Eo^XB^XD^XF^XG^XG+1 6ACEFGH^RABCD;Eo^XB^XD^XF^XG^XH+O(^X6=R4); where^XArepresentstheinitialnormalcoordinatesandisconstructedfromthebackgroundcoordinatesxa=(t;r;;)viaEqs.( 4-131 )togetherwith( 4-85 ),( 4-136 )and( 4-137 ).Also,ABandhABarecomputedviaEqs.( 4-117 )and( 4-118 )alongwith( 4-138 ),and^ABC;Do,^RABCDo,^ABC;DEoand^RABCD;EoareevaluatedusingEqs.( 4-93 )-( 4-96 )alongwith( 4-91 )and( 4-92 ).Theactualexpressionof2inthebackgroundcoordinatesxa=(t;r;;)wouldbeverylengthytotheorderspeciedinEq.( 4-143 ),andherewespecifyitonlytothecubicorderinorderto

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describethemainfeaturesofSforthenextSection f(tto)(rro)2EJ(tto)(o)+1 f(rro)(o)+(r2o+J2)(o)2ME_r r2o(tto)3+M r2o1+2E2 r2o(tto)2(o)ME_r f2r2o(tto)(rro)22(roM)EJ fr2o(tto)(rro)(o)+roE_r(tto) f2r2o1+_r2 f2r2o(rro)2(o)+ro1_r2 wherePIV(t;r;;)andPV(t;r;;)representthequarticandquinticorderterms,respectively.AllthesetermscanbespeciedusingMAPLEandGRTENSOR. Itisimportanttonotethatonlytheminimuminformationabouttheback-groundspacetimeisrequiredforconstructing^XAinordertodetermine2toanyhighorderwedesire.Inotherwords,wespecify^XAonlytothequadraticorderasinEq.( 4-131 )foritsuseinEq.( 4-143 ),andthespecicationof^XAtocubicorhigherorderwouldnotmakeanydierencein2. 4.3 ,wesawthatanapproximationofSisparticularlysimpleinTHZnormalcoordinates,

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where2=X2+Y2+Z2andRrepresentsalengthscaleofthebackgroundgeometry.FollowingRefs.[ 17 ]and[ 18 ],theregularizationparameterscanbedeterminedfromevaluatingthemultipolecomponentsof wherealabelstheSchwarzschildbackgroundcoordinatesxa=(t;r;;).TheremainderO(2=R3)intheaboveapproximationisdisregardedsinceitgivesnocontributiontoraSaswetakethe\coincidencelimit",xa!xao,wherexadenotesapointinthevicinityoftheparticleandxaothelocationoftheparticleintheSchwarzschildbackground. InevaluatingthemultipolecomponentsofraSviaEqs.( 4-146 )and( 4-144 ),singularitiesareexpectedtooccurwithcertainterms.Tohelpidentifythosesingularities,weintroduceanorderparameterwhichistobesettounityattheendofthecalculation:weattachntoeachnthorderpartof2inEq.( 4-144 )andre-express2as wherePII,PIII,PIVandPVrepresentthequadratic,cubic,quarticandquinticorderpartsof2,respectively. Weexpress@a(1=)inaLaurentseriesexpansion,andeverydenominatorofthisexpansiontakestheformofPn=2II(n=3;5;7;9;).Thus,PIIplaysanimportantroleinthemultipoledecomposition,butthequadraticpartPII,directlytakenfromEq.( 4-144 ),isnotyetfullyreadyforthistask.First,omustbedecoupledfromrrosothateachappearsonlyasanindependentcompletesquare.Couplingbetweenttoandodoesnotcreatedicultyinthedecomposition.

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Thus,wereshapethequadraticpartofEq.( 4-144 )into with whererro,andanidentity_r2=E2f(1+J2=r2o)isusedforsimplifyingthecoecientof2.Here,takingthecoincidencelimit!0,wehave0!o.ThissameideaisfoundinMino,Nakano,andSasaki[ 26 ].Also,forthemultipoledecompositionthequadraticpartmustbeanalyticandsmoothovertheentiretwo-sphere,andwewrite Herewehaveusedtheelementaryapproximations0=sin(0)+O[(0)3]and1=sin+O[(=2)2]. ToaidinthemultipoledecompositionwerotatetheusualSchwarzschildcoordinatesbyfollowingtheapproachofBarackandOri[ 27 ]suchthatthecoordinatelocationoftheparticleismovedfromtheequatorialplane==2tothenewpolaraxis.ThenewanglesanddenedintermsoftheusualSchwarzschildanglesare sincos(0)=cossinsin(0)=sincoscos=sinsin:

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Also,underthiscoordinaterotation,asphericalharmonicY`m(;)becomes wherethecoecients`mm0dependontherotation(;)!(;)aswellason`,mandm0,andtheindex`ispreservedundertherotation[ 28 ].AsrecognizedinRef.[ 27 ],thereisagreatadvantageofusingtherotatedangles(;):afterexpanding@a(q=)intoasumofsphericalharmoniccomponents,wetakethecoincidencelimit!0,!0.Then,nallyonlythem=0componentscontributetotheself-forceat=0becauseY`m(0;)=0form6=0.Thus,theregularizationparametersofEq.( 4-33 )arejust(`;m=0)sphericalharmoniccomponentsof@a(q=)evaluatedatxao. Now,withtheserotatedangles,PIIisre-expressedas wheretheelementaryapproximationsin2=2(1cos)+O(4)isused.Wemaynowdene ~2(E2f)(tto)22E_rr2o Inparticular,whenxingt=to,wedene ~2o~2t=to=2r2o+J22+1cos(4-155)

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with and NowwerewriteEq.( 4-147 )byreplacingtheoriginalquadraticpartPIIwith~2, wherePIVnowincludestheadditionalquarticordertermsthathaveresultedfromthereplacementofPIIby~2throughEqs.( 4-150 )and( 4-153 ).ALaurentseriesexpansionof@a(1=)jt=toisthen 2@a(~2)jt=to 2@aPIIIjt=to 4[@a(~2)]PIIIjt=to 2@aPIVjt=to 4[@a(~2)]PIVjt=to+(@aPIII)PIIIjt=to 16[@a(~2)]P2IIIjt=to 2@aPVjt=to 4[@a(~2)]PVjt=to+(@aPIII)PIVjt=to+(@aPIV)PIIIjt=to 162[@a(~2)]PIIIPIVjt=to+(@aPIII)P2IIIjt=to 32[@a(~2)]P3IIIjt=to AfterthederivativesinEq.( 4-159 )aretaken,thedependenceupon,andrmayberemovedinfavorof~o,andbyuseofEqs.( 4-155 )-( 4-157 ).Thenthethreestepsof(i)aLegendrepolynomialexpansionforthedependence,whilerandareheldxed,followedby(ii)anintegrationover,whilerisheldxed,andnally(iii)takingthelimit!0,togetherprovidetheregularization

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87 parameters.ThetechniquesinvolvedintheLegendrepolynomialexpansionsand theintegrationoveraredescribedindetailinAppendicesCandDofRef.[ 18 ]. BelowinSubsections 4.4.1 4.4.4 ,wepresentthekeystepsofcalculatingthe regularizationparameters A a B a C a and D a inEqs.( 4-34 )-( 4-44 ). 4.4.1 A a -terms Wetakethe 2 termfromEq.( 4-159 )anddene Q a [ 2 ] q 2 2 @ a (~ 2 ) j t = t o ~ 3 o : (4-160) Wemayexpressthisinagenericform Q a [ 2 ]= X k =0 ; 1 k X p =0 kp ( a ) 1 k ( o ) k p 2 p ~ 3 o ; (4-161) where r r o ,and kp ( a ) isthecoecientofeachindividualtermthatdepends on k and p aswellasonthecomponentindex a ,withadimension R k for a = t;r and R k +1 for a = ; .Thebehaviorof Q a [ 2 ],accordingtothepowersofeach factorinEq.( 4-161 ),is Q a [ 2 ] ~ 3 o 1 k ( o ) k p 2 p R s ; (4-162) where s = k for a = t;r and s = k +1for a = ; .WerecallfromEqs. ( 4-148 )and( 4-149 )thattherstofthestepstoleadto~ 2 o isreplacing o by( 0 ) J r =f ( r 2 o + J 2 )toeliminatethecouplingterm( o )in P II Thismakesasumofindependentsquareformsofeachofand 0 ,whichis anecessarysteptoinducetheLegendrepolynomialexpansionslater.Thus,tobe consistentwiththismodicationmadefor~ 2 o ,theremaining o inEq.( 4-162 )

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shouldbealsoreplacedby(0)J_r=f(r2o+J2).Then, (o)kp=(0)J_r whereabinomialexpansionovertheindexi=0;;kpisassumedwiththecoecientckpi1=RkpiinEq.( 4-163 ),andinEq.( 4-164 )(0)iisreplacedby[sin(0)]i+O[(0)i+2]|thetermO[(xxo)kp+2]attheendresultsfromthisO[(0)i+2],thenthecoordinatesarerotatedusingthedenitionofnewanglesbyEq.( 4-151 ).Also,byEq.( 4-151 )again UsingEqs.( 4-164 )and( 4-165 ),thebehaviorofQa[2]inEq.( 4-162 )nowlookslike wheres=p+ifora=t;rands=p+i+1fora=;,andanycontributionsfromO[(xxo)kp+2]inEq.( 4-164 )andfromO[(xxo)p+2]inEq.( 4-165 )havebeendisregarded:byputtingthesepiecesintoEq.( 4-162 )wesimplyobtain0termsorO(2),whichwouldcorrespondtoCa-termsorO(`4)inEq.( 4-33 ),andlaterinthisSectionitisprovedthattheyalwaysvanish.Qa[2]inEq.( 4-166 )thencanbecategorizedintothefollowingcases: (i) Theintegrandfortheintegration-averagingprocessoverisF()(sin)p(cos)i=(sin)p(cos),andithasthepropertiesthatF(+)=

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(ii) TheintegrandF()=(sin)phasthepropertiesthatF()=1forp=0andthatF(+)=F()forp=1.Thus,theonlynon-zerocontributiontoQa[2]comesfromthecasep=0,i.e. Thesignicanceofthisanalysisisthatthenon-vanishingAa-termsshouldal-waystaketheformofEq.( 4-168 )andthereforeincalculatingtheregularizationparametersweneedtosortoutonlythiskind. Then,weproceedwithourcalculationsoftheregularizationparametersonecomponentatatime. FirstwecompletetheexpressionforQt[2]byrecallingEqs.( 4-154 )and( 4-155 ) @2+1cos1=2; where@=@jmeansthatisheldconstantwhilethedierentiationisper-formedwithrespectto.

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AccordingtoAppendixDofRef.[ 18 ],wehavethefollowingLegendrepolyno-mialexpansionsof(2+1cos)p1=2:forp1 andforp=0 Then,byEq.( 4-170 )forp=1,andbyEqs.( 4-171 )and( 4-157 ),inthelimit!0(equivalently!0)Eq.( 4-169 )becomes lim!0Qt[2]=sgn()q2_rro1 2P`(cos)q2EJ3=2cos (r2o+J2)3=21X`=0@ @P`(cos): Then,weintegratelim!0Qt[2]overanddivideitby2(henceforth,wedenotethisprocessbytheanglebrackets\hi") 2P`(cos);(4-173) whereweexploitthefactthat3=2cos=0togetridofthesecondterminEq.( 4-172 ). 18 ]providesh1i=2F11;1 2;1;F1=(1)1=2,whereJ2=(r2o+J2).SubstitutingthisintoEq.( 4-173 ),theregularizationparameterAtisthecoecientofthesumontherighthandsidein @P`(cos)=0as!0,toshowthatthispartdoesnotsurviveattheend.

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thecoincidencelimit!0 Similarly,wehave Here,beforecomputing@r(~2)jt=towereversethestepsofEqs.( 4-148 ),( 4-150 ),( 4-153 )and( 4-154 )toobtaintherelation ~2=PII+O[(xxo)4];(4-176) wherePIIisnowbacktoEq.( 4-148 ).DierentiatingthiswithrespecttorandgoingthroughthestepsofEqs.( 4-150 )and( 4-151 ),Eq.( 4-175 )canbeexpressedwiththehelpofEq.( 4-155 )as Then,therestofthecalculationiscarriedoutinthesamefashionasinthecaseofAt-termabove.Weobtain f(1+J2=r2o):(4-178) Firstwehave

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TakingthesamestepsasusedforAr-termaboveviaEqs.( 4-176 ),( 4-150 )and( 4-151 )inorder,weobtain Then,inasimilarmannertothatemployedinthepreviouscases,inthelimit!0Eq.( 4-179 )becomes lim!0Q[2]=q23=2cos (r2o+J2)1=21X`=0@ @P`(cos):(4-181) Therighthandsidevanishesthrough\hi"processbecause3=2cos=0.Hence, Itisevidentfromtheparticle'smotion,whichisconnedtotheequatorialplaneo= with Then,similarlyasinthecaseofA-termabove lim!0Q[2]=q2r2o3=2sin (r2o+J2)3=21X`=0@ @P`(cos):(4-185)

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Again,via\hi"process,therighthandsidevanishesbecause3=2sin=0.Thus, 4-159 )anddene 2@aPIIIjt=to 4[@a(~2)]PIIIjt=to whereforcomputing@a(~2),Eq.( 4-176 )shouldbereferredto,andPIIIisthecubicparttakendirectlyfromEq.( 4-144 ).InagenericformEq.( 4-187 )canbeexpressedas whererro,andnkp(a)isthecoecientofeachindividualtermthatdependsonn,kandpaswellasonthecomponentindexa,withadimensionRk1fora=t;randRkfora=;.AswesawalreadyfromtheanalysisgiveninSubsection 4.4.1 ,wereplaceoby(0)J_r=f(r2o+J2)toeliminatethecouplingterm(o)inPIIastherstofthestepstoleadto~2oforthedenominatorontherighthandsideofEq.( 4-188 ).Forconsistency,ointhenumeratorshouldbealsoreplacedby(0)J_r=f(r2o+J2).Then,asweexpandthequantity[(0)J_r=f(r2o+J2)]raisedtothe(kp)-thpower,anumberofadditionaltermsapartfrom(0)kpwillbecreated,andthecomputationwillbeverycomplicated. ByanalyzingthestructureoftherighthandsideofEq.( 4-188 ),onecanprovethatomaybejustreplacedby0withoutJ_r=f(r2o+J2)inthenumerator(thesameideaisfoundinMino,Nakano,andSasaki[ 26 ]).Thevericationfollows.Thebehaviorofthequantityontherighthandsideof

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94 Eq.( 4-188 ),accordingtothepowersofeachfactor,is Q a [ 1 ] ~ (2 n +1) o 2 n k ( o ) k p 2 p R s ; (4-189) where s = k 1for a = t;r and s = k for a = ; .Further, ( o ) k p = ( 0 ) J r f ( r 2 o + J 2 ) k p ( 0 ) i k p i = R k p i (4-190) (sin) i (cos) i k p i = R k p i + O [( x x o ) k p +2 ] ; (4-191) whereabinomialexpansionovertheindex i =0 ; 1 ; ;k p isassumedin Eq.( 4-190 ),and 2 p =(sin) p (sin) p + O [( x x o ) p +2 ](4-192) asalreadyshownintheanalysisofSubsection 4.4.1 .Usingthese,thebehaviorof Q [ 1 ]inEq.( 4-189 )lookslike Q a [ 1 ] ~ (2 n +1) o 2 n p i (sin) p + i (sin) p (cos) i R s ; (4-193) where s = p + i 1for a = t;r and s = p + i for a = ; .Herewehavedisregarded anycontributionsfrom O [( x x o ) k p +2 ]inEq.( 4-191 )and O [( x x o ) p +2 ]inEq. ( 4-192 ):whenthesearesubstitutedintoEq.( 4-189 ),weobtain 1 termsor O ( 3 ), whichwouldcorrespondto 2 p 2 D a = (2 ` 1)(2 ` +3)or O ( ` 6 )inEq.( 4-33 ). 3 Q a [ 1 ]thencanbecategorizedintothefollowingcases: (i) i =2 j +1( j =0 ; 1 ; 2 ; ) Theintegrandfor\ hi "process, F () (sin) p (cos) i =(sin) p (cos) 2 j +1 hasthepropertiesthat F (+ )= F ()for p =evenintegersandthat 3 While O ( ` 6 )isconsideredtobecompletelyvanishing,these 1 termsarenot neglectedandwillbeincorporatedintothecalculationsof D a -termslater.

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(ii) Forp=oddintegers,theintegrandF()=(sin)p(cos)2jhasthepropertythatF(+)=F(),hence Forp=evenintegers,usingEqs.( 4-155 )and( 4-157 ),wecanexpress(sin)p+iinEq.( 4-193 )aboveintermsof~oandviaabinomialexpansion (sin)p+2j=[2(1cos)]p=2+j+O[(xxo)p+2j+2]=p=2+jXq=0dpjq~2qop+2j2q+O[(xxo)p+2j+2](p=2+j)~2qop+2j2q=Rp+2j+O[(xxo)p+2j+2]; whereq=0;1;;1 2p+jistheindexforabinomialexpansion,andthecoecientsdpjq(p=2+j)=Rp+2j.WhenEq.( 4-196 )issubstitutedintoEq.( 4-193 ),thecontributionfromO[(xxo)p+2j+2]canbedisregardedsinceitwouldcorrespondto1termsorO(3)again.Then,wehave wheres=1fora=t;rands=0fora=;,andwecanguaranteethatnq0alwayssince0q1 2p+j=1 2(p+i),0ikpandpk2n.Then,Eq.( 4-197 )canbesubcategorizedintothefollowingtwocases; (ii-1)

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ByEqs.( 4-155 ),( 4-157 )and( 4-170 ) wheres=3fora=t;rands=2fora=;. (ii-2) ByEqs.( 4-155 ),( 4-157 )and( 4-171 ) wheres=2fora=t;rands=1fora=;. Therefore,byanalyzingthestructureofQa[1]wendthatthe1termsvanishinallthecasesexceptwhennq=0.Thenon-vanishingBa-termsarederivedonlyfromthiscase.Then,by0q1 2p+j=1 2(p+i),0ikpandpk2ntogetherwithn=qonecanshowthat SubstitutingthisresultintoEq.( 4-190 ),itisconrmedthatinthenumeratorofQ[1]inEq.( 4-188 )onecansimplysubstitute (o)kp!(0)kp:Q:E:D:(4-201) ThesignicanceofthisproofdoesnotlieintheresultgivenbyEq.( 4-201 )only,butalsointhefactthatthenon-vanishingcontributioncomesonlyfromthecasen=qforEq.( 4-197 ).By0q1 2p+j=1 2(p+i),0ikp,pk2nandn=qwendk=2nandthen2j=2np.Therefore,Eq.( 4-197 )nally

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97 becomes Q a [ 1 ] (sin) p (cos) 2 n p n ~ 1 o R s ; (4-202) where n =1 ; 2and p =0 ; 2,and s = 1for a = t;r and s =0for a = ; .Only thisformwillbetakenintheactualcalculationsofregularizationparameters. TocalculatetheregularizationparametersusingEq.( 4-202 ),rst,change Q a [ 1 ]intotheexpressionin .FromEq.( 4-156 )wehave sin 2 = ( r 2 o + J 2 )(1 ) J 2 ; (4-203) cos 2 = ( r 2 o + J 2 ) r 2 o J 2 : (4-204) AndfromEq.( 4-155 ) ~ 1 o = 1 p 2 r 2 o + J 2 1 = 2 1 = 2 2 +1 cos 1 = 2 : (4-205) However,byEq.( 4-171 )thiscanbewrittenas ~ 1 o = r 2 o + J 2 1 = 2 1 = 2 1 X ` =0 P ` (cos) ; 0 : (4-206) Then,substitutingalltheresultsofEqs.( 4-203 ),( 4-204 )and( 4-206 )into Eq.( 4-202 ),our Q a [ 1 ]canberewritteninagenericform Q a [ 1 ]= X n =1 ; 2 n X m =0 nm ( a ) m n 1 = 2 1 X ` =0 P ` (cos) ; (4-207) where nm ( a ) dependson n and m aswellasonthecomponentindex a ,witha dimension 1 = R 2 for a = t;r and 1 = R for a = ; ,andisdeterminedbythe coecientsofthetermsinthepolynomials P II and P III ofEq.( 4-144 ).Thenal stepsofourcalculationsaretakingtheintegralaverageof Q a [ 1 ]over0 2 andthentakingthecoincidencelimit 0,i.e. D lim 0 Q a [ 1 ] E 0 = X n =1 ; 2 n X m =0 nm ( a ) m n 1 = 2 1 X ` =0 1 : (4-208)

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Then,ourBa-termsarereadofromthecoecientofthesumP1`=01, wheretheidentityhpi=Fp2F1p;1 2;1;J2=(r2o+J2)wastakenfromAppendixCofRef.[ 18 ]. BelowarepresentedthecalculationsofBa-termsoftheregularizationparame-tersbycomponent. Webeginwith 2@tPIIIjt=to 4[@t(~2)]PIIIjt=to ThesubsequentcomputationwillbeverylengthyanditwillbereasonabletosplitQt[1]intotwoparts.First,let where roEJ AsprovedatthebeginningofthisSubsection,every(o)minthenumeratorsofthe1termcanbereplacedby(0)mwithoutaectingtherestofcalculation.Then,followedbytherotationofthecoordinatesviaEq.( 4-151 ) roEJ

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whereanapproximationsin2=2(1cos)+O[(xxo)4]isusedtoobtainthelastterminsidetherstbracket.HerewemaydropothetermO[(xxo)4=~3o],whichisessentiallyO(1),andits1partwillbeincorporatedintothecalculationsofDa-termslater.Then,usingthesametechniquesasusedtondAa-terms,wecanreduceEq.( 4-213 )to roEJ3=2cos @2+1cos1=2q2E_rro1 Aswesawbefore,byEq.( 4-170 )(2+1cos)3=21inthelimit!0andthersttermontherighthandsidewillvanish.Thesecondtermwillalsogivenocontributiontotheregularizationparametersbecause3=2cos=0.Onlythelastterm,whichis~1o,willgivenon-zerocontributionaccordingtotheargumentintheanalysispresentedabove(seeEq.( 4-202 )).UsingEq.( 4-171 )inthelimit!0andtaking\hi"process,Eq.( 4-214 )becomes 2q2 TheidentityhpiD1sin2pE=2F1p;1 2;1;Fp,withJ2=(r2o+J2)istakenfromAppendixCofRef.[ 18 ],andwetakethelimit!0 2q2 Nowtheremainingpartis

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where roJ_r2 f+2J2 Takingasimilarprocedureasabove,thenon-vanishingcontributionsturnouttobe 2q2EJ2_rro~5ocos2sin4=lim!03 2q2 2q2 whereallothertermsthan~1oagainhavebeendroppedoduringtheproceduresincetheyvanisheitherinthelimit!0orthroughthe\hi"process.Then,usingtheidentityhpi2F1p;1 2;1;Fp,wehave 2q2 BycombiningEqs.( 4-216 )and( 4-220 ),wenallyobtain

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FromEq.( 4-187 )westartwith 2@rPIIIjt=to 4[@r(~2)]PIIIjt=to Then,followingthesamestepsastakenforthecaseofBt-termabove,weobtain Again,fromEq.( 4-187 )wetake 2@PIIIjt=to 4[@(~2)]PIIIjt=to Then,similarlywecanderive 2(1+J2=r2o)3=2#:(4-225) AsAvanishes,soshouldB.Byworkingout 2@PIIIjt=to 4[@(~2)]PIIIjt=to inthesamemannerasabove,onendsthatthereisnotermlike~1o:alltermsareeitherlike2n=~2n+1oorlike2n1sincos=~2n+1o(n=1;2),whichvanishinthelimit!0orthroughthe\hi"process.Thus 4-159 ),alwaysvanish.Thiscanbeprovedbyanalyzingthestructureof0

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term.TakethisfromEq.( 4-159 )anddene 2@aPIVjt=to 4[@a(~2)]PIVjt=to+(@aPIII)PIIIjt=to 16[@a(~2)]P2IIIjt=to Generically,thiscanbewrittenas whererro,andnkp(a)isthecoecientofeachindividualtermthatdependsonn,kandpaswellasonthecomponentindexa,withadimensionRk2fora=t;randRk1fora=;. ThebehaviorofQa[0],accordingtothepowersofeachfactorontherighthandsideofEq.( 4-229 ),is wheres=k2fora=t;rands=k1fora=;.FollowingthesameprocedureasintheanalysisgiveninSubsection 4.4.2 ,Eq.( 4-230 )becomes whereabinomialexpansionovertheindexi=0;1;;kpisassumed,ands=p+i2fora=t;rands=p+i1fora=;.Herewehavedisregardedanyby-productslikeO[(xxo)kp+2]andO[(xxo)p+2],whichoriginatefrom(o)kpand 4-230 )wesimplyobtainO(2)terms,whichwouldcorrespondtoO(`4)inEq.( 4-33 )andshouldvanishwhensummedover`inournalself-forcecalculationbyEq.( 4-32 ).

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Then,therestoftheargumentisdevelopedinthesamewayasintheanalysisofSubsection 4.4.2 .WecategorizeQa[0]inEq.( 4-231 )intothefollowingcases: (i) Theintegrandfor\hi"process,F()(sin)p(cos)i=(sin)p(cos)2j+1hasthepropertiesthatF(+)=F()forp=evenintegersandthatF(+=2)=F()forp=oddintegers.Thus (ii) Forp=oddintegers,theintegrandF()=(sin)p(cos)2jhasthepropertythatF(+)=F(),hence Forp=evenintegers,usingEqs.( 4-196 ),wehave whereq=0;1;;1 2p+jistheindexforabinomialexpansionof(sin)p+2jands=2fora=t;rands=1fora=;.Herewecanguaranteethat2(nq)+10since0q1 2p+j=1 2(p+i),0ikpandpk2n+1.Then,Eq.( 4-234 )canbesubcategorizedintothefollowingtwocases; (ii-1) ByEqs.( 4-155 ),( 4-157 )and( 4-170 ) wheres=4fora=t;rands=3fora=;.

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(ii-2) ByEqs.( 4-155 ),( 4-157 )and( 4-171 ) wheres=3fora=t;rands=2fora=;. Clearly,inanycasesthequantityQa[0]doesnotsurvive,thereforewecancon-cludethatalways 4-159 )anddene 2@aPVjt=to 4[@a(~2)]PVjt=to+(@aPIII)PIVjt=to+(@aPIV)PIIIjt=to 162[@a(~2)]PIIIPIVjt=to+(@aPIII)P2IIIjt=to 32[@a(~2)]P3IIIjt=to Inagenericform,wemayexpressEq.( 4-238 )as whererro,and#nkp(a)isthecoecientofeachindividualtermthatdependsonn,kandpaswellasonthecomponentindexa,withadimensionRk3fora=t;randRk2fora=;.

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AccordingtothepowersofeachfactorontherighthandsideofEq.( 4-239 ),Qa[1]behaveslike wheres=k3fora=t;rands=k2fora=;.SimilarlytotheanalysisgiveninSubsection 4.4.2 ,Eq.( 4-240 )developsinto whereabinomialexpansionovertheindexi=0;1;;kpisassumed,ands=p+i3fora=t;rands=p+i2fora=;.ThetermslikeO[(xxo)kp+2]andO[(xxo)p+2]havebeendisregarded,whichoriginatefrom(o)kpand 4-240 ),theseordertermsmakeO(3)thatwouldcorrespondtoO(`6)inEq.( 4-33 )andshouldvanishwhensummedover`inournalself-forcecalculationbyEq.( 4-32 ).Then,Qa[1]canbecategorizedintothefollowingcases: (i) Theintegrandfor\hi"process,F()(sin)p(cos)i=(sin)p(cos)2j+1hasthepropertiesthatF(+)=F()forp=evenintegersandthatF(+=2)=F()forp=oddintegers.Ineithercases (ii) Forp=oddintegers,theintegrandF()=(sin)p(cos)2jhasthepropertythatF(+)=F(),hence

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Forp=evenintegers,usingEqs.( 4-196 ),wehave whereq=0;1;;1 2p+jistheindexforabinomialexpansionof(sin)p+2jasseeninEq.( 4-196 )ands=3fora=t;rands=2fora=;.Itisguaranteedthat2(nq)+20ornq1since0q1 2p+j=1 2(p+i),0ikpandpk2n+2.Then,Eq.( 4-244 )canbesubcategorizedintothefollowingthreecases; (ii-1) ByEqs.( 4-155 ),( 4-157 )and( 4-170 ) wheres=5fora=t;rands=4fora=;. (ii-2) ByEqs.( 4-155 ),( 4-157 )and( 4-171 ) wheres=4fora=t;rands=3fora=;. (ii-3) Wehave wheres=3fora=t;rands=2fora=;. Therefore,throughtheanalysiswendthatthenon-vanishingQa[1]resultsonlyfromthecaseofnq=1intheformofEq.( 4-247 ).By0q1 2p+j=1 2(p+i),

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0ikp,pk2n+2andn=q1wehavek=2n+2andthen2j=2(n+1)p.ThentheultimateformofQa[1]is wheren=1;2;3;4andpisapositiveevenintegerin[0;2(n+1)],ands=3fora=t;rands=2fora=;. AsinthecalculationsoftheBa-termsinSubsection 4.4.2 ,tocalculatetheDa-termsweneedchangeEq.( 4-248 )intotheexpressionin.FromEq.( 4-156 )wehave ~o=p However,accordingtoAppendixDofRef.[ 18 ] (2`1)(2`+3)P`(cos);!0:(4-250) Thusour~ocanbewrittenas ~o=p (2`1)(2`+3)P`(cos);!0:(4-251) SubstitutingEqs.( 4-203 ),( 4-204 )and( 4-251 )intoEq.( 4-248 ),wemayrewriteourQa[1]inagenericform (2`1)(2`+3)P`(cos);(4-252) where#nm(a)dependsonnandmaswellasonthecomponentindexa,withadimension1=R2fora=t;rand1=Rfora=;,andisdeterminedbythecoecientsofthetermsinthepolynomialsPII,PIII,PIVandPVofEq.( 4-144 ).Finally,wetaketheintegralaverageofQa[1]over02andthentakethecoincidencelimit!0,i.e. (2`1)(2`+3):(4-253)

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Then,ourDa-termsarereadofromthecoecientofthesumP1`=0(2p wheretheidentityhpi=Fp2F1p;1 2;1;J2=(r2o+J2)wasused. IntheactualcalculationsofDa-termswemustincludenotonlythemainframeworkQa[1]asrepresentedbyEq.( 4-238 ),butalsothe1termsasby-productsthatoriginatefromQa[1]throughthestepsofEqs.( 4-191 )and( 4-192 ). TheactualcalculationsofDa-termsaretremendouslytediousduetothelengthinessofPIVandPVinEq.( 4-144 ).ThecalculationscanbeimplementedusingMAPLE,andweprovideonlytheresultsbelow. .

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. 2(f1)1+J2=r2o3=2F1=2+[(f1)_r2+(1f)E2f]F1=2

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4-32 )wethenhave viawhichwecomputeourself-force. 3-10 )alongwithEq.( 3-11 ).ThepracticaldetailsforthistaskaredescribedwellinAppendixEofRef.[ 18 ].AndfromEq.( 4-33 )wehaveadescriptionforthesingularpartFS`rinthecoincidencelimitr!ro, limr!roFS`r=`+1 2Ar+Br2p wheretheregularizationparametersaresimpliedfromthoseforageneralorbitEqs.( 4-34 )-( 4-44 )duetotheconditionthat_r=0: r4o(ro2M)1=2F1=2(ro3M)F3=2

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Table4{1. ThettedparametersofFret`r(bycourtesyofDetweileretal,2003[ 18 ]) ThefourthtermofEq.( 4-260 )wasintroducedtoextrapolatethehigherorderregularizationparametersthanDr-term.Ekrareindependentof`,andaretobedeterminedbynumericaltting.TheuseoftheseadditionalparameterswillhelptoincreasedramaticallytheeectiveconvergencetoournalresultofFselfr(seeTable 4{1 andFigure 4{1 ).MoretechnicaldetailsregardingthisarefoundinRef.[ 18 ]. FromEqs.( 4-259 )and( 4-260 ),weseethatourself-forceiscomputedbysummingtheresidualsafterremovingthetermsofAr,Br,DrandadditionaltermsofEkrfromthenumericalsolutionFret`r.RemovingthecontributionofEkrimprovesthefallooftheresidualsbyanadditionaltwopowersof`.Nu-mericallydeterminedEkrcoecientsandtheircontributionstotheself-forceFselfraregiveninTable 4{1 .Figure 4{1 summarizestheresultsofthisnumer-icalanalysis.ThecurvelabeledFoutrepresentsFret`rasafunctionof`.ThecurvesA,BandDshowFret`r(`+1=2)Ar,Fret`r[(`+1=2)Ar+Br]andFret`r(`+1=2)Ar+Br2p 18 ].

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112 Figure4{1. Self-forceofascalareldintheSchwarzschildspacetime(bycourtesy ofDetweileretal,2003[ 18 ]) Theself-force F self r =1 : 37844828(2) 10 5 wasobtainedbysumming F ret `r ( ` +1 = 2) A r + B r 2 p 2 D r = (2 ` 1)(2 ` +3) overtherangeofourdata [ 18 ].Theremainderofthesumto ` = 1 wasapproximatedbythecontributions ofthe E 1 r E 2 r ::: sumsfrom41to 1 afterdeterminingthe E k r coecients[ 18 ]. Table 4{1 showsthecontributionsofeach E k r totheself-force F self r .Thisresultis inagreementwiththatofBurko[ 29 ].

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InthisChapterweprovideamethodtodeterminetheeectsofthegravi-tationalself-forceonapointmassorbitingaSchwarzschildblackhole.First,weaddressthegaugeissuesinrelationtoMiSaTaQuWaGravitationalSelf-force[ 4 5 ].ThenwefollowarecentanalysisbyDetweiler[ 10 ]todescribethegravitationaleld,whichistheperturbationcreatedbythepointmassfromthebackgroundspacetime.Toavoidthegaugeproblem,ratherthancalculatingtheself-forcedi-rectly,wefocusongaugeinvariantquantitiesanddeterminetheirchangesduetotheself-forceeects.Techniquesinvolvedincalculatingtheregularizationparam-etersforthegravitationaleldcasearemorecomplicatedthanforthescalareldcase.WefollowanalysesbyDetweilerandWhiting[ 11 ]tondthemethodsforcalculatingtheregularizationparameters. 3.1 webrieyreviewedthegravitationalself-forceduetotheperturbationhabcreatedbyapointmassmfromthebackgroundspacetimegab,whichischaracterizedbyEq.( 3-12 )andoftenreferredtoas\MiSaTaQuWa"self-force.Afteramappingtothebackgroundspacetime,MiSaTaQuWaequationstaketheform[ 30 ] 2rbhtailcd;(5-1) where 2gabGretdda0b0(z();z(0))_za0_zb0d0:(5-2) 113

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Whiletheequationsofmotionfortheotherelds,( 3-6 )and( 3-9 )aregenerallycovariant,Eq.( 5-1 )isnotandreectsaspecicchoiceofcoordinatesystemandwouldnotpreserveitsformunderaninnitesimalcoordinatetransformation.Thatistosay,thegravitationalself-forcecalculatedviaMiSaTaQuWaequationsisnotgaugeinvariantanddependsuponthegaugeconditiongivenbyEq.( 3-16 ).AccordingtoRef.[ 31 ],underacoordinatetransformation wherexaarethecoordinatesofthebackgroundspacetimeandaisasmoothvectoreldofO(m),theparticle'saccelerationchangesaccordingly za!za+z[]a;(5-4) where z[]a=(ab+_za_zb)b+Rbcde_zcd_ze(5-5) isthe\gaugeacceleration"andb=b;a_za;c_zcisthesecondcovariantderivativeofbinthedirectionoftheworld-line.Thisimpliesthatagaugetransformationcanaltertheparticle'sacceleration:aspecialchoiceofacouldevenmakeza=0,whichisjustbacktotheoriginalgeodesicofmotion[ 30 ]. FromthisobservationwecometotheconclusionthattheMiSaTaQuWaequationsofmotionarenotgaugeinvariantandcannotprovidebythemselvesameaningfulinterpretationtothegravitationalself-force.Toobtainaphysicallymeaningfulanswertothegravitationalself-forceproblem,weshouldbeaimingatthequantitiesthatmustbedescribableinamannerwhichisgaugeinvariant[ 30 ]. 10 ].Supposeaparticleofsmallmassmovesalongageodesicof

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backgroundgeometrygab.TheperturbationbeginswithabackgroundmetricgabwhichisavacuumsolutionoftheEinsteinequationsGab(g)=0.Theparticlethendisturbsthegeometrybyanamounthab(g)=O()whichisdeterminedbytheperturbedEinsteinequationswiththestress-energytensorTab=O()oftheobjectbeingthesource, Here,Eab(h)isthelineardierentialoperatordenedby andGabistheEinsteintensorofgab,sothat 2Eab(h)=r2hab+rarbh2r(archb)c+2Racbdhcd+gabrcrdhcdr2h; withhhabgab.IfhabisasolutionofEq.( 5-6 ),thenitfollowsfromEq.( 5-7 )thatgab+habisanapproximatesolutionoftheEinsteinequationswithsourceTab, FromEq.( 5-8 ),usingtheBianchiidentity,wehave foranysymmetrictensorhab.Thus,anintegrabilityconditionforEq.( 5-6 )isthatthestress-energytensorTabbeconservedinthebackgroundgeometry[ 10 ] Thesecondorderperturbationanalysisisnomoredicultthantherstorder.Themaindierenceistheintegrabilitycondition:forthesecondorderequationsTabmustbeconservednotinthebackground,butintherstorderperturbed

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geometry.Thisrequiresustochangethestress-energytensorinawaywhichisdependentupontherstordermetricperturbationsbeforesolvingthesecondorderequations.ThiscorrectiontoTabissaidtoresultfromthe\self-force"ontheobjectfromitsowngravitationaleldandincludesthedissipativeeectsof\radiationreaction"aswellasothernonlinearaspectsofgeneralrelativity[ 10 ]. Tofocusonthosedetailsoftheself-forcewhichareindependentofthestructureoftheobject,wemodeltheobjectbyanabstractpointparticlewithnospinangularmomentumorotherinternalstructure.Thestress-energytensorofapointparticleis g4(xaXa(s))ds;(5-12) whereXa(s)describestheworld-lineoftheparticleinsomecoordinatesystemasafunctionofthepropertimesalongtheworld-line.Thismodelingofasmallobjectbyadelta-functiondistributionforthestress-energytensorissatisfactoryintherstorderperturbationanalysis.TheintegrabilityconditionattherstorderasdescribedbyEq.( 5-11 )impliesthattheworld-lineoftheparticleisapproximatelyageodesicofthebackgroundgeometrygab,withanaccelerationubrbua=O().Thiscanbeprovedasbelow[ 10 32 ] (gca+ucua)rbTab=(gca+ucua)Z1(rbua)ub g4(xaXa(s))+uarbub g4(xaXa(s))ds=Z1(rbuc)ub g4(xaXa(s))ds; wherethesecondequalityfollowsfrompropertiesoftheprojectionoperatorgca+ucua.IfrbTab=0,thenwemusthaveubrbua=0fromthisequation.Theintegrabilityconditionatthesecondorder,however,presentsadiculty.Atthesecondorder,theparticleistomovealongageodesicofgab+hab,buthabissingularatthelocationoftheparticleandnotdierentiableon.Thisdicultycanbe

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resolvedviaadecompositionofhab,inwhichthesingularparthSabisidentiedandremovedfromhabtoleavetheremaininghRabonly.Thepointparticlewouldthenmovealongageodesicofgab+hRab[ 10 ]. wherehSabistermedtheSingulareldandhRabistermedtheRegulareld.ThisfollowsfromthesamespiritaswehadforthescalareldinSection 4.1 ofChap-ter 4 .AnalogoustoSandR,hSabandhRabarenaturalsolutionsoftheperturbedEinsteinequations( 5-6 )inaneighborhoodof:hSabhasonlythemassasitssource,whilehRabisavacuumsolution[ 10 ],i.e. Analternativewayofsplittinghactabis However,ifhtailabwereinsertedintoEq.( 5-6 ),itwouldyieldaphysicallyinfeasiblequantityTtailab.Further,aspointedoutinRef.[ 16 ],unlessRacbducud=0atthelocationoftheparticle,htailabisnon-dierentiablethere.Thus,theapproachbasedonthisdecompositiondoesnotclearlyexplaintheself-forceintermsofgeodesicmotioninanactualgravitationaleld. Ourcalculationsofgravitationalself-forceeectswillbebasedonthedecom-position( 5-14 ),andasinthecaseofscalareldself-forcethesingularparthSabisregardedasnon-contributingtotheself-force,whiletheremaininghRabisseen

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toberesponsiblefortheself-force.DescriptionsofthetwoeldsaregiveninthefollowingSubsections. 4.3 ofChapter 4 willservethispurposebestalsoinourgravitationalself-forceproblem.WithTHZcoordinatesthebackgroundmetricis where2HABisdenedbyEq.( 4-50 )and(X2+Y2+Z2)1=2withX,Y,ZbeingthespatialTHZcoordinatesandRrepresentsalengthscaleofthebackgroundgeometry. ToavoidthesingularityinhabinourperturbationanalysisofSection 5.2 ,wereplacethepointparticlemodelbyasmallSchwarzschildblackhole.Thenthedicultycausedbytheformalsingularityisreplacedbytherequirementofboundaryconditionsattheeventhorizon.WhenasmallSchwarzschildblackholeofmassmovesthroughabackgroundspacetime,themetricofthesmallblackholeisperturbedbytidalforcesarisingfrom2HABinEq.( 5-18 ),andtheactualmetricneartheblackholeiswritteninTHZcoordinatesas[ 10 ] wherethequadrupolemetricperturbation2hABisasolutionoftheperturbedEinsteinequations( 5-6 )withtheappropriateboundaryconditionsthattheperturbationbewellbehavedonthefutureeventhorizonandthat2hAB!2HABinthebuerregion[ 20 ],whereR.ForR,2hABisgovernedby

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theRegge-Wheeler[ 33 ]orZerilli[ 34 ]waveequationwithapotentialbarrier.Forsimplicity,consideringonlythetimeindependentlimit,whichisrelevanttotheexpansionofthebackgroundinEq.( 5-18 ),thisadmitsananalyticsolution 3KPQBQIXPXI(12=)dTdXK; where(T;;#;')isthesphericalpolarrepresentationofTHZcoordinates(T;X;Y;Z).Thisiswellbehavedontheeventhorizonandmatches2HABwhen. Inthisregion,theactualmetricgactABisequallywelldescribedeitherbythebackgroundmetricbeingperturbedbythesmallmass,orbytheleading=termsoftheSchwarzschildmetricbeingperturbedbyweaktidalforces[ 10 ].WithTHZcoordinatesthebackgroundmetricisexpressedbyEq.( 5-18 )andtheactualmetricis whereeachhAB[n]isthepartofthemetricperturbationwhichisproportionalton.ThelinearparthAB[]isresponsibleforthesingularsourceeld[ 10 ], where dT2+d2(5-23) isthe=partoftheSchwarzschildmetricgSchwAB,and EIJXIXJdT28 isthe=R2partof2hABfromEq.( 5-20 ).

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5-22 )-( 5-24 )isintheRegge-Wheelergauge.WemaytransformthisintotheLorenzgaugevia[ 10 ] wherethegaugevectorAisgivenby Thisresultsin 1+EIJXIXJdT2+1EIJXIXJKLdXKdXL4EIJdXIdXJ+4 Forcompleteness,thetraceofhS(lz)ABisgivenby +O(2=R3);(5-28) anditstrace-reversedformhS(lz)ABhS(lz)AB1 2g0ABg0CDhS(lz)CDis hS(lz)ABdXAdXB=4 1+EIJXIXJdT24EIJdXIdXJ4 whichsatisestheLorenzgaugecondition[ 10 ]

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5-14 ),theregularremaindereldisdenedby inaneighborhoodofwherehRabsatisesthevacuumEinsteinequations( 5-16 ).hRabdoesnotchangeoveranO()lengthscale,andwiththeO()correctionsincluded,theworld-lineofapointmassthroughthebackgroundisageodesicofg0ab+hRab.Thisstatementisjustiedbytheconsistencyofthematchedasymptoticexpansions,andadetaileddiscussionofthisisfoundinDetweiler[ 10 ]. 35 ]presentsanelementaryexampleofaself-forceeectusingNewtoniananalysis.AsmallmassrevolvingaroundamoremassiveobjectMinacircularorbitofradiusRhasanangularfrequencygivenby 2=M R3(1+=M)2:(5-32) Whenisinnitesimal,thelargermassMdoesnotmove,theradiusoftheorbitRisequaltotheseparationbetweenthemassesand2=M=R3.However,whenisstillsmallbutnite,thetwomassesorbittheircommoncenterofmasswithaseparationofR(1+=M),andtheangularfrequencyisasgiveninEq.( 5-32 ).TheniteinuencesthemotionofM,whichtheninuencesthegravitationaleldwithinwhichmoves.Thisbackactionofuponitsownmotionisthetypicalcharacteristicofaself-force,andthedependenceofEq.( 5-32 )isproperlydescribedasaNewtonianself-forceeect[ 10 ].ForM,Eq.( 5-32 )canbeexpandedas 2M R312=M+O(2=M2):(5-33)

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Thenwendthatthenitemassratio=Mbringsachangeintheorbitalfre-quencybyafractionalamount = M:(5-34) TheextensionofthisNewtonianproblemtogeneralrelativitywouldbethesimplestandinterestingexampleforourrelativisticgravitationalself-forceproblem.InthisSectionwefocusonasmallmassinacirculargeodesicaboutaSchwarzschildblackholeofmassMandstudytheeectsofself-forceonsomegaugeinvariantquantitiessuchastheangularfrequencyandthecombinedquantityEJwhereEandJarereminiscentoftheparticle'senergyandangularmomentumperunitrestmass|astheperturbationbreaksthesymmetriesoftheSchwarzschildgeometry,thereisnonaturallydenedenergyorangularmomentumfortheparticle.Ourattemptofevaluatingthechangesinthesegaugeinvariantquantitiesastheeectsofself-forcewillavoidtheambiguityposedbythegaugefreedominMiSaTaQuWa'sapproachasdescribedinSection 5.1 10 ]presentstheexamplesofgaugeinvariantquantitiesinafewdierentcategories.Amongthem,werstfocusontheangularfrequency,whichwillcorrespondtoadirectobservablemeasuredatinnity.Ref.[ 10 ]givestheexpressionofforacircularorbitasmeasuredatinnity 2=(d=dt)2=u=ut2=M R3R3M whereuarepresentsthefour-velocityoftheparticlemovinginacircularorbitontheequatorialplane,whosecomponentsaredenedby AnothergaugeinvariantquantityweareinterestedinisEJ,whichisthecontractionofuawiththeKillingvectora.Itsexpressionforacircularorbitis

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[ 10 ] (EJ)2=13M R1uaubhab+1 2Ruaub@rhab:(5-37) 5-35 )and( 5-37 )containthetermsthatdependonhabor@rhab,whichimplytheeectsofself-forceduetotheperturbationofthegeometrybythepresenceofthepointmass.Thecontributionsfromthesetermscanbeevaluatedviathesametechniqueofmode-sumregularization,whichwaspioneeredbyBarackandOri[ 9 ],asusedforthescalareldprobleminSection 4.2 ReggeandWheeler[ 33 ]andZerilli[ 34 ]showhowtoobtainthemetricper-turbationsofSchwarzschildviasphericalharmonicdecomposition.BothTabandhabarefourieranalyzedintime,withfrequency!,anddecomposedintermsofsphericalharmonics,withmultipoleindices`andm.Linearcombinationsofthecomponentsofh`m;!absatisfyordinarydierentialequationswhichcanbenumeri-callyintegrated.Theperiodicityofacircularorbitmakesadiscretesetfrequencies!m=m[ 10 ]. Assumingthath`m;!ab(r)canbedeterminedforany`andm,thesumoftheseoverall`andmwillconstitutehactab.Ifevaluatedatthelocationof,however,thissumwilldiverge.SubtractingthesingularityhS(`m;!)abfromhact(`m;!)abandsummingthedierenceover`andm,weobtainaconvergentsum[ 10 ] whichistheregularremaindereld.Similarly,wecandetermineitsderivativevia InReggeandWheeler'sanalysis[ 33 ],theindividualcomponentsh`mabareclassiedaccordingtotheirangularmomentumpropertiesunderarotationofthe

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framearoundtheorigin,wheretherotationisconnedtothetwo-dimensionalsubmanifold:ft=const;r=const;;g.Theyareseparatedintothethreegroups: (i) Scalars`m `m=constY`m(;);(5-40) (ii) Vectors`m^a `m^a(even)=constY`m;^a;parity()`; `m^a(odd)=const^a^bY`m;^b;parity()`+1; (iii) Tensors`m^a^b `m^a^b(scalar)=const^a^bY`m; `m^a^b(even)=constY`m;^a^b;parity()`; `m^a^b(odd)=const^a^cY`m;^c^b+^b^cY`m;^c^a;parity()`+1; wherethelabels^aand^brunoverand,andthesemicolondenotescovariantdierentiation,and^a^brepresentsthemetrictensoronthetwo-dimensionalsphere,denedby and^a^bisthealternatingtensoron,denedby

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and Dependinguponthecoordinatelabelsaandbofhactab,i.e.uponwhetherhactabisascalar,avectororatensor,wehavecorrespondingexpansionbases,whicharescalar,vectorortensorsphericalharmonics.Then,takingthesumsovermrstinEqs.( 5-38 )and( 5-39 )usingthesebases,wesimplifythetwo-indexmode-sumstotheone-indexmode-sumsover`only, 5-49 )and( 5-50 )canbedeterminedfullyanalytically,andaredescribedbyregularizationparameters.Themethodstodeterminetheregularizationparametersaresimilartothatforthescalareldproblem.ThedierenceisthathSab(or@chSab)istreatedasascalar,avectororatensor,dependingonthecoordinatelabels,tocomplywiththeanalysisbyReggeandWheeler[ 33 ]above,andforeachdierenttypeweneeddevelopadierentstrategyforcalculation. TheregularizationparametersforhSabcanbecalculated,forexampleintheLorenzgaugeasinEq.( 5-27 ),inasimilarmannertothatinSection 4.4 .First,weneedtondthefunctionalexpressionsofhSabinthebackgroundcoordinatesxa=(t;r;;)usingEqs.( 5-27 )andEqs.( 4-129 )-( 4-141 )viathetransformation whereAlabelsTHZcoordinatesXA=(T;X;Y;Z).ThenwecategorizethefunctionshSab(t;r;;)intothethreegroups:

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(i) Scalarsourcefunctions;fhStt(t;r;;);hStr(t;r;;);hSrr(t;r;;)g, (ii) Vectorsourcefunctions;fhSt(t;r;;);hSt(t;r;;);hSr(t;r;;);hSr(t;r;;)g, (iii) Tensorsourcefunctions;fhS(t;r;;);hS(t;r;;);hS(t;r;;)g. TheregularizationparametersforthethreescalarsfhStt;hStr;hSrrgcanbecalculateddirectlyfromthescalarsourcefunctions(i)above.However,thecasesofvectorsfhSt;hSt;hSr;hSrgandtensorsfhS;hS;hSgaremorecomplicated.Theirregularizationparameterscannotbedetermineddirectlyfromthesourcefunctions(ii)and(iii)above.Theymustreectthedistinctpropertiesunderarotationonthetwo-dimensionalsubmanifoldasshowninthepreviousSubsection.Thentheproperfunctionalformsforvectorsandtensors,fromwhichourregularizationparametersarecalculated,mustbedeterminedbyconsideringtheirgeometricalproperties.DetweilerandWhiting[ 11 ]presentclearanalysesonthis,whichwillprovideuswiththeframeworkforthecalculationsofvectorandtensorregularizationparameters.Thebriefsummaryfollows: (A) Vectors Anarbitraryvectoreld^a(2fhSt;hSt;hSr;hSrg),denedonthetwo-dimensionalsubmanifold,mayberepresentedbytwoscalareldsevandodas whereevandodarecalledtheevenandoddparitypotentialsof^a,andareeachuniqueuptotheadditionofaconstant.Thecomponentsevandodaredeterminedby and

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(B) Symmetrictensors AnarbitrarysymmetrictensoreldF^a^b(2fhS;hS;hSg)mayberepre-sentedbyascalareldFTraceandavectoreldF^aas 2^a^bFTrace+2r(^aF^b)^a^br^cF^c;(5-55) where andthevectorF^bisasolutionof 2RF^b=r^aF^a^b1 2^a^bFTrace;(5-57) andRisthescalarcurvatureofthegeometry^a^b.F^bisuniqueuptotheadditionofavectoreldk^awhichsatisestheconformalKillingequation, Further,F^a^bmayberepresentedintermsofthescalareldsFevandFodbysubstitutingEq.( 5-52 )intoEq.( 5-55 ), 2^a^bFTrace+2r(^ar^b)^a^br2Fev+2(^a^cr^b)r^cFod;(5-59) wherethethreescalareldsFTrace,FevandFodarereferredtoasthetrace,theevenparitypotentialandtheoddparitypotential,respectively,andtheyaredeterminedbyEqs.( 5-56 ),( 5-53 )and( 5-54 )viaEqs.( 5-57 )and( 5-52 ). Oncetheirproperfunctionalformsarefound,theregularizationparametersforvectorsfhSt;hSt;hSr;hSrgandtensorsfhS;hS;hSgarecalculatedsimilarlytothecaseofscalarsfhStt;hStr;hSrrg.BelowwepresentthekeystepsofcalculatingtheleadingorderregularizationparametersforhSabforageneralorbit. Scalars:fhStt;hStr;hSrrg

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However,eachcomponenthS`mab(scalar)maybeexpressedintermsofsphericalharmon-icsalso, whered0sin0d0d0andhSab(scalar)(;)2fhStt(t;r;;);hStr(t;r;;),hSrr(t;r;;)g. Now,substitutingEq.( 5-61 )intoEq.( 5-60 ),wehave Ref.[ 28 ]showsthat 2`+1 4P`(cos)=XmY`m(0;0)Y`m(;);(5-63) where cos=coscos0+sinsin0cos(0):(5-64) However,inthecoincidencelimitxa!xao,wemayhave(;)!(=2;o)!(=2;),whereoJ_r=f(r2o+J2)accordingtoEq.( 4-149 ).Then,cosabovebecomes,inthecoincidencelimit, cos!sin0cos(0)=cos;(5-65) bythedenitionoftherotatedangles(;)giveninEq.( 4-151 ).ByEqs.( 5-63 )and( 5-65 ),hSab(scalar)inEq.( 5-62 )becomesinthecoincidencelimit 2Id 2hSab(scalar)(;)P`(cos);(5-66) wherewerotatedtheanglesintheintegral,fromd0sin0d0d0todsindd.Now,weseparatethevariablesintheintegrandhSab(scalar)(;)and

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decomposeitintozonalharmoniccomponents Then,substitutingEq.( 5-67 )intoEq.( 5-66 )andusingtheidentityfromRef.[ 28 ] whereXcos,wecanrewriteEq.( 5-66 )as 2F`():(5-69) Thisissimply usingthenotation\hi"fortheintegration-averagingprocess.Then,wendour`-modesingulareldhS`ab(scalar)fromEq.( 5-70 ), whereF`()isthe`-thcomponentoftheLegendrepolynomialexpansionsofhSab(scalar)(;)asinEq.( 5-67 ),andhSab(scalar)(;)isobtainedfromthescalarsourcefunctions,fhStt(t;r;;);hStr(t;r;;);hSrr(t;r;;)g,viathecoordinaterotation. To0-order,hSab(scalar)hasthestructure ~o1+linearpolynomialsinxa wherexadenotestheSchwarzschildcoordinates(t;r;;).Fromthiswemaydevelopstructureanalysesfor1-termand0-termsimilartothoseinSection 4.4 tondthenon-vanishingregularizationparametersviaEq.( 5-71 ). Theresultsofthecalculationsare

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where and Vectors:fhSt;hSt;hSr;hSrg FromEq.( 5-52 )wedene and wherethelabelcrunsovertandr,andthelabel^arunsoverand. Takingthecasewithc=tand^a=forexample,wehave

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ThescalarpotentialsevandodmustsatisfythePoisson'sequations accordingtoEqs.( 5-53 )and( 5-54 )alongwithEq.( 5-80 ).However,onecanalsodecomposeevandodintosphericalharmoniccomponents, Then,wemaysolvethePoisson'sequations( 5-85 )and( 5-86 )intermsofsphericalharmonicsviaEqs.( 5-87 )and( 5-88 ),andobtaintheexpressionsfor`mevand`mod whered0sin0d0d0. Now,substitutingEqs.( 5-87 )and( 5-88 )intoEqs.( 5-83 )and( 5-84 )alongwithEqs.( 5-89 )and( 5-90 ),respectively,wehave @"XmY`m(0;0)Y`m(;)#; @"XmY`m(0;0)Y`m(;)#:

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FromEq.( 5-63 )wemayderive 2`+1 4@ @P`(cos)=@ @"XmY`m(0;0)Y`m(;)#; 2`+1 4sin@ @P`(cos)=sin@ @"XmY`m(0;0)Y`m(;)#: wherecos=coscos0+sinsin0cos(0).Inthecoincidencelimitxa!xao,wehavecos!sin0cos(0)=cos,whereoJ_r=f(r2o+J2),withthedenitionoftherotatedangles(;)giveninEq.( 4-151 ).UsingthechainrulealongwithEq.( 4-151 ),wemayrewriteEqs.( 5-93 )and( 5-94 )inthecoincidencelimitas @"XmY`m(0;0)Y`m(;)#xa!xao=2`+1 4sincosd dXP`(X);(5-95) sin@ @"XmY`m(0;0)Y`m(;)#xa!xao=2`+1 4sinsind dXP`(X);(5-96) whereXcos.UsingEqs.( 5-95 )and( 5-96 )forEqs.( 5-91 )and( 5-92 ),respectively,inthecoincidencelimitxa!xao,wehave 4`(`+1)IdF(;)sincosd dXP`(X); 4`(`+1)IdG(;)sinsind dXP`(X); wheretheangleswererotatedintheintegrals,fromd0sin0d0d0todsindd,andaccordinglytheintegrandschangedtheirvariables, OneshouldnotethatEqs.( 5-97 )and( 5-98 )containdP`(X)=dXratherthanP`(X)unlikeEq.( 5-66 )inthecaseofscalars.TheserstderivativesoftheLegendrepolynomialsareorthogonalover(1;1)withweightingfunction1X2

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andsatisfy[ 28 ] dXPn(X)d dXP`(X)=2n` (`1)!:(5-101) Inordertotakeadvantageofthisidentity,wemayrewriteEqs.( 5-97 )and( 5-98 )as 2`(`+1)Z20d 2Z11dX(1X2)F(;)sincos 1X2d dXP`(X);(5-102) 2`(`+1)Z20d 2Z11dX(1X2)G(;)sinsin 1X2d dXP`(X):(5-103) Now,weseparatethevariablesinF(;)sincos=(1X2)andG(;)sinsin=(1X2)andexpandthemindPn(X)=dX, 1X2=XnFn()d dXPn(X); 1X2=XnGn()d dXPn(X): Then,substitutingEqs.( 5-104 )and( 5-105 )intoEqs.( 5-102 )and( 5-103 ),respec-tively,weobtainviaEq.( 5-101 ), 2F`()=X`hF`()i; 2G`()=X`hG`()i: Fromthesewendour`-modesingulareldshSev`tandhSod`t, andbycombination

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whereF`()andG`()arethe`-thcomponentsoftheexpansionsofF(;)sincos=(1X2)andG(;)sinsin=(1X2)indPn(X)=dX,asinEqs.( 5-104 )and( 5-105 ),respectively,andF(;)andG(;)areobtainedfromEqs.( 5-99 )and( 5-100 )togetherwiththevectorsourcefunctions,fhSt(t;r;;);hSt(t;r;;);hSr(t;r;;);hSr(t;r;;)g,viathecoordinaterotation. Tothersttwohighestorders,hSthasthestructure ~o33+linearpolynomialsinxa wherexadenotestheSchwarzschildcoordinates(t;r;;).Eq.( 5-111 )hastheexpansionbasisdP`(X)=dX,and3-termand2-termherewouldcorrespondto1-termand0-termoftheexpansionwiththebasisP`(X)asinEq.( 5-72 ),sincetheweightingfunction1X2=sin2playingintoEqs.( 5-101 ),( 5-102 )and( 5-103 )makesupfor2inthelimit!0.Structureanalysesfor3-termand2-termmaybedevelopedinasimilarmannertothatinSection 4.4 sothatwecanndthenon-vanishingregularizationparametersviaEqs.( 5-108 )and( 5-109 ). TheothercasesforhSt,hSrandhSrcanbetreatedinthesamemannerasabove.Theresultsofthecalculationsare where

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Tensors:fhS`;hS`;hS`g FromEq.( 5-59 )wedene and 2^a^bFTrace; wherethelabels^a,^band^crunoverand. Takingthecasewith^a=and^b=forexample,wehave 2FTrace; HereFTraceisdenedbyEqs.( 5-56 )and( 5-123 )as

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andthescalarpotentialsFevandFodmustsatisfythedierentialequationsaccordingtoEq.( 5-57 )alongwithEq.( 5-52 ), 2^a^bFTrace;(5-131) whereRisthescalarcurvatureofthegeometry^a^b.However,onecanalsodecomposeFevandFodintosphericalharmoniccomponents, Then,wemaysolvethedierentialequations( 5-131 )intermsofsphericalharmon-icsviaEqs.( 5-132 )and( 5-133 ),andobtain 2^a^bFTrace: TosingleouteachexpressionofF`mevandF`mod,wecontractEq.( 5-134 )withr^band^b^dr^d,respectively,andobtain 2^a^bFTrace;(5-135) 2^a^bFTrace:(5-136) IntegratingEqs.( 5-135 )and( 5-136 )overdY`0m0(;),wefurtherobtain 2^a0^b0FTraceY`m(0;0);(5-137)

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2^a0^b0FTraceY`m(0;0);(5-138) whered0sin0d0d0,andR=2sincethescalarcurvatureR=2=r2foratwo-sphereofradiusrandourtwo-spherehastheunitradius[ 11 19 ]. Now,substitutingEqs.( 5-132 )and( 5-133 )intoEqs.( 5-128 )and( 5-129 )alongwithEqs.( 5-137 )and( 5-138 ),respectively,wehave (`1)`(`+1)(`+2)Id0r^b0r^a0F^a0^b01 2^a0^b0FTrace@2csc2@2cot@"XmY`m(0;0)Y`m(;)#; (`1)`(`+1)(`+2)Id0^b0^d0r^d0r^a0F^a0^b01 2^a0^b0FTrace2csc(@@cot@)"XmY`m(0;0)Y`m(;)#; whereweusedtheequality`2(`+1)22`(`+1)=(`1)`(`+1)(`+2).FromEq.( 5-63 )wemayderive 2`+1 4@2csc2@2cot@P`(cos)=@2csc2@2cot@"XmY`m(0;0)Y`m(;)#; 2`+1 2csc(@@cot@)P`(cos)=2csc(@@cot@)"XmY`m(0;0)Y`m(;)#; wherecos=coscos0+sinsin0cos(0).Inthecoincidencelimitxa!xao,wehavecos!sin0cos(0)=cos,whereoJ_r=f(r2o+J2),withthedenitionoftherotatedangles(;)giveninEq.( 4-151 ).UsingthechainrulealongwithEq.( 4-151 ),wemayrewriteEqs.( 5-141 )and( 5-142 )inthe

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coincidencelimitas 41X2cos(2)d2 2csc(@@cot@)"XmY`m(0;0)Y`m(;)#xa!xao=2`+1 41X2sin(2)d2 whereXcos.UsingEqs.( 5-143 )and( 5-144 )forEqs.( 5-139 )and( 5-140 ),respectively,inthecoincidencelimitxa!xao,wehave 4(`1)`(`+1)(`+2)IdF(;)1X2cos(2)d2 4(`1)`(`+1)(`+2)IdG(;)1X2sin(2)d2

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wheretheangleswererotatedintheintegrals,fromd0sin0d0d0todsindd,andaccordinglytheintegrandschangedtheirvariables, 2^a0^b0FTrace(;)=1 2@201 2csc20@20+3 2cot0@01hS00+4csc20cot0(@0@0+@0)hS00+csc201 2@20+csc20@20+5cot20+2cot0@05cot20hS00(;); 2^a0^b0FTrace(;)=csc0(@0@0+cot0@0)hS00+csc0@20+csc20@20cot0@0+16csc0cot0+7cot20hS00+csc30(@0@0+cot0@0)hS00(;): OneshouldnotethatEqs.( 5-145 )and( 5-146 )containd2P`(X)=dX2ratherthanP`(X)unlikeEq.( 5-66 )inthecaseofscalars.ThesesecondderivativesoftheLegendrepolynomialsareorthogonalover(1;1)withweightingfunction(1X2)2andsatisfy[ 28 ] (`2)!:(5-149)

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Inordertotakeadvantageofthisidentity,wemayrewriteEqs.( 5-145 )and( 5-146 )as 2(`1)`(`+1)(`+2)Z20d 2Z11dX1X22F(;)cos(2) 1X2d2 2(`1)`(`+1)(`+2)Z20d 2Z11dX1X22G(;)sin(2) 1X2d2 Now,weseparatethevariablesinF(;)cos(2)=(1X2)andG(;)sin(2)=(1X2)andexpandthemind2Pn(X)=dX2, 1X2=XnFn()d2 1X2=XnGn()d2 Then,substitutingEqs.( 5-152 )and( 5-153 )intoEqs.( 5-150 )and( 5-151 ),respec-tively,weobtainviaEq.( 5-149 ), 2F`()=X`hF`()i; 2G`()=X`hG`()i: However,hSTrace1 2FTraceinEq.( 5-127 )mustbetreatedasascalaraccordingtoEq.( 5-130 ).Aswedidpreviously,werstrotatethecoordinates(;)!(;)intheexpressionofhSTracesothat 2hS+csc2hS(;):(5-156)

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Next,weseparatetheangularvariables(;)inhSTrace(;)anddecomposeitintothecomponentsoftheLegendrepolynomialexpansions Then,takingthesamestepsasinEqs.( 5-66 )-( 5-70 ),weobtain FromEqs.( 5-157 ),( 5-154 )and( 5-155 )wendour`-modesingulareldshSTrace`,hSev`andhSod`,respectively, andbycombination whereH`()isthe`-thcomponentoftheexpansionsofhSTrace(;)inPn(X)asinEq.( 5-157 ),andF`()andG`()arethe`-thcomponentsoftheexpansionsofF(;)cos(2)=(1X2)andG(;)sin(2)=(1X2)ind2Pn(X)=dX2asinEqs.( 5-152 )and( 5-153 ),respectively,andhSTrace(;),F(;)andG(;)areobtainedfromEqs.( 5-156 ),( 5-147 )and( 5-148 )togetherwiththetensorsourcefunctions,fhS(t;r;;);hS(t;r;;);hS(t;r;;)g,viathecoordinaterotation. 5-72 ),thusthesamestructureanalysesfor1-termand0-termapplytondthenon-vanishingregularizationparametersviaEq.( 5-159 ).hSev=odhasthestructuretothersttwohighestorders,

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~o55+linearpolynomialsinxa wherexadenotestheSchwarzschildcoordinates(t;r;;).Eq.( 5-163 )hastheexpansionbasisd2P`(X)=dX2,and5-termand4-termherewouldcorrespondto1-termand0-termoftheexpansionwiththebasisP`(X)asinEq.( 5-72 ),sincetheweightingfunction(1X2)2=sin4playingintoEqs.( 5-149 ),( 5-150 )and( 5-151 )makesupfor4inthelimit!0.Structureanalysesfor5-termand4-termmaybedevelopedinasimilarmannertothatinSection 4.4 sothatwecanndthenon-vanishingregularizationparametersviaEqs.( 5-160 )and( 5-161 ). TheothercasesforhSandhScanbetreatedinthesamemannerasabove.Theresultsofthecalculationsare

PAGE 153

where

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Abinaryinspiralofasmallblackholeofsolarmassandasupermassiveblackholeof105to107solarmass,calledanextrememass-ratiosystem,isoneofthepossibletargetsourcesofgravitationalwavesforLISAdetection[ 1 ].Anaccuratedescriptionoftheorbitalmotionofthesmallblackhole,includingtheeectsofradiationreactionandtheself-forceisessentialtodesigningthetheoreticalwaveformfromthisbinarysystem. Inthisdissertationwehavepresentedspecicmethodsforcalculatingtheeectsofradiationreactionandtheself-forceforthetwomodelsofsuchsystems:thecaseofascalarparticleorbitingaSchwarzschildblackholeandthecaseofapointmassorbitingaSchwarzschildblackhole.Inbothcasesourcalculationshavebeenimplementedviathe\mode-sum"methodpioneeredbyBarackandOri[ 9 ],inwhichtheself-forceortheeectsofself-forceareevaluatedfromthedierencebetweentheparticle'sowneldanditssingularpartviamode-decomposedmultipolemomentsofeachasinEq.( 4-32 )orEqs.( 5-38 )and( 5-39 ). Themode-decomposedmultipolemomentsofthesingulareldaredescribedbyRegularizationParametersasrepresentedinEqs.( 4-33 )-( 4-44 )forthecaseofthescalareldandinEqs.( 5-73 )-( 5-79 ),( 5-112 )-( 5-122 )and( 5-164 )-( 5-171 )forthecaseofgravitationaleld.Thedeterminationofregularizationparametershasinvolvedthetwomainanalyticaltasks:alocalanalysisofspacetimegeometryandstructureanalysesofthesingulareld.Thelocalanalysisofspacetimegeometryhasprovidedpowerfultools,suchasTHZnormalcoordinatesasshowninEqs.( 4-129 )-( 4-141 )alongwithEqs.( 4-117 )and( 4-118 ).WiththesewehaveobtainedthesimplestexpressionforthesingulareldasinEq.( 4-45 )forthe 144

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scalareldandasinEq.( 5-27 )forthegravitationaleldintheLorenzgauge.Then,thestructureanalysesofthesingulareld,asshowninEq.( 4-159 )alongwithEqs.( 4-160 )-( 4-168 ),( 4-187 )-( 4-209 ),( 4-228 )-( 4-237 )and( 4-238 )-( 4-254 )forthecaseofthescalareld,havefacilitatedthecalculationsoftheregularizationparameters.Withtheseanalysesthestrategiesofcalculationshavebeendevelopedforeachregularizationparameter,Aa,Ba,CaandDa.Similaranalysesapplytothecaseofthegravitationaleld.However,techniquesofcalculationsinvolvedinthegravitationaleldcasearemorecomplicatedthaninthescalareldcase.Thisisduetothedistinctgeometricalpropertiesofhabunderarotationonthetwo-sphereinaRegge-WheelerstyledecompositionasshowninSubsection 5.4.2 .ThedicultieswithregardtothegeometricalpropertieshavebeenresolvedbyfollowingtheanalysesbyDetweilerandWhiting[ 11 ],andthedierenttechniquesofcalculatingtheregularizationparametersforeachdierentgroup,scalarsfhStt;hStr;hSrrg,vectorsfhSt;hSt;hSr;hSrgandtensorsfhS;hS;hSg,alldenedonthetwo-spherehavebeendevelopedasinSubsection 5.4.3 Theapplicationsoftheseregularizationparametersinthemode-sumself-forcecalculationhaveshownniceresultsasinSection 4.5 forthecaseofthescalareld.Inparticular,theuseofDa-termsinthemode-sumcomputationhasprovidedmorerapidconvergenceandmoreaccuratenalresults.Calculationsofthegravitationalself-forceeectsarecurrentlyinprogress,anddeterminationsoftheregularizationparametersfor@chSabandthehigherorderregularizationparametersDabforhSabwillbedemanded,suchasforthemode-sumcalculationsofthechangesingaugeinvariantquantitieslikeandEJasinEqs.( 5-35 )and( 5-37 ). Also,theregularizationparametersfortheKerrspacetimewillbecalculatedviasimilarstrategiestotheSchwarzschildcasewithslightmodicationstothelocalanalysisofspacetimeforthedierentbackgroundgeometry.

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InSection 4.4 wedene with Andweuse 2;1;Fp: Inparticular,forthecasesp=1 2andp=1 2wehavethefollowingrepresentations 2;1 2;1;=2 and 2;1 2;1;=2 where^K()and^E()arecalledcompleteellipticintegralsoftherstandsecondkinds,respectively. IfwetakethederivativeofF1=2withrespecttokp A3 ),weobtain 146

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orusingEq.( A4 ) @k=^K k+ However,Ref.[ 36 ]showsthat @k=^E k(1k2)^K k:(A8) Thus,bycomparingEq.( A7 )andEq.( A8 )wendtherepresentation Further,wecanalsondtherepresentationforF5=2.First,takingthederiva-tiveofF3=2withrespecttokp A3 )gives Also,usingEq.( A9 )togetherwithEqs.( A3 )-( A5 ),anotherexpressionforthesamederivativeisobtainedsolelyintermsofcompleteellipticintegrals k(1k2)2:(A11) Then,byEqs.( A9 ),( A10 )and( A11 )wend 3"2(2)^E Now,usingEqs.( A4 ),( A9 )and( A12 ),wemayrewritethenon-zeroBaregularizationparameters,Eqs.( 4-37 )-( 4-39 )inSection 4.2 as

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whichareidenticaltotheresultsofBarackandOri[ 27 ].

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InStep(ii)ofSubsection 4.3.3 weobtainedtheexpressionsofFerminormalcoordinatesintermsofthestaticinertialcoordinates^XAas whereI=1;2;3and 6^APQ;RoPBQCRD+3QBRChPD+3RBhPChQD+hPBhQChRD; 24^APQ;RSoPBQCRDSE+4QBRCSDhPE+6RBSChPDhQE+4RBhPChQDhRE+hPBhQChRDhSE; with ThetransformationviaEqs.( B1 )and( B2 )reproducesthedesiredgeome-tryofFerminormalcoordinates.Onecanverifythisbyexaminingthemetric 149

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perturbationsofg(FN)ABderivedfromtheseequations.FromRef.[ 24 ]wehave CombiningthiswithEqs.( B1 )-( B4 )and( 4-90 )onends Toviewthenewgeometryproperly,oneshouldexpressitsmetricperturbationsintermsofthecoordinatesofthenewgeometryitself.Thus,wetaketheinverseof^XAviaEqs.( B1 )and( B2 )andspecifyittotherstorder ^XA=^uAoTFN+^nAo(I)XIFN+O(X3FN=R2);(B11) whichisessentiallythesameexpressionasEq.( 4-104 )consideringthatthequadratic-ordertermsareabsentfromtheexpansionsasaresultoftheintegration

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inEq.( 4-100 ).Also,fromEqs.( 4-93 )and( 4-95 )wederive ^HABCD=1 2^ABC;Do+^BAC;Do; ^HABCDE=1 6^ABC;DEo+^BAC;DEo; where^ABCAF^FBC.Then,substitutingEqs.( B3 ),( B4 ),( B11 ),( B12 )and( B13 )intoEqs.( B8 )-( B10 ),andexploitingthefactsthatAB^nBo(I)=0andthathAB^uBo=0,weobtain 3^RACDB;Eo^uAo^uBo^nCo(K)^nDo(L)^nEo(M)XKFNXLFNXMFN+O(X4FN=R4); 3^RACDBo^uAo^n(I)Bo^nCo(K)^nDo(L)XKFNXLFN+1 4^RACDB;Eo^uAo^n(I)Bo^nCo(K)^nDo(L)^nEo(M)XKFNXLFNXMFN+O(X4FN=R4); 3^RACDBo^n(I)Ao^n(J)Bo^nCo(K)^nDo(L)XKFNXLFN1 6^RACDB;Eo^n(I)Ao^n(J)Bo^nCo(K)^nDo(L)^nEo(M)XKFNXLFNXMFN+O(X4FN=R4); wheretheidentitiesusedare ^RABCDo=^ABD;Co^ABC;Do; ^RABCD;Eo=^ABD;CEo^ABC;DEo: Thelocaltetradvectors^Ao(P)n^uAo;^nAo(1);^nAo(2);^nAo(3)o,whereP2(0;1;2;3)isthelabelforeachvectorofthetetrad,essentiallyyieldtheinverse-LorentzboostbetweentheFerminormalframeandthestaticinertialframeviaEqs.( B1 )and

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( B2 ).Theseareinfactthetransformationfactorsbetweenthetwocoordinateframesevaluatedatthelocationoftheparticle ^uAo=0A=@^XA ^nAo(I)=IA=@^XA wherePArepresentstheinverseoftheLorentzboostPAsuchthatPAQA=PQ.Usingthese,wetransformtheRiemanntensoranditsrstderivativeas Then,ourmetricperturbationsEqs.( B14 )-( B16 )arenallyexpressedas 3R(FN)0K0L;MoXKFNXLFNXMFN+O(X4FN=R4); 3R(FN)0KILoXKFNXLFN1 4R(FN)0KIL;MoXKFNXLFNXMFN+O(X4FN=R4); 3R(FN)IKJLoXKFNXLFN+1 6R(FN)IKJL;MoXKFNXLFNXMFN+O(X4FN=R4); orwithdownstairsindicesas 3R(FN)0K0L;MoXKFNXLFNXMFN+O(X4FN=R4);

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3R(FN)0KILoXKFNXLFN1 4R(FN)0KIL;MoXKFNXLFNXMFN+O(X4FN=R4); 3R(FN)IKJLoXKFNXLFN1 6R(FN)IKJL;MoXKFNXLFNXMFN+O(X4FN=R4): Tothequadraticorder,theseresultsagreewithMisner,Thorne,andWheeler[ 19 ]andalsowithManasseandMisner[ 23 ]apartfromthesignsinfrontoftheRiemanntensors. 37 ]and[ 38 ].

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InStep(iii)ofSubsection 4.3.3 thetransformationbetweenFerminormalandTHZnormalcoordinateswasfoundtobe 168_EKLXKFNXLFN2FN+O(X5FN=R4); 6EIKXKFN2FN+1 3EKLXKFNXLFNXIFN1 6_EIKXKFN2FNTFN+1 3_EKLXKFNXLFNXIFNTFN1 24EIKLXKFNXLFN2FN+1 12EKLMXKFNXLFNXMFNXIFN2 63IMK_BMLXKFNXLFN2FN+O(X5FN=R4): OnecanverifythatEqs.( C1 )and( C2 )properlyconvertthemetricoftheFerminormalgeometryintothatoftheTHZnormalgeometrywiththehelpofsomepropertiesoftheRiemanntensorsforvacuumspacetime. FollowingRef.[ 24 ],weexamine 154

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viaEqs.( B23 )-( B25 )togetherwith( C1 )and( C2 ).Then,themetricfortheTHZcoordinatestakestheform 3R(FN)0K0L;MoXKFNXLFNXMFN+O(X4FN=R4); 3R(FN)0KILoXKFNXLFN1 4R(FN)0KIL;MoXKFNXLFNXMFN+3 28_EIKXKFN2FN11 28_EKLXKFNXLFNXIFN+O(X4FN=R4); 3R(FN)IKJLoXKFNXLFN+1 6R(FN)IKJL;MoXKFNXLFNXMFN1 3EIJ2FN+2 3E(IKXjKjFNXJ)FN+2 3IJEKLXKFNXLFN1 6EIJKXKFN2FN+5 12E(IKLXjKjFNXjLjFNXJ)FN4 63(IPK_BJ)PXKFN2FN8 63(IPK_BjPjLXjKjFNXjLjFNXJ)FN+O(X4FN=R4): ItisclearfromEqs.( C1 )and( C2 )thattheinversetransformationisidentityatthelinearorder, Then,ourtransformationfactors,whichareequivalenttothesetoflocaltetradvectorsintheTHZcoordinates,aresimply TheRiemanntensorsandtheirderivativesevaluatedatthelocationoftheparticlefollowcovarianttransformation,

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However,duetoEq.( C8 )wehave TheTHZcoordinatesdescribetheexternalmultipolemomentsofavacuumsolutionoftheEinsteinequationsaswecanseefromthemetricinEqs.( 4-49 )-( 4-51 ).Then,usingtheseequations,wecanalsoexpresstheRiemanntensorsandtheirderivativesintheTHZcoordinatesintermsoftheseexternalmultipolemoments.Atthelocationoftheparticletheyturnouttobe 3JKP_BPI+IKP_BPJ; 3IJPBPKL+1 3JK_EILIK_EJL; 3LMP_BPJ+JMP_BPL+JLEIKM+1 3KMP_BPI+IMP_BPKILEJKM+1 3KMP_BPJ+JMP_BPKJKEILM+1 3LMP_BPI+IMP_BPL; whereI;J;K;M;P=1;2;3.ThesearealltheconsequencesofthesolutionofthevacuumEinsteinequationswithdeDondergaugeconditions.ThegeneralalgorithmforndingthesolutionisdiscussedintheAppendixofRef.([ 20 ])andRef.([ 21 ]).

PAGE 167

SubstitutingEqs.( C13 )-( C18 )into( C4 )-( C6 )via( C11 )and( C12 )andusingEq.( C7 ),wenallyobtain 3EKLMXKXLXM+O(X4=R4); 3IKPBPLXKXL10 21_EKLXKXLXI+4 212_EKIXK+1 3IKPBPLMXKXLXM+O(X4=R4); 21IKP_BPLXKXLXJ+1 212IKP_BJPXK+1 3IJEKLMXKXLXM+O(X4=R4) (C21) or 3EKLMXKXLXM+O(X4=R4); 3IKPBPLXKXL10 21_EKLXKXLXI+4 212_EKIXK+1 3IKPBPLMXKXLXM+O(X4=R4); 21IKP_BPLXKXLXJ1 212KPI_BJPXK1 3IJEKLMXKXLXM+O(X4=R4); whichexactlymatchesEqs.( 4-49 )-( 4-51 ).

PAGE 168

REFERENCES [1] LaserInterferometerSpaceAntenna(LISA),http://lisa.jpl.nasa.gov/,AccessedJuly2005. [2] P.A.M.Dirac,Proc.R.Soc.(London) A167 ,148(1938). [3] B.S.DeWittandR.W.Brehme,Ann.Phys. 9 ,220(1960). [4] Y.Mino,M.Sasaki,andT.Tanaka,Phys.Rev.D 55 ,3457(1997). [5] T.C.QuinnandR.M.Wald,Phys.Rev.D 56 ,3381(1997). [6] T.C.Quinn,Phys.Rev.D 62 ,064029(2000). [7] B.S.DewittandC.M.Dewitt,Physics(LongIslandCity,NY) 1 ,3(1964). [8] M.J.PfenningandE.Poisson,Phys.Rev.D 65 ,084001(2002). [9] L.BarackandA.Ori,Phys.Rev.D 61 ,061502(R)(2000). [10] S.Detweiler,Class.QuantumGrav. 22 ,S681-S716(2005). [11] S.DetweilerandB.F.Whiting,DepartmentofPhysics,UniversityofFlorida, Gainesville,\Metricperturbationsandgaugetransformationsofspherically symmetricgeometries"(inpreparation)(2005). [12] G.A.Schott,Phil.Mag. 29 (1915). [13] J.L.Synge, Relativity:TheGeneralTheory (North-Holland,Amsterdam, 1960). [14] J.Hadamard, LecturesonCauchy'sProbleminLinearPartialDierential Equations (YaleUniversityPress,NewHaven,Connecticut,1923). [15] J.M.Hobbs,Ann.Phys.(NY) 47 ,141(1968). [16] S.DetweilerandB.F.Whiting,Phys.Rev.D 67 ,024025(2003),grqc/0202086. [17] D.-H.Kim,\Regularizationparametersfortheself-forceofascalar particleinageneralorbitaboutaSchwarzschildblackhole"(2004), http://arxiv.org/abs/gr-qc/0402014v2,AccessedJuly2005. [18] S.Detweiler,E.Messaritaki,andB.F.Whiting,Phys.Rev.D 67 ,104016 (2003). 158

PAGE 169

159 [19] C.W.Misner,K.S.Thorne,andJ.A.Wheeler, Gravitation (Freeman,San Fransisco,1973). [20] K.S.ThorneandJ.B.Hartle,Phys.Rev.D 31 ,1815(1985). [21] X.-H.Zhang,Phys.Rev.D 34 ,991(1986). [22] K.S.ThorneandS.J.Kovacs,Astrophys.J. 200 ,245(1975). [23] F.K.ManasseandC.W.Misner,J.Math.Phys. 4 ,735(1963). [24] S.Weinberg, GravitationandCosmology (Wiley,NewYork,1972). [25] F.deFeliceandC.J.S.Clarke, RelativityonCurvedManifolds (Cambridge UniversityPress,Cambridge,1990). [26] Y.Mino,H.Nakano,andM.Sasaki,Prog.Theor.Phys. 108 ,1039(2002), gr-qc/0111074. [27] L.BarackandA.Ori,Phys.Rev.D 66 ,084022(2002),gr-qc/0204093. [28] J.MathewsandR.L.Walker, MathematicalMethodsofPhysics (W.A. Benjamin,NewYork,1970),2nded. [29] L.M.Burko,Phys.Rev.Lett. 84 ,4529(2000). [30] E.Poisson,LivingRev.Relativity 7 ,6(2004), http://www.livingreviews.org/lrr-2004-6,AccessedJuly2005. [31] L.BarackandA.Ori,Phys.Rev.D 64 ,124003(2001). [32] A.P.Lightman,W.H.Press,R.H.Price,andS.A.Teukolsky, ProblemBook inRelativityandGravitation (PrincetonUniversityPress,Princeton,1979). [33] T.ReggeandJ.A.Wheeler,Phys.Rev. 108 ,1063(1957). [34] F.J.Zerilli,Phys.Rev.D 2 ,2141(1970). [35] S.DetweilerandE.Poisson,Phys.Rev.D 69 ,084019(2004). [36] G.B.ArfkenandH.J.Weber, MathematicalMethodsforPhysicists (Harcourt/AcademicPress,SanDiego,2001),5thed. [37] W.-Q.LiandW.-T.Ni,J.Math.Phys. 20 ,1473(1979). [38] P.L.FortiniandC.Gauldi,NuovoCimento 71 ,B37-B54(1982).

PAGE 170

Dong-HoonKimwasborninSeoul,thecapitalcityofKorea,onJune19,1970.HereceivedhisBachelor'sdegreeinPhysicsfromSogangUniversityin1996.ThenhewenttoEnglandtostudyMathematicalphysicsforhisMasterofScienceattheUniversityofDurham,andreceivedhisMasterofSciencedegreein1999.Infallof2000,hejoinedthePh.D.programinPhysicsattheUniversityofFlorida.Sincefallof2001,hehaspursuedhisresearchonGeneralRelativityunderthesupervisionofProf.SteveDetweiler. 160


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RADIATION REACTION IN CURVED SPACETIME


By

DONG-HOON KIM

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005


































Copyright 2005

by

Dong-Hoon Kim
















To my family.















ACKNOWLEDGMENTS

First and foremost I would like to thank my research advisor Professor Steven

Detweiler, for his constant encouragement and guidance throughout the entire

course of my research work. I would also like to thank Professor Bernard Whiting

and Professor Richard Woodard for valuable discussions. I am honored and grateful

to Professor James Fry, Professor David Reitze, and Professor Ata Sarajedini for

serving on my supervisory committee.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

FIGURE ...................... ................ viii

ABSTRACT ...................... ............. ix

CHAPTER

1 INTRODUCTION .................... ....... 1

2 GENERAL FORMAL SCHEMES FOR RADIATION REACTION ... 5

2.1 Dirac: Radiating Electrons in Flat Spacetime ............ 5
2.1.1 The Fields Associated with an Electron ............ 5
2.1.2 The Equations of Motion of an Electron ............ 8
2.2 Dewitt and Brehme: Electromagnetic Radiation Damping in Curved
Spacetime ...................... ........ 12
2.2.1 Bi-tensors .................. ......... .. 12
2.2.2 Green's Functions in Curved Spacetime . . ... 18
2.2.3 Electrodynamics in Curved Spacetime . . ... 21
2.2.4 Derivation of Equations of Motion for an Electric Charge
via World-tube Method ............ ... .. .. 25

3 CALCULATIONS OF SELF-FORCE: REVIEW OF GENERAL SCHEMES
AND ANALYTICAL APPROACHES .................. .. 33

3.1 General Formal Schemes Revisited ................. .. 34
3.1.1 Dirac: Radiating Electrons in Flat Spacetime . ... 34
3.1.2 Dewitt and Brehme: Electromagnetic Radiation Damping in
Curved Spacetime ....... . . ... 35
3.1.3 Quinn: Radiation Reaction of Scalar Particles in Curved Space-
time ..... ........ .... ... ...... 36
3.1.4 Mino, Sasaki, and Tanaka; Quinn and Wald: Gravitational
Radiation Reaction of Particles in Curved Spacetime . 37
3.2 Analytical Calculations of Self-force ................. .. 38
3.2.1 Dewitt and Dewitt: Falling Ci'! ges . . 38
3.2.2 Pfenning and Poisson: Scalar, Electromagnetic, and Gravi-
tational Self-forces in Weakly Curved Spacetimes . 40









4 PRACTICAL SCHEMES FOR CALCULATIONS OF SELF-FORCE
(A): SCALAR FIELD ............................ 43

4.1 Splitting the Retarded Field .......... .. .... ...... 45
4.1.1 Conventional Method of Splitting the Retarded Field .... 45
4.1.2 New Method of Splitting the Retarded Field . ... 49
4.2 Mode-sum Decomposition and Regularization Parameters ..... 51
4.3 Description of Singular Field and THZ Coordinates . ... 56
4.3.1 Introduction of THZ Coordinates . . . 56
4.3.2 Approximation for the Singular Field in THZ Coordinates .60
4.3.3 The Determination of THZ Coordinates . . ... 65
4.3.4 Determination of the Singular Field . . ..... 80
4.4 Determination of Regularization Parameters ..... . 82
4.4.1 Aa-terms ............... ......... .. 87
4.4.2 Ba-term s .................. .......... .. 93
4.4.3 Co-term s .. ... .. .. .. ... .. .. .. ... .. 101
4.4.4 D,-terms . . ... .......... 104
4.5 An Example: Self-force on Circular Orbits about a Schwarzschild
Black Hole ................ . . .. 110

5 PRACTICAL SCHEMES FOR CALCULATIONS OF SELF-FORCE
(B): EFFECTS OF GRAVITATIONAL SELF-FORCE . ... 113

5.1 MiSaTaQuWa Gravitational Self-force and Gauge Issues ..... ..113
5.2 First Order Perturbation Analysis .................. .. 114
5.3 Decomposition of the Perturbation Field hab ........... 117
5.3.1 Singular Field hb ................... ..... 118
5.3.2 Regular Field hAb ............... .. .... .. 121
5.4 An Example: Self-force Effects on Circular Orbits in the Schwarzschild
Geometry ..... .. ........ ...... 121
5.4.1 Gauge Invariant Quantities .................. 122
5.4.2 Mode-sum Regularization .............. .. 123
5.4.3 Regularization Parameters .................. 125

6 CONCLUSION .................. ............. 144

APPENDIX

A HYPERGEOMETRIC FUNCTIONS AND REPRESENTATIONS OF
REGULARIZATION PARAMETERS .................. 146

B THE GEOMETRY OF FERMI NORMAL COORDINATES ....... 149

C THE TRANSFORMATION BETWEEN FERMI NORMAL AND THZ
NORMAL GEOMETRIES .................. ....... 154

REFERENCES .................. ................ 158









BIOGRAPHICAL SKETCH ................... ........ 160















FIGURE
Figure page

4-1 Self-force of a scalar field in the Schwarzschild spacetime . ... 112















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RADIATION REACTION IN CURVED SPACETIME

By

Dong-Hoon Kim

August 2005

C('! i: Steven L Detweiler
Major Department: Physics

A binary inspiral of a small black hole of solar mass and a supermassive black

hole of 105 to 107 solar mass, called an extreme mass-ratio system, is one of the

possible target sources of gravitational waves for LISA (Laser Interferometer Space

Antenna) detection. An accurate description of the orbital motion of the small

black hole, including the effects of radiation reaction and the self-force is essential

to designing the theoretical waveform from this binary system.

One can calculate the effects of radiation reaction and the self-force for the

two models of such systems: the case of a scalar particle orbiting a Schwarzschild

black hole and the case of a point mass orbiting a Schwarzschild black hole. As

for the former, the interaction of a scalar point charge with its own field results in

the self-force on the particle, which includes but is more general than the radiation

reaction force. In the vicinity of the particle in curved spacetime, one may follow

Dirac and split the retarded field of the particle into two parts: (1) the singular

source field which resembles the Coulomb potential near the particle, and (2) the

regular remainder field. The singular source field exerts no force on the particle,

and the self-force is entirely caused by the regular remainder. As for the latter, a









point mass interacts with the metric perturbations created by itself when it moves

through the background geometry. Similarly, the perturbation field can be split

into two parts: (1) the singular source field which resembles the Coulomb potential

near the particle, tidally distorted by the local Riemann tensor of the background

and exerts no force back on the particle itself, and (2) the regular remainder field

which is entirely responsible for the self-force as the particle moves along a geodesic

of the perturbed geometry.

In this dissertation we describe systematic methods for finding multiple

decompositions of the singular source fields for both cases. This important step

leads to the calculation of the self-force on a scalar-charged particle or a point mass

orbiting a Schwarzschild black hole.















CHAPTER 1
INTRODUCTION

Einstein's General Theory of Relativity is a fundamental theory of gravitation

and spacetime. It has described with great accuracy and precision many phenom-

ena in our physical universe that classical physics has not been able to explain

successfully, such as the perihelion motion of planets and the bending of starlight

by the Sun. It has also made many significant predictions such as the existence of

gravitational waves, black holes and the expansion of the universe.

Among other predictions, gravitational waves might be the most exciting

problem these di ;- since the possible detection of them could help reveal infor-

mation about the very structure of their origins and about the nature of gravity,

thus would open up a new window for our understanding of the universe both from

physics and from astronomy. Gravitational waves can be described as ripples in

the fabric of spacetime caused by violent .,-lr i .!r, v-i I1 events in the distant uni-

verse, for example the coalescence of binary black holes or the inspiral of compact

objects into the supermassive black holes. Although the detection of gravita-

tional waves has been known to be technically challenging, scientists are eager to

implement experiments which propose to detect gravitational waves. Currently,

several ground-based detectors are in operation or under construction, including

LIGO (USA), VIRGO (Italy/France), GEO (Germany/Great Britain) and TAMA

(Japan), and the space-based observatory LISA is scheduled to launch in 2011.

Since there are so many sources at a given time, in order to detect gravita-

tional waves, it is necessary to model the gravitational waveform which is based

upon a detailed theoretical study of the target sources. Then the theoretical models









of gravitational waves would help scientists to sort out what to look for from a

seemingly huge mess of observational data.

As an example of the possible sources of gravitational waves for LISA detec-

tion [1], a binary inspiral of a small black hole of solar mass and a supermassive

black hole of 105 to 107 solar mass, what we call an extreme mass-ratio system, can

be taken. Such black holes are now believed to reside in the cores of many galaxies,

including our own.

Designing the theoretical waveform from this binary system would require an

accurate description of the orbital evolution of the small black hole. The orbital

motion can be modeled by considering a pointlike test particle moving in the

gravitational field which results from combining the field of the large black hole

with the much smaller field of the small black hole using perturbation techniques.

The resulting motion then includes the effects of radiation reaction and the self-

force.

This dissertation presents specific methods for calculating the effects of

radiation reaction and the self-force for the extreme mass-ratio systems. We explore

two models of such systems in the main body of the dissertation. The case of a

scalar particle orbiting a Schwarzschild black hole is investigated first, and the

case of a point mass orbiting a Schwarzschild black hole follows. The study of the

former itself might not provide physical interpretations as directly applicable to our

gravitational wave physics, but it provides valuable computational tools with which

we can approach the latter. The entire dissertation can be outlined as follows.

In ('!i lpter 2 we introduce general formal schemes on radiation reaction.

Two main articles on this subject by Dirac [2] and by Dewitt and Brehme [3] are

reviewed.

In ('! Ilpter 3 we revisit the general formal schemes and review briefly the

structure of the equations of motion for the self-force for each case from Dirac to









Mino, Sasaki, and Tanaka, and Quinn and Wald [2, 3, 4, 5, 6]. Then, we provide

two examples of the purely analytic attempts to the self-force calculations by

Dewitt and Dewitt [7] and Pfenning and Poisson [8].

In ('!i lpter 4 we introduce a hybrid of both analytical and numerical methods,

known as the "mode-- iin method devised by Barack and Ori [9], in order to

handle more general problems than the purely analytical approaches can. We

then work on the case of a scalar particle orbiting a Schwarzschild black hole via

this method. The self-force calculations for this case involve analytical work for

determining R.i /J ;,.,, .: ,/.:.n Parameters, which refer to the mode-decomposed

multiple moments of the singular part of the scalar field. The computations of the

regularization parameters are facilitated via a local analysis of spacetime, and an

elaborate perturbation analysis of the local geometry is developed for this purpose.

The regularization parameters are calculated to sufficiently high orders so that

their use in the mode sums for the self-force calculation will result in more rapid

convergence and more accurate final results. These analytical results are then

combined with the numerical computations of the retarded field to provide the

self-force ultimately.

In C'!i lpter 5 we provide a method to determine the effects of the gravitational

self-force on a point mass orbiting a Schwarzschild black hole. First, we address the

gauge issues in relation to MiSaTaQuWa Gravitational Self-force [4, 5]. Then we

follow a recent analysis by Detweiler [10] to describe the gravitational field, which

is the perturbation created by the point mass from the background spacetime. To

avoid the gauge problem, rather than calculating the self-force directly, we focus

on gauge invariant quantities and determine their changes due to the self-force

effects. Techniques involved in calculating the regularization parameters for the

gravitational field case are more complicated than for the scalar field case. We






4


follow analyses by Detweiler and Whiting [11] to find the methods for calculating

the regularization parameters.














CHAPTER 2
GENERAL FORMAL SCHEMES FOR RADIATION REACTION

Historically, Dirac gave the first formal analysis of the radiation reaction effect

for the electromagnetic field of a particle moving in flat spacetime in 1938 [2]. In

the equation of motion for a moving electron, he was able to obtain the additional

force term, named the "Abraham-Lorentz-Dirac (ALD) damping term," apart from

the Lorentz force due to the external electromagnetic field. But this ALD damping

term eventually turns out to vanish in free fall, leaving the particle's motion in

geodesic, and no radiation damping or "self-force" effect occurs in flat spacetime.

However, Dirac's pioneering idea was succeeded and generalized to curved

spacetime in similarly formal approaches by the following generations. Dewitt

and Brehme [3] extended Dirac's analysis to curved spacetime. Mino, Sasaki, and

Tanaka [4] developed a similar analysis for the gravitational tensor field. Quinn and

Wald [5] and Quinn [6] worked out similar schemes for the radiation reaction of the

gravitational, electromagnetic, and scalar fields by taking axiomatic approaches.

All these generalized versions of the radiation reaction problem show the obvious

existence of non-vanishing damping terms in addition to the ALD damping term,

which would eventually cause radiation reaction in curved spacetime.

In this ('C Ilpter we review the two main articles on this subject, one by Dirac

[2] and the other by Dewitt and Brehme [3].

2.1 Dirac: Radiating Electrons in Flat Spacetime

2.1.1 The Fields Associated with an Electron

The problem to deal with is a single electron moving in an electromagnetic

field in flat spacetime (following the signature convention (+1, -1, -1, -1) as in

Dirac's original note).









Let us describe the world-line of the electron in spacetime by the equation


Za= Za(s), (2-1)


where za(s) is a function of the proper-time s, and dzo/ds > 0. The electromagnetic

potential at the point Xa satisfies the Maxwell's equations


Ad,
A = 0, (2-2)

OA, = 47rJ, (2-3)

where Ja is the charge-current density vector. With our present model of the

electron, Ja vanishes everywhere except on the world-line of the electron, where it

is infinite
/dz
Ja e Cj 6(xo zo)6(xi zi)6(x2 z2)6(x3 z3)ds (2-4)

for an electron of charge e. The electromagnetic field tensor Fabcan be derived from

the potential Aa

Fab aAb abA (2-5)

Eqs (2-2) and (2-3) have many solutions and thus do not fix the field uniquely.

One may use a solution provided by the well-known retarded potentials of Li6nard

and Wiechert. We call the field derived from these potentials Fai. One can obtain

other solutions by adding to this one any solution of Eq. (2-2) and


Aa, = 0, (2-6)


representing a field of radiation. Then, the actual field F$,b for our one-electron

problem will be the superposition of the field from the retarded potentials and the

field from the solutions of Eq. (2-6) that represent the incoming electromagnetic

waves incident on our electron


Fab Fab + ab.
act ret in


(2-7)









Also we have the field Fbv derived from another solution of Eqs. (2-2) and

(2-3), which is provided by the advanced potentials. F.db is expected to p1 li a

symmetrical role to Fr in all questions of general theory. Thus, corresponding to

Eq. (2-7) one may put
Fab ab + Fab (2-8)
act (2adv out-)

where a new field Fbt is expected to p1' v a symmetrical role in general theory

to Fib, and should be interpretable as the field of outgoing radiation leaving the

neighborhood of the electron. The difference
Fab ab pab -9)
rad out in (2-9)

would then be the field of radiation produced by the electron. Alternatively, from

Eqs. (2-7) and (2-8), this difference may be expressed as


Fbd = F F ,, (2-10)


which shows that F^ is completely determined by the world-line of the electron.

Through some calculations, it is found to be

4e d3a dzb d3XbdX, ^on\
Fabrad =- d3 db 3 tb ) (2-11)
3 ( ds" ds ds" ds

near the world-line, and is free from singularity.

With the attained symmetry between the use of retarded and advanced fields,

one defines a field

fab (Fab ab a Fbv) (2-12)

which will be used to describe the motion of the electron. This field is derivable

from potentials satisfying Eq. (2-6) and is free from singularity on the world-line of

the electron. From Eqs. (2-7) and (2-8), it is in fact just the mean of the incoming









and outgoing fields of radiation,

1
fab 2 ( ab ab (2-13)
2 (i +outj (2

2.1.2 The Equations of Motion of an Electron

The interaction between an electron and the electromagnetic field can be

examined from the equations of motion for the electron, i.e., the equations to

determine the world-line of the particle in motion. The laws of conservation of

energy and momentum are used to get information on this question. First, one

surrounds the singular world-line of the particle by a thin world-tube, whose radius

is much smaller than the range of interaction between the particle and the field in

consideration. Then, one calculates the flow of energy and momentum across the

surface of this world-tube, using the stress tensor Tac of Maxwell's theory, which is

calculated from the actual field Fab via

1 bd
47rTac FabFcb + 4gacFbd bd. (2-14)
4

By the conservation laws, the total flow of energy (or momentum) out from the

surface of any finite length of world-tube must be equal to the difference in the

energy (or momentum) residing within the tube at the two ends of this length:

depending only on conditions at the two ends of this length, the rate of flow

of energy (or momentum) out from the surface of the tube must be a perfect

differential.

The information obtained in this manner is independent of shape and size

of the world-tube provided that it is much smaller than the realm of the Taylor

expansions used in the calculations. If we take two world-tubes surrounding

the singular world-line, the divergence of the stress tensor aTac/ x will vanish

everywhere in the region of spacetime between them, since there are no singularities









in this region and Eq. (2-6) is satisfied throughout it. The integral


fIm 8 (8Tac/8xc) dxodx dx2dx3 (2-15)

over the region of spacetime between the two world-tubes of a certain length can
be expressed as a surface integral over the three-dimensional surface of this region.
Then the difference in the flows of energy (or momentum) across the surfaces of
the two tubes should depend only on conditions at the two ends of the length
considered. Thus the information provided by the conservation laws is well defined.
For easier calculations, the simplest configuration of the world-tube is chosen,
with a spherical surface and of a constant radius c for each instant of the proper
time in that Lorentz frame of reference in which the electron is at rest. Also, we
note the following elementary equations for later use

VaV' = 1, (2-16)

Vai = 0, (2-17)

v +vv =a 0, (2-18)

where Va dza/ds and dots denote differentiations with respect to s. After rather
lengthy calculations with the integral of the stress tensor Tac over the world-tube,
one can show that the flow of energy and momentum out from the surface of any
finite length of tube is given as


I 2 2 -1" a Vbfab ds, (2-19)

where terms that vanish with c are neglected. Since this integral must depend only
on conditions at the two ends of the length of tube, the integrand must be a perfect
differential, i.e.,
1 = B. (2-20)
te2 C_- ia CVb fab Ba. (2-20)
2









This is all one can get from the laws of conservation of energy and momentum.

To develop this further into the equation of motion for the electron, one needs to

fix the vector Ba by making some assumptions. Taking a dot product of the both

sides of Eq. (2-20) with v', we have


VaBa -= e2- vaa -(, *', fab 0, (2-21)
2

by Eq. (2-17) and from the antisymmetry of the tensor fab. Then we may assume

that Ba could be any vector function of Va and its derivatives. The simplest choice

that satisfies Eq. (2-21) would be


Ba = k,, (2-22)


where k is a constant.

Substituting Eq. (2-22) into the right hand side of Eq. (2-20), one sees that the

constant k must be of the form


k = t2-1 m, (2-23)


where m is another constant independent of e, in order that our equations may

have a definite limiting form when c tends to zero. Then one gets


mia = eVbfab, (2-24)


as the equations of motion for the electron. This is the usual form of the equation

of motion of an electron in an external electromagnetic field, with m being the

rest-mass of the electron and fa F ba t (Fa et + Fbadv) being the external

field.









In practical problems, however, we are given not fab but the incident field F1in.

These two fields are connected via Eqs. (2-12), (2-7) and (2-10) ,

fb Fa +F1 b
fab Fon + 2 arad
4 3C ('nav ) (2-25)3
= Fin 1e i -'b ba) (2-25)

with the help of Eq. (2-11). Substituting this into Eq. (2-24) and using Eqs. (2-16)

and (2-18), one obtains


mnia 2 2a + i2Va) = CbFin, (2-26)


where v2 =- Da ,a. Eq. (2-26) would be equal to the equation of motion derived

from the Lorentz theory of the extended electron by equating the total force on

the electron to zero, if one neglects terms involving higher derivatives of Va the the

second.

To discuss the physical interpretations of Eq. (2-26), one needs to examine

the equation for a = 0 component, describing the energy balance. The right hand

side gives the rate at which the incident field does work on the electron, and is

equated to the sum of the three terms mvo, -2. 7', and -'ii, The first two of

these are the perfect differentials of the quantities mvo and 2.;,, respectively,

and may be considered as intrinsic energies of the electron: the former is the usual

expression for a particle of rest-mass m and the latter the .... 1 I. i. ( i 1-,,,

of the electron [12]. C'!i i ,. in the acceleration energy correspond to a reversible

form of emission or absorption of the field energy near the electron. However, the

third term ( I I,, corresponds to irreversible emission of radiation and gives the

effect of radiation damping on the motion of the electron. According to Eq. (2-

17), this term must be positive since ia is orthogonal to the time-like vector Va

and is thus a space-like vector, and hence its square is negative (in the signature

convention (+1,-1,-1,-1)).









Later, we will compare Eq. (2-26) with the equations of motion for a particle

moving in electromagnetic [3], scalar [6] and gravitational fields [4, 5] in curved

spacetime. Then, it would be more convenient to write Eq. (2-26) in the alternative

signature convention (-1, +1, +1, +1) to be consistent with the other equations of

motion in sign, namely


ma = ev bbin + 2e2 (a '2a). (2-27)


2.2 Dewitt and Brehme: Electromagnetic Radiation Damping in
Curved Spacetime

2.2.1 Bi-tensors

As Dirac's work on the classical radiating electron in Section 2.1 was developed

under Lorentz invariance throughout, Dewitt and Brehme's curved-spacetime

generalization of Dirac's is carried out under general covariance throughout. This

covariant generalization involves non-locality questions, and it is essential to

introduce bi-tensors, which are a generalization of ordinary tensors. A bi-tensor is

a set of functions of two spacetime points, each member of which transforms under

a coordinate transformation like an ordinary local tensor, with the difference that

the transformation indices do not all refer to the same point, but rather to the two

separate points. The simplest example of a bi-tensor is the product of two local

vectors, Aa(x) and Bb'(z), taken at different spacetime points, x and z with the

indices a and b' running from 0 to 3:


Cab(X,z ) =Aa(x) Bb' (z). (2-28)


Here the convention is that the usual, non-primed indices are alv--,v- to be associ-

ated with the point x, while the primed indices are ahv--, to be associated with

the point z. Then the coordinates of the points themselves are expressed as x' and
zb









The coordinate transformation law for this bi-tensor is given by

:Cr 8gb'
CCd O Cab,. (2-29)

In addition, the usual operations such as contraction and covariant differentiations

may be immediately extended to bi-tensors with the precautions: (i) contraction

may be performed only over the indices referring to the same point, (ii) in taking

covariant derivatives all indices except those referring to the variable in question

should be ignored. One may take covariant derivatives with respect to either

variable,


Cab; ab + CICebl, (2-30)

Ca b'd' b',d' Ffi /d- C (2-31)


where the semicolon denotes covariant differentiation and the comma denotes

ordinary differentiation. Indices associated with covariant differentiation at different

points commute, while the usual commutation laws hold for indices referring to the

same point.

One may define a bi-scalar which is an invariant bi-tensor bearing no indices.

One may also introduce a u.:-.1 ,h.:'/; and its most elementary example is the

four-dimensional delta function


6() (x,) 6(x0 )6(x1 1)6(x2 2)6(x3 Z3) 6()(,x). (2-32)


In general, the delta function may be regarded as a density of weight w at the

point x and weight 1 w at the point z, where w is arbitrary. One may choose

w = 1/2 for the sake of symmetry, and the transformation law for the delta

function may be give in the form

(4) 1/2 z 1/2
ax(1" )a -.46 (x,z). (2-33)










One may introduce a bi-scalar of geodetic interval, which is of fundamental

importance in the study of the non-local properties of spacetime. It is defined as

the magnitude of the invariant distance between x and z as measured along the

geodesic joining them. Denoting it by s(x, z), one may express its basic properties

in the equations
ab a'b'
gS;a;b S;a'S;b' = 1, (2-34)

lims = 0, (2-35)
X-Z

where the signature of the metric is taken as (-1, +1, +1, +1) (compare this

with Dirac's convention in Section 2.1). The interval between x and z is said to

be spacelike when the sign is + and timelike when the sign is in Eq. (2-34).

However, the bi-scalar itself is taken non-negative. When s = 0, the locus of points

x define the light cone through z.

Geodesics joining x and z may not necessarily be unique, and the bi-scalar of

geodetic interval can be multiple-valued. However, there will be a region in which

the geodetic interval is single valued, and our attention is confined to this region

in developing our argument: the geodetic interval in this single-valued region can

serve as the structural element of covariant expansion techniques later. And in

order to avoid Ii i~1i I! point" problems, instead of s, it will be more convenient to

work with the quantity, which is known as Synge's world function [13],


S- =s2, (2-36)
2

which satisfies

11 a'b'
t abq ;;b 'bqa ;a':;b' = a, (2-37)
2 2
lima = 0, (2-38)
where he interval is said o be spacelike wih sign and imelike wih sign.
where the interval is said to be spacelike with + sign and timelike with sign.










Using a, a bi-tensor Ta'b', whose indices all refer to the same point z, can be

expanded about z in the covariant form

1
Ta/'b = Aa/b/ + Aa/b/'c/'Uc + Aa'/b/cd'/;c' ;d' + O(S3), (2-39)
2

where the expansion coefficients Aa'b', Aa/bc'/, Aa'b'c'd/, etc. are ordinary local tensors

at z. These coefficients can be determined in terms of the covariant derivatives of

Ta/b' :


Aa/b/ = lim Ta/b/, (2-40)
x-z
Aab/c/ = lim Ta'b';c' Aa'b/;c (2-41)
X-Z
Aa'b'c'd' = lim Ta'b/;c/'d Aa'b';c'd' Aa'b'c';d' Aa'b'd';c'. (2-42)
x-z

A particular example of such expansions to note is


J;a'b' ga'b' ,+ R 'c'b'd' c' d'; + ( 0S3). (2-43)
3

One can develop the expansions to higher orders and obtain further


a'b'c' (Ra'c'b'+ Rad'b'c/ ;,d' +0(s2), (2-44)
1
;a'b'c'd' (Ra'cb'd' + Ra'd'b'c') + 0(s). (2-45)
3

For expanding a bi-tensor whose indices do not all refer to the same point,

for example Tab', one introduces a device called the bi-vector of geodetic paral-

lel displacement and denotes it by gab'(x, z). This bi-vector has the significant

geometrical interpretation in the defining equations


gab';cgdd 0, (2-46)

gab';c'gcd';d' = 0, (2-47)


lim gab' gab' or lim gab' 6ab.
X-z X-z


(2-48)









From Eqs. (2-46) and (2-47) it is inferred that its covariant derivatives vanish in

the directions tangent to the geodesic joining x and z, while Eq. (2-48) states that

it reduces to the ordinary metric (or Kronecker delta) in the coincidence limit.

Also, this bi-vector has symmetric reciprocity


gab'(x, gba(z, x). (2-49)

The role of the bi-vector gab' is to in !!',, i,,. ." the indices. For instance, a

local vector Ab' at the point z transforms into the local vector Aa at the point x

by parallel displacement. The application can also be extended to local tensors of

arbitrary order. In particular, one has


gab'g cd'b'd' gac, (2-50)

gab'g */, gb'd', (2-51)

gab' ;b' -.;a, (2-52)

gab ;a = --;b', (2-53)

gab' cb' 6c, (2-54)

gab' gad' = bd. (2-55)

Tensor densities are also subjected to a geodesic parallel displacement by

means of the bi-vector gab'. One can introduce its determinant


6= gJa (2-56)

This determinant is a bi-scalar density, having weight 1 at the point x and weight

-1 at the point z. It satisfies the equations


9;agaba;b = 0, (2-57)

;a' a'bl' ;b = 0, (2-58)

lim 6 1. (2-59)
X~z









Eqs. (2-57)-(2-59) have the unique solution


6(x, z) = g/2(X)g1/2() -1(z, x), (2-60)


where

9 = 9ab (2-61)

A local vector density Ab, of weight w transforms into the local vector Aa along the

geodesic from z to x by parallel displacement in the manner


A, w-gab'Ab,. (2-62)


The transformation by parallel displacement can be extended to the general case.

A bi-scalar of fundamental importance in the theory of geodesics is the Van

Vleck determinant, given by

A -g- 1 ;ab' (2-63)

where

= -I9ab' (2-64)

with the property

g(x,z) g/2(x)g1/2(z) =g(z,x). (2-65)

Differentiating Eq. (2-37) repeatedly and using Eq. (2-63), one can show that


A-1 (A"7) = 4. (2-66)


Also important is the expansion of this determinant, known to be


A R a'b'(;a';b' + O(s3). (2-67)
6









2.2.2 Green's Functions in Curved Spacetime

Looking for the solutions of the covariant vector wave equation


V2A, RabAb = 0, (2-68)


one follows Hadamard [14], according to which an elementary solution can be

written in the form


G),,a = (2w U + vaa, n 1 + aa), (2-69)
(2))2 I

where the functions Uaa', Vaa', Waa' are bi-vectors. If Eq. (2-69) is substituted into

Eq. (2-68), the first function is uniquely determined, using the boundary condition

at x z,

Uaa = A1/2 aa' (2-70)

while the other two are most easily obtained by expanding the functions in a power

series
OO
Vaa' = Y Vnaa'g, (2-71)
n=0
oo
Waal' = 'oaO'Cn, (2-72)
n=0

and obtaining the recurrence formulae for the coefficients. Using Eq. (2-67) for

Eq. (2-70), one obtains


Uaa' 1 Rb'bc;b2';c + O(S3) gaa. (2-73)


By repeatedly differentiating gaa', however, one finds

1 dl
gaa';bc = gda'Rbca + 0(s). (2-74)


Then, differentiating Eq. (2-73) repeatedly and using Eq. (2-74), one also finds

1
V2U aa' gaa'R + O(s). (2-75)
6









Also, inserting Eqs. (2-71) and (2-72) into the equation

V72G1)aa, RabG()ba = 0 (2-76)

and making use of Eq. (2-75), one arrives at

lim Vaa' = b Rab ga'b'R (2-77)
x-z 2 6

One introduces the Feynman propagator

GF' (2~r 1/2 a + Vaa' n (7 + io) + Waa, (2-78)
(27)2 7 + O

which can be separated into real and imaginary parts,

GFaa' = G(1)aa 2iGaa,. (2-79)

Using the identities

(7 + i0)-1 Pe1- 7i(), (2-80)

ln(a + iO) = In la + i0(-a), (2-81)

where P denotes the principal value and

0, a < 0
0(a) = (2-82)
1, a > 0,

one finds for the "symmetric" Green's function, Gaa',


(2-83)


Gaa' (87T)-1 [A1/2gaa'u) Vaa'O(- )]









The various Green's functions are now defined,


Gretaa, (x, z)

Gadvaa'(x, z)

Gaa' (x, z)


20 [e(x), z] Guaa(X, z),

20 [z, E(x)] Gaa,(x, z),

Gadvaa'(X, z) Gretaa,(X, Z),


where E(x) is an arbitrary spacelike hypersurface containing x, and 0 [E(x), z]

1 0 [z, E(x)] is equal to 1 when z lies to the past of E(x) and vanishes when z lies

to the future. These Green's functions satisfy the equations


a (Gret + Gadvaa)
2aa -' F (Lr


(2-87)


V2 Gaa RobGaao


V2 Gret aa

V2Gadvaa,


Rab Gretaa,

SabGadvaa,


g- 1/2 ga (4),


V2 Gaa RabGaa, = 0.

Also, they have the symmetry properties


Gaa (x, z)

G 'et,(x, z)

Gaai(x, z)

Vaa,(X, z)


-Gaia(Z, x),

Gadva'a(z, X),

Gala(Z, x),

Va'a(Z, x).-


Finally, one can note that the substitution of Eq. (2-69) into Eq. (2-76) via

Eq. (2-72) leaves wo0 arbitrary in the solution for Waa,, which corresponds to

adding to G(1) any singularity-free solution of the wave equation. However, this

arbitrariness disappears in the solution for the symmetric Green's functions as it is

evident from Eq. (2-83).


(2-84)

(2-85)

(2-86)


(2-88)


(2-89)


(2-90)

(2-91)

(2-92)

(2-93)









2.2.3 Electrodynamics in Curved Spacetime

Stationary action principle .
The Lagrangian density for a point particle of charge e and bare mass mo,

interacting with an electromagnetic field Fab in a spacetime with metric gab, can be
written as



C C source + interaction + re.m.

-mo -T ( Ia' b' )1/2 (4)dT + e Aa,a'(4)T (167)-1 g12Fbab

= Lo(4 T + Aaj (167)-1 g1/2FabFab, (2-94)

where

Fab Ab;a Aab, (2-95)

Lo = -mo -ga'b' b) 12 (2-96)

ja e / a2ga' (4) dr. (2-97)


Here, the world-line of the particle is described by a set of functions za'('r), with

7 representing an arbitrary parameter, and the dot over z denotes differentiation
with respect to 7. Multiple dots will be used to denote repeated absolute covariant

differentiation with respect to 7,

a' = dza/d-, (2-98)

a-' = ad'"/d+r,,, bc', (2-99)

"d = ac'/dT+r,/b', (2-100)


The action for the system is given by











S= J d4x, (2-101)

where the integration is performed over the region between any two spacelike

hypersurfaces. With variations taken in the dynamical variables za'and Aa which

vanish on these hypersurfaces, the action suffers the variation


6S = (-moga'b' + CeFab ) 6za'dr

+ i [- (47)-1 g12Fab;b + ] j Aad4x, (2-102)

provided that r is taken to be the proper time of the particle (and will henceforth

be assumed) such that


ga'b' ab' = -1, (2-103)

ga'b hab' = 0, (2-104)

ga'b/ a'iZb' g= -a'b' 2. (2-105)

Application of this action principle yields the dynamical equations

mnoxa' eFib' b', (2-106)

gl/2 Fabb 47a. (2-107)

By the fact that


1
Fabba (Rbac Fb + RbabcF a) 0, (2-108)

one can show via Eq. (2-107) that the current density is conserved.

Conservation of the stress-energy tensor.

The stress-energy tensor of the system is given by


Tab tickle + TIabd,
Particle field'


(2-109)












no /j /2gaa',gbb' a'/b' (4) dT,


(47)-1 g1/2 (FacFbc


The divergences of these tensors are found to be


-m0 a' g a'b (g1/2(4);b, d

e 12 /2 aF a'b' b'(4)dT


mo J /S2ga'"a'd (4)dT


Fa bb,



(47r) 1 ;b b (Fd;b Fdb;c + Fbc;d) Fcd

-Fabb.


Combining these results, one obtains the conservation law



Ta;b = 0.

Vector potentials and electromagnetic fields.


(2-112)


(2-113)


(2-114)


In the Lorenz gauge


gabAa;b = 0,


(2-115)


the electromagnetic field equation (2-107) may be rewritten as an inhomogeneous

vector wave equation


(2-116)


Particular solutions of this equation are given by


where


Tab
particle

Trab
field


1 ab aFcdF
4 )


(2-110)

(2-111)


rab
particle;b


47a g12 (gbcAabc RabA b)











A (t(x) 47 G"ret(x, z) Ja'(z)d4z, (2-117)

adv(x) 4 Gad(, z ()d4z, (2-118)

by which the advanced and retarded fields of the particle are written

re ret A Aret (2-119)
b b;a a;b,

Fadv Aadv Aadv (2-120)
b -a b;a a;b

The total field may be expressed in the forms


Fab F + Fret out adv. (2-121)
Fab Fab + ab ab + Fab (2

Alternatively, one may express the total field in the form


Fab fab + Fab, (2-122)

where


Fab 2 Ft+ F2adv) (2-123)

fab Fa + Fout

S l1" rad Rout 1 rad (-124)
iab + 2Fab ab ab (2-124)
2 2
F rad Fret _adv (2-125)
ab ab Fa- .(2-125)

Similarly to Eqs. (2-119) and (2-120), the fields Fab and Fard may be expressed in

terms of potentials Aa and Arad which are defined by the integral expressions of

the form (2-117) and (2-118), involving the functions Gaa' and Grada,, respectively.

The various fields defined thus satisfy the equations


gl/2Fretab;b gl/2Fadvab;b gl/2Fab;b 4= 4


(2-126)









Finab;b Foutab;b ab;b Fradab;b = 0. (2-127)

Substituting Eqs. (2-83), (2-84), (2-85) and (2-97) into Eqs. (2-117) and

(2-118), one obtains

Aadv/ret = 4 G adv/reta/ 'd
-oO

S6 [Uaa',(7) Vaa',(-Cr)] a'(dT
Jr2
r / \ ~-11 *0
Te Uaa/ / v b' ,a'ZdT, (2-128)
ST= Tadv/ret Tadv/ret

where in the second line T- is the value of the proper time at the intersection of the

world-line of the particle with an arbitrary spacelike hypersurface E(x) containing

x, and in the third line Tadv/ret denotes the advanced or retarded proper time of the

particle relative to the point x. These potentials are the covariant Lienard- Wiechert

potentials. Corresponding to these, the field strength tensors is expressed as



F adv/ret ) "a \ [" b' c' 'bl' ( d')-3
Fvb = Te UbaI,;a -Uaa J;b) ~ ;b'c + cr;b 'Z (;d' Z


[(Ubai';a Uaa'/;b).b. za'%b' + (Uba/J;a Uaa'/;b) a I (u;cd^') 2

+ (Uba'u;a Uaa';b + I' I ,; Vaa';b) Z' (,0bl') 1TTjadv ret
T =adv/ret

e / ( ,, UVa;b) a 'd, (2-129)
Tadv/ret

where the last term is generally named I i!" term, which involves integration over

the entire past or future history of particle.

2.2.4 Derivation of Equations of Motion for an Electric Charge via
World-tube Method

Construction of world-tube.

In order to determine the effect of radiation reaction on the particle one must

keep a record of the energy-momentum balance between the particle and the field.









This effect is examined via the equations of motion of the particle which describe

its local behavior, and they can be obtained only if one keeps an instantaneous

record in the immediate neighborhood of the particle. For this purpose one

constructs a three-dimensional hypersurface around the world-line of the particle,

or the world-tube, which is generated by a small sphere surrounding the particle as

time varies. In terms of Synge's world function, the generating sphere of radius e,

as time varies, produces a hypersphere defined by




2
a= (2-130)

;o, = -en(i)a' i, (2-131)

alza' = 0, (2-132)


where n') (I = 2, 3) denotes spatial basis vectors which are orthogonal to each

other and span the hypersurface orthogonal to the world-line of the particle,


)Jn(j)a/, 6, (2-133)

n(j)aa' = 0, (2-134)

and fi represents a set of direction cosines which satisfy


ii = 1. (2-135)

In terms of i one can specify the direction relative to na of an arbitrary unit

vector which is perpendicular to the world-line at z. Then, starting in the direction

of this arbitrary vector, one constructs a geodesic emanating from z extending out

to a fixed distance C to a point x. The coordinates of x will depend on the direction

cosines i and on the proper time r which is the parameter for the point z.









A variation 6Qi in the direction cosines produces a variation in the point x,

which is via Eq. (2-131) given by


J;aa/,xI -_n(i),a'6i. (2-136)

A pair of independent variations 61Qj and 652i in the direction cosines define an

element dQ of solid angle by the relation


QidQ = Cijk61Sj62Qk, (2-137)


where eijk is the three-dimensional Levi-Civita. This solid angle defines an element

of two-dimensional area on the surface of the sphere, enclosed by the parallelogram

formed from the corresponding displacements Jix" and 62x'. However, one is rather

interested in a three-dimensional surface element of the world-tube generated by

the sphere as proper time r varies. Then one shall construct a general displacement

of the point x on the world-tube, which is produced by independent variations of

- and Qi, with a linear combination of 61x", 62xa and the third displacement 63 x

orthogonal to the first and second, forming a parallelepiped:


gab1Xa63xb = 0, gab2X as3b = 0. (2-138)

Later, integrals over the world-tube will be evaluated to compute the energy-

momentum flow, and for this purpose one defines the directed surface element dEa,

which is a vector density, formed from independent displacements 61x", 62Xa and

63Xa

da = EabcdlX6b62X63Xd. (2-139)

In terms of the radius of tube c, variation of solid angle dQ and variation of proper

time dr, the surface element at x is expressed as


dEy(x) = 22 g-1/2 ()gaa'a' ( + C Rb'c' b' c dQdr + 0(e5), (2-140)
\ 6









where
Qa a' ni' (2-141)

K -j;a'b'Za'' a;.) 1/2 (2-142)

The equations of motion .
The conservation law of energy and momentum, whose differential form was
given by Eq. (2-114), can be expressed in integral form using the bi-vector of the
geodetic parallel displacement, in which the contributions to the integral at the
variable point x is referred back to some fixed point z. This integral is a local
covariant vector at z, and Gauss's theorem can be employ, -1 Then, one may write



0 gaa' ab;b 4

ga+ + ) g dbb 9a ;bg ab 4x, (2-143)

where Ei and E2 are the hypersurfaces or "caps" at the proper times r1 and T2,
respectively, and E represents the surface of the world-tube between E1 and E2,
and V is the volume of the tube, enclosed by Ei, E2 and E. Now by taking the
limit c -i 0, the integrals over Ei, E2 and V will retain contributions only from the
particle stress-energy tensor. Furthermore, taking the fixed point z to lie on the
particle's world-line at a proper time T, which is r- < r < T2, will give

0 limr r'
0 -lim 9 gaa" abd + in0 [gb"Ia ((z"),_ -)) 9" (,T)] T//
-0 J4 T r"=T1

-mo gb1a';," (z(r"), z(r)) b" (-") c" (r")dr", (2-144)

where the replacement has been made,


I j j (2-145)
//S J 47r









such that the integral over the surface E can be computed explicitly in terms of an

integral over proper time and an integral over solid angle. By letting r and T2 both

approach r, Eq. (2-144) becomes


0 mo dr + lim ga'TabdEb.
e->0 J4,


(2-146)


One shall focus on the evaluation of the second term of this equation to derive the

equations of motion of the electric charge.

First, the retarded and advanced field strength tensors of Eq. (2-129) must

be expressed in the form of expansions. After a very tedious algebra involving a

number of perturbations one finds


F _g 1 -1 2Xa'Qb' + 1 -3.a' b' 1 5 '2
2ega[a'9lbb'] -2 a5 a/ + 2
L 2
-1 1-3* a''b2' _2 -4-a'b' + lnah'2c 'R
2 3 12
+-1/a RbK-I c, s if ~-- a' b' c' ,c + -1la'Qb' Pc'd' c' d'

2 12
1 1 b' d ,c' da -lb cL ,c' d
2 12
t --K 3 a c'd'e' ,c'd'Q 1 -2a Z bc',
6 3


Tadv/ret


where the tail term has been written in terms of the Green's function G /ret (x, z(r))

rather than the Hadamard expansion term Vaa' (x, z(r)) for later convenience.

From Eq. (2-147) it follows that the field Fradab is everywhere finite. At the

location of the particle, it is described as


Fret a'b' adv a'b'

14, (a *' + 42b' K-2[a')R b]cc
3 3
+2e I V[b'Gret a'] Cc" (Tr")dr" d [badva'] c,"
7-0o J ,Tadv
(2-148)


F adv/ret /
Iab r )


Frad a'b'









On the other hand, for the mean of the retarded and advanced fields one has the

expression

Fb ( ret + Fadv
Fab ab + r'ab )

e (gaa'gbb' ba' gab')
1 1 1
x e-2~-1a'Qb' + -32a'+b' + t-5 2 -3**ab'
2 8 2
+ terms linear and cubic in the Q's involving the Riemann tensor
/Tret OO
+e V[bG",'e t + [b ( ()d + O(C). (2-149)
-o Tadv

By splitting the total electromagnetic field as in Eq. (2-122), one can now

compute the stress-energy tensor via Eq. (2-149). Taking advantage of the fact that

fab is singularity-free, one may write


ga'T abdb


(47)-1 g12 gaa (FacF + f + Facfbc) d b

- FcdFd + fdFd) gaa dEa + 0(C).
4 2


Using Eqs. (2-103), (2-104), (2-105), (2-140) and the expansion

Kn2 = 1+ C 2a a' + 0(C2),

one computes the right hand side of Eq. (2-150) and finds


(2-150)


(2-151)


-1 1 2 a' 1 31a' -13a'.b b' a' 2
(47r) e + -z + -b -b z z Q + 2
S2 2 4 2
+ terms of odd degree in the Q's involving the Riemann tensor

t/ rt
b' f r V [b, G-reta'] .c" //) ,,d 11
--O
+ j V[b, Gadv a'] cc""// T// T} dQdT
+(4 gIa' b'dT + 0(cr")dr". (2-52)d
"Tadv
-(4x)-lef 'zb dbdr + O(e). (2-152)


gal Tabdyb









Carrying out the integration, one eliminates all the terms containing odd degree in

the direction cosines and obtains


a 'Tabdzb
/ 4x


a' 2 b [ ret a'] 'c"


+ j V[bGadva'] c" (Tr)d/ "- efa'b' dr + 0(e).
T(2-53dv
(2-153)


The divergent term in Eq. (2-153) has the same kinematical structure as the

mass term in Eq. (2-146). Therefore, it has the effect of an unobservable mass

renormalization, and by introducing the observed mass


S02 -1
m = mo + lim -e
e-o 2


(2-154)


one may now rewrite Eq. (2-146) as


/f bIZ'

+e2 b/ rt
-0C


V[bGreta'] c"1 (1)"dr" + J v, [badv a'] c" 1
(2-dv55)
(2-155)


Then, substituting Eqs. (2-124) and (2-148) together with (2-151) into Eq. (2-155)

and using Eqs. (2-103) and (2-105), one finally obtains the equations of motion for

the electric charge 1 :



1 The result shown here is the modified version by Hobbs [15] and is slightly dif-
ferent from the original, Eq. (5.26) in Dewitt and Brehme [3]. This is due to the
corrections made to Eqs. (5.12) and (5.14) in Dewitt and Brehme, whose modified
forms are now Eqs. (2-147) and (2-148), respectively.


mz












2 1
m" = ebF a +bin 2 e2(a 2a) + 2 (R b + aRbc bc)
3 3
ret
+2b V[bGret a]C (z (T), z (T')) c' (T')dT'. (2-56)


The integral term involving this Green's function in Eq. (2-156), often referred

to as the "tail" term, gives an implication that the motion of the particle is affected

by the entire history of the particle itself. This, together with the third term on

the right hand side will make the particle deviate from its original world-line to

the order of e2, even in the absence of an external incident field Fabin, which means

that radiation damping is expected to occur even for a particle in free fall in curved

spacetime.















CHAPTER 3
CALCULATIONS OF SELF-FORCE: REVIEW OF GENERAL SCHEMES AND
ANALYTICAL APPROACHES

In C!i ipter 2 we studied general formal schemes of radiation reaction in a vari-

ety of contexts, from Dirac's radiating electrons in flat spacetime to Mino, Sasaki,

and Tanaka and also Quinn and Wald's gravitational radiation reaction in curved

spacetime [2, 3, 4, 5, 6]. These formal schemes are theoretically well developed and

provide a good foundation for radiation reaction in curved spacetime. However,

the practical, quantitative calculations of radiation reaction remain a challenge.

The difficulty lies in the "tail" integral terms appearing in the equations of motion:

it is extremely difficult to determine precisely the retarded Green's functions in

the integrals for general geometry and for general geodesic of particle's motion.

Some attempts were made to evaluate the self-force by computing those 1 I!"

integral terms directly, but their applications had to be limited to the problems

having certain symmetries and conditions that would simplify the Green functions

in the integrals [7, 8]. Hence, for more realistic physical problems, in which special

conditions and restrictions might not be alv--,v- expected, different schemes of

calculations would be demanded to compute the I il" integral terms, thence the

self-force.

In Section 3.1 we revisit the general formal schemes and review briefly the

structure of the equations of motion for the self-force for each case from Dirac to

Mino, Sasaki, and Tanaka, and Quinn and Wald [2, 3, 4, 5, 6]. Then, Section 3.2

presents two examples of the purely analytic attempts to the self-force calculations,

in which the tail integral terms are directly calculated as the retarded Green's

functions are simplified by some special conditions. Dewitt and Dewitt [7] and









Pfenning and Poisson [8] are provided as the examples. An alternative scheme

for the self-force calculations, which has been devised to work for more general

problems, is a hybrid of both analytical and numerical methods. This will be

the main approach that this dissertation is going to take, and we leave its full

discussion for the next two C'! lpters.

3.1 General Formal Schemes Revisited

3.1.1 Dirac: Radiating Electrons in Flat Spacetime

For an electron of mass m moving in the flat spacetime region with the

incident electromagnetic field, Dirac [2] derived the following equation of motion

using the conservation of the stress-energy tensor inside a narrow world-tube

surrounding the particle's world-line,


m-" = eibFbi + e2 (- -a ) (3-1)_

where Fifb aAb ObAa represents the incident electromagnetic field and the

second term on the right hand side, known as the Abraham-Lorentz-Dirac (ALD)

force, results from the radiation field produced by the moving electron. In this

analysis, the retarded electromagnetic field is decomposed into two parts:

1 1
ab.&_ / ab ab ab ab Nt7& av i( 2 (FTb F (3-)
ret + adv () + Fret Fadv) (i) (3-2)

The first term (i) on the right hand side of Eq. (3-2) is the solution of the inhomo-

geneous equation

OA, = -47Ja, (3-3)

with the charge-current density


Ja I a(T)6(4) (x z(r)dr, (3-4)

and corresponds to the field resembling the Coulomb q/r piece of the scalar

potential near the particle, which does not contribute to the force on the particle









itself. And the the second term (ii), defined as the radiation field, comes from the

homogeneous solution of the equation


]A, 0, (3-5)


and is completely responsible for the ALD force.

It turns out that in the absence of the incident field Fib, the only physical

solution of Eq. (3-1) is v' = 0, i.e. geodesic motion, hence there is no self-force on

the particle.

3.1.2 Dewitt and Brehme: Electromagnetic Radiation Damping in
Curved Spacetime

Dewitt and Brehme [3] generalized Dirac's approach [2] to the general curved

spacetime. Using the Hadamard expansion techniques for the vector field in curved

spacetime, the equation of motion for the charged particle turns out to be

2 1
mI" = ezbF bin + -e2 (a 32 a) + 1 2 (Rabb + %aRbcb c)
3 3
+ lim e2 b V[bVGret a]c/ (z(-T), z(-T)) c' (T)d-', (3-6)


where Ge (z(7), z(7')) is a bi-vector retarded Green's function for the vector wave

equation in curved spacetime


V2A et R R b et -47,J (3-7)


with the current density


Ja e I gaa (x, z(7)) va')(4) (x z(r)) d, (3-8)


(g9aa (x, z): bi-vector of geodesic parallel displacement).
The integral term involving this Green's function in Eq. (3-6) is often called

the I i!" part, giving an implication that the particle's motion is affected by the

entire history of the source. This tail term, together with the third term on the









right hand side of Eq. (3-6) results from the particle's motion and the curvature

of spacetime, and will make the particle deviate from its original world-line to the

order of e2, even in the absence of an external incident field Fi2b. Hence, radiation

damping is expected to occur even for a particle in free fall.

3.1.3 Quinn: Radiation Reaction of Scalar Particles in Curved Space-
time

In a similar manner to Dewitt and Brehme, using the Hadamard expansion

techniques for the scalar field in curved spacetime, Quinn [6] was able to derive the

equation of motion for a scalar point particle moving in curved spacetime as



1 1 1
m a qaV'in + q2( a 2Za) + tq (R b + aRbcb ) q2R a
3 6 12
+ lim q2 VaGret (z(r), z(T')) dT, (3-9)

where Gret (z(r), z(r')) is a bi-scalar retarded Green's function for the scalar wave

equation in curved spacetime

v2 ret = -47, (3-10)

with the scalar charge density


=q j(-g)-1/2(4)(x- z())dr. (3-11)

Again, we have a 1 i!" term involving the Green's function in Eq. (3-9), which

gives the same implication as that of Dewitt and Brehme's. Similarly as in the case

of electromagnetic vector field, the last three terms including this tail term on the

right hand side of Eq. (3-9) result from the particle's motion and the curvature of

spacetime, and will be responsible for radiation reaction of the scalar particle in

free fall.









3.1.4 Mino, Sasaki, and Tanaka; Quinn and Wald: Gravitational
Radiation Reaction of Particles in Curved Spacetime

Extending Dewitt and Brehme's approach to the gravitational tensor field

hab, the perturbation created by a point mass m from the background spacetime

gab, Mino, Sasaki, and Tanaka [4] and Quinn and Wald [5] obtained the following

equation of motion



a1 ( h 21ad
m a = m V 7a bcin Vbh acin ad
2 2

+lim m2 b c [Va Grret (z(T) b ()) V bGret"a a'b' ((), ))

_- a dVdGe (b ), b)) a'()b' (')d', (3-12)


where Gab, (z( ), z(r')) is a bi-tensor retarded Green's function for the tensor

wave equation in curved spacetime


V2ab + 2Rabdhcd = -1Tab, (3-13)


with the trace-reversed field defined by

1
hab hab (hcC) gab (3-14)
2

and the stress-energy tensor given by


Tab = aa ga (x, z(T)) gbl, z(X, )) a')b')(4) (x z(T)) d, (3-15)
Jr

and with the harmonic gauge condition


hab;b = 0. (3-16)


It should be noted that the Abraham-Lorentz-Dirac (ALD) term ~ ('" x2a)

is absent from the equation of motion (3-12) unlike the other cases: it has been

effectively dropped off by the reduction of order procedure. The "tail" term









is present, giving the same implication as in the other cases, and is the only

contribution to the self-force for the gravitational field.

3.2 Analytical Calculations of Self-force

3.2.1 Dewitt and Dewitt: Falling Charges

Dewitt and Dewitt [7] computed the self-force on the electric charge falling

freely in the non-relativistic limit of small velocities in a static weak gravitational

field which is characterized by (in harmonic coordinates)


hb 2GM b, (3-17)


where G is the gravitational constant and M is the total mass contained in the

spacetime. By simplifying the bi-vector retarded Green's function in this limit,

they were able to evaluate the I l! integral" in Eq. (3-3) directly. The result of

calculation shows that, in this limit, the force separates naturally into the following

two parts (looking into the spatial components of the force):


F = F + FNC
CMr + 22 ( ) (GM) (3-18)
4 3 r

where Fc is a conservative force which arises from the fact that the mass of the

particle is not concentrated at a point but is partly distributed as electric field

energy in the space surrounding the particle, and FNC is a non-conservative force

which gives rise to radiation la,,n',:,j having a linear dependence on both the

velocity of the charge and the curvature of the background geometry.

FC corresponds to a repulsive inverse square potential

e2GM
c 2GM (3-19)
2r2









This is shown to make the retrograde contribution to the perihelion precession,


60- 7- (3-20)
a(l e2)

where a is the semi-i i' ',r axis of the orbit, e is its eccentricity, and re = e2/m, the

classical radius of the particle.

From Eq. (3-17)

hoo =2GM (3-21)

and this leads to


1 a2 (GM>
Rlioo = hooij = i (3-22)

Then with this, FNC can be rewritten in the form


FNCei e2RC,, ,,, (3-23)
3

which shows directly that the damping effect comes from the curvature of space-

time. FNC may be written in another form by making use of the undamped

equation of motion

= GM) (3-24)

as the first approximation. Then, one gets

2 2"
FNC e= er, (3-25)


which is in agreement with the flat spacetime theory. From this, an integration by

parts gives

AEorbit = -22 bit dt, (3-26)
3 orbit
which expresses either the total energy loss for an unbound orbit or the loss in

one period for a bound orbit. This would be identical with the effect from the

traditional damping term of Eq. (3-6) which is used for accelerations caused by









nongravitational forces. When gravitational forces are present alone, it is important

to note that the phenomenon of preacceleration does not occur as it would be

argued by Eq. (3-25), since Eq. (3-23) shows that the nonconservative force

depends on the velocity of the particle rather than its r'.

From Eq. (3-18), the effect of radiation damping represented by FNC is

negligibly small in magnitude compared to the conservative force Fc, owing to

the dependence of FNC on the velocity of the particle. Hence, its experimental

detection would be virtually impossible, and all the discussions above on radiation

damping would be of conceptual interest only.

3.2.2 Pfenning and Poisson: Scalar, Electromagnetic, and Gravitational
Self-forces in Weakly Curved Spacetimes

Pfenning and Poisson [8] calculated the self-force experienced by a point scalar

charge q, a point electric charge e, and a point mass m moving in a weakly curved

spacetime characterized by a time-independent Newtonian potential 4. As it was

in Dewitt and Dewitt [7], the matter distribution responsible for this potential is

assumed to be bounded, so

-M (3-27)
r

at large distances r from the matter, whose total mass is M (with the convention

G = c = 1). The procedure of calculating the self-force is similar to Dewitt

and Dewitt [7], i.e. first computing the retarded Green's functions for scalar,

electromagnetic, and gravitational fields in the weakly spacetime, and then for each

case of field evaluating the I i! integral" over the particle's past world-line.

For the scalar charge, the result is


Fs = Fsc + Fsc NC

2q2 + iq 2d, (3-28)









where ( is a dimensionless constant measuring the coupling of the scalar field to

the spacetime curvature, and P is a unit vector pointing in the radial direction, and

g = -t4 is the Newtonian gravitational field. Here, is introduced to imply that

the conservative term disappears when the field is minimally coupled.

For the electric charge, the same result to Dewitt and Dewitt [7] is reproduced,


Fem Fem N + Fem NC
M 2 dg
e2 + 2 d (3-29)
r 3 3 dt

For the point mass particle, the conservative force vanishes and only the non-

conservative (radiation-reaction) force is present,

11 -dg
Fgr = Fgr NC = dt (3-30)

where (-) sign implies radiation "antidampini However, this result for the grav-

itational self-force has some problems of interpretation: (i) A radiation-reaction

force should not appear in the equation of motion at this level of approxima-

tion, which corresponds to 1.5 post-Newtonian order. (ii) It should not give rise

to radiation antidamping. These problems can be resolved by incorporating a

i II, I r-mediated force" into the equation of motion: the matter-mediated force

originates from a disturbed spacetime which has become locally non-vacuum due

to the changes in its mass distribution induced by the presence of a particle in the

region. It is obtained as

11 dg
Fmm = 6g + 1PN + 2d (3-31)
3 dt

where the first term represents the change in the particle's Newtonian gravitational

field associated with its motion around the fixed central mass, the second term

is a post-Newtonian correction to the Newtonian force mg, and the third term is

a radiation damping term. When the two forces from Eqs. (3-30) and (3-31) are






42


combined, the radiation damping force cancels out the antidamping force, so the

equation of motion is conservative, and it agrees with the appropriate limit of the

standard post-Newtonian equation of motion.















CHAPTER 4
PRACTICAL SCHEMES FOR CALCULATIONS OF SELF-FORCE (A):
SCALAR FIELD

In C!I ipter 3 we reviewed the analytic approaches to the self-force calculations

by Dewitt and Dewitt [7] and Pfenning and Poisson [8]. From the reviews, it was

meaningful to see that their calculations show desired correspondence limit to

the flat spacetime theory or the agreement with the low-order post-Newtonian

approximations. In their calculations of the self-force, however, special conditions

such as non-relativistic velocities of particles and a weak gravitational field had

to be imposed to enable the entire calculations to be treated fully analytically.

More realistic self-force problems having more general conditions would require

completely different approaches for the calculations, in which we combine both

analytical and numerical methods.

Here, we introduce an alternative approach to the self-force calculations,

known as the "mode-- iin method, which was originally devised by Barack and

Ori [9]. Employing both analytical and numerical techniques, this method does not

limit the particle's velocities and the field's strength, and should practically work

for all kinds of fields under consideration, whether it is scalar, electromagnetic,

or gravitational. It is particularly powerful for the problems in a spherically

symmetric spacetime, such as Schwarzschild. In principle, we take advantage of the

spherical symmetry of the background geometry to decompose the retarded Green's

function in the I il!" term into spherical-harmonic modes which can be computed

individually. Then, from the mode-decomposition of the retarded Green's function

we obtain a mode-decomposition of the retarded field, and from this subtract a

mode-decomposition of the -.:,lu;,lar field, which is locally well described. The









i,. ,,lar, remainder, which is the difference between the retarded field and singular

field, is believed to be smooth and well behaved on the world-line of the particle,

and is summed over all modes to provide the self-force or the radiation reaction

effect. The mode-decomposed solution of the retarded field is obtained numerically

while the mode-decomposed piece of the singular field is determined in a purely

analytic manner. The latter is generically referred to as BR.l, ,lri, .: -i.:..n Parameters.

In this ('!h plter we deal with the self-force of a scalar charge orbiting a

Schwarzschild black hole. Section 4.1 introduces a recent method to split the

retarded scalar field in curved spacetime -i '-.- -1. 1 by Detweiler and Whiting

[16]. This follows Dirac's idea in his flat spacetime problem [2], and gives good

interpretations not only in the singular behavior of the retarded field, but also in

its differentiability. In Section 4.2 we give the overview of the mode-sum method

originated by Barack and Ori [9], and present the analytic results for the singular

field, i.e. the regularization parameters, which were obtained by Kim [17]. The

regularization parameters are locally well defined and should describe the singular

behavior and the differentiability of the field precisely. Higher order expansions of

the singular field will generate higher order regularization parameters, and their use

in the mode sums for the self-force calculation will result in more rapid convergence

and more accurate final results. To facilitate the computations of the regularization

parameters, an in-depth analysis of the local spacetime would be demanded, and in

Section 4.3 we develop an elaborate perturbation analysis of the local geometry for

this purpose. Then, Section 4.4 is devoted to the calculations of the regularization

parameters. These results are then combined with the numerical computations of

the retarded field to provide the self-force ultimately. This final task is done in the

Section 4.5.









4.1 Splitting the Retarded Field

4.1.1 Conventional Method of Splitting the Retarded Field

Historically, Dirac [2] first gave the analysis of the self-force for the electromag-

netic field of a particle in flat spacetime. He was able to approach the problem in

a perturbative scheme by allowing the particle's size to remain finite and invoking

the conservation of the stress-energy tensor inside a narrow world-tube surrounding

the particle's world-line. In his analysis, the retarded field is decomposed into two

parts: (i) The first part is the ii,,, i of the advanced and retarded fields" which

is a solution of the inhomogeneous field equation resembling the Coulomb q/r

piece of the scalar potential near the particle. (ii) The second part is a i 1 h II i i1""

field which is a homogeneous solution of Maxwell's equations. Dirac describes the

self-force as the interaction of the particle with the radiation field, a well-defined

vacuum field solution.

In the analyses of the self-force in curved spacetime, first by Dewitt and

Brehme [3], and subsequently by Mino, Sasaki, and Tanaka [4], by Quinn and Wald

[5] and by Quinn [6], the Hadamard form of the Green's function is employ, .1 to

describe the retarded field of the particle. Traditionally, taking the scalar field

case for example, the retarded Green's function Gret(p,p') is divided into "direct"

and "tail" parts: (i) The first part has support only on the past null cone of the

field point p. (ii) The second part has support inside the past null cone due to the

presence of the curvature of spacetime. Accordingly, the self-force on the particle

consists of two pieces: (i) The first piece comes from the direct part of the field

and the acceleration of the world-line in the background geometry; this corresponds

to Abraham-Lorentz-Dirac (ALD) force in flat spacetime. (ii) The second piece

comes from the tail part of the field and is present in curved spacetime. Thus, the

description of the self-force in curved spacetime reduces to Dirac's result in the flat

spacetime limit.









In this approach, the self-force is considered to result via


Fa qV= V (4-1)

from the interaction of the charge with the field

.1 = ret direct. (4-2)


Now is investigated in the following manner. We have the scalar field equation

V2 -47Q, (4-3)

where

Q(P) = q (g)-1/2(4) (- p'()) d (4-4)

is the source function for a scalar charge q moving along a world-line F, described

by p'(r), with 7 representing the proper time along the world-line. This field

equation is solved in terms of a Green's function,


V2G(p,p') (-g)-1/ (4)(p- p'). (4-5)

The scalar field of this charge is then


b(p) 47 fG(p, p') Q(p')dp'
(p)

S 4q G [p, p'()] dr. (4-6)
Jr

Dewitt and Brehme analyze the scalar field in curved spacetime using the

Hadamard expansions of the Green's function near F. A bi-scalar quantity a(p,p'),

termed Synge's .- i Id function" [13] is defined as half of the square of the distance

measured along a geodesic from p to p', and a < 0 for a timelike geodesic, a = 0 on

the past and future null cones of p, and a > 0 for a spacelike geodesic. The usual

symmetric scalar field Green's function is derived from the Hadamard form to be











G (p p') [u(p,p')(a) v(p,p')(-a)] (4-7)
87
where u(p,p') and v(p,p') are bi-scalars described by Dewitt and Brehme, and their

expansions are known to be convergent within a finite neighborhood of F if the

geometry is analytic. In the vicinity of F, Dewitt and Brehme show that


u(p,p') = 1 + RabVaVba + O(p3/3), (4-8)
12

and that



v(p,p') = R(p') + O(p/R3), (4-9)
12
where p is the proper distance from p to F measured along the spatial geodesic

which is orthogonal to F, and R represents a length scale of the background

geometry (the the smallest of the radius of curvature, the scale of inhomogeneities

and time scale for changes in curvature along F). The ((-o) guarantees that only

when p and p' are timelike-related is there a contribution from v(p,p'). In any

Green's function the terms containing u and v are frequently referred to as the

"direct" and I i!" parts, respectively. Also, the retarded and advanced Green's

functions are expressed in terms of Gsym(p,p') as,

Gret(p,p/) 20 [E(p),p'] Gsym(p,p/),

Gadv (p p') = 20 [p', (p)] Gym(p,p'), (4-10)

respectively, where 0 [E(p),p'] = 1- 0 [p', (p)] equals 1 if p' is in the past

of a spacelike hypersurface E(p) that intersects p, and is 0 otherwise. As Dirac

decomposed the retarded electromagnetic field Fa'" into two parts as in Eq. (3-2),

we may try to decompose our scalar field 'ret into











r direct + tail direct + (' c det 1 tai) (4-11)
I eret+t ~tai ,
where irct 2 direct + ict) and dnect di- iect d- iret) such that

direct direct dect. We separate Ydirect from the rest on the right hand
side of the above equation since this term is singular and exerts no force on the

particle. Then, we single out ', ;C1 and ret from Eq. (4-11), which are the only

contributions to the force on the particle, and may write down


= ect ,- (4-12)


With the help of Eqs. (4-6), (4-7) and (4-10) together with the definition of ', ;jct

above, one can express Eq. (4-12) as


"_ ( qu r[p, '(T )] 1dv T ret
2(P) [- a -_q v [p,p'(r) ]dr, (4-13)
L 2a J et -
which gives our self-force via Eq. (4-1).

Although this traditional approach provides adequate methods to compute

the self-force, it does not share the physical simplicity of Dirac's analysis where the

force is described entirely in terms of an identifiable, vacuum solution of the field

equations [16]: unlike Dirac's radiation field, the ', '" in Eq. (4-13) is not a solution

of the vacuum field equation V2 = 0.

In addition, the ', '" is not fully differentiable on the world-line F. The first

term in Eq. (4-13) is finite and differentiable in the coincidence limit, p -- F. This

term, in fact, provides the curved spacetime generalization of the ALD force, and

is eventually expressed in terms of the acceleration of F and components of the

Riemann tensor via local expansions of u(p,p') and &(p,p') as in Refs. [3, 4, 5, 6].

The integral term in Eq. (4-13) comes from the tail part of the Green's function.









Taking its derivative with respect to x', the coordinates for p, one obtains [6]


qv [p, p'(rt)] Varet = q [vT1 Vaa
qR(p) X
R (p) xa ) + O(p/3, p (4-14)
12p

where Eq. (4-9) was used for v [p,p'(Qret)] near F. The spatial part of the right

hand side of Eq. (4-14) is not defined when p is on F, thus the differentiability is

not guaranteed in general on the world-line if the Ricci scalar of the background

is not zero-similarly, the electromagnetic potential At1il and the gravitational

metric perturbation h1ai are not differentiable at the point of the particle unless

(Rab 'gabR) Ub and R ,,, R ,i', respectively, are zero in the background [16].
Therefore, in order to obtain a well defined contribution to the self-force out of the

tail part, one first averages Va', "' over a small, spatial two-sphere surrounding the

particle, thus removing the spatial part of Eq. (4-14), then takes the limit of this

average as the radius of the two-sphere tends to zero [3, 4, 5, 6].

4.1.2 New Method of Splitting the Retarded Field

According to Detweiler and Whiting [16], a new symmetric Green's function

can be constructed by adding to the first in Eq. (4-7) any bi-scalar which is

a homogeneous solution of Eq. (4-5). Dewitt and Brehme [3] show that the

symmetric bi-scalar v(p,p') is a solution of the homogeneous wave equation,


V2(p,p') = 0. (4-15)

Then, using this we generate a new symmetric Green's function


Gs(p,p') GY(p,p') + -v(p,p')
87
S[u(p,p') () + v (p,p')e(a)]. (4-16)
87

This new symmetric Green's function has support on the null cone of p just as

Gsym does, and has support outside the null cone, but not within the null cone,









unlike Gsym. We consider GS(p,p') only in a local neighborhood of the particle,

thus the use of Gs(p,p') is not complicated by the need for knowledge of the entire

past history of the source and is amenable to local analysis. By Eqs. (4-6) and

(4-16), the corresponding field is


s(p)= -q ,() + q ) + v [p, p'(r)] dr, (4-17)
2 o t| 2 I L r2jt

which is an inhomogeneous solution of Eq. (4-3) just as i, is, and is analogous

to Dirac's singular field (Fb + F'dv). Following Dirac's pioneering idea, one can

define

GR(p,p') Gret(p,p) GS(p,p'). (4-18)

It is remarkable that like Gret(p,p'), GR(p,p') has no support inside the future null

cone. Corresponding to GR(p,p'), we construct

rR ret- S
qu [, p'(r)] Tadv (Tret 1 /Tadv)
q v [p, p'(T)] dT, (4-19)
2( T t 2 rett
1 b (F Fab >
which is analogous to Dirac's radiation field (Fe Fadv)

As both ', 1' and s are inhomogeneous solutions of the same differential

equation, Eq. (4-3), consequently, QR, as defined by the first line of Eq. (4-19)

is a homogeneous solution and therefore expected to be differentiable on F. The

relation between QR and is

q Tadv v [p, p'(Q)] d. (4-20)
VTret

Here is observed a result of great significance: qR can replace ', for an ex-

plicit computation of the self-force, since the integral term in Eq. (4-20) gives

no contribution to a self-force. For a field point p near F, via Eq. (4-9) and









Tadv Tret = 2p + O(p2/R7), the integral term in Eq. (4-20) is

1
qpv(p,p) + O(p2/R3) qpR(p) + O(p2/R3), p F. (4-21)
12

Taking the derivative of the right hand side of this equation gives

1 qR(p)
qR(p)Vp + O(p/R) ( x') + O(p/R), p (4-22)
12 12p

When this result is combined with Eq. (4-14) via Eq. (4-20), the troublesome part

of Va', '" in Eq. (4-14) is canceled by its negative counterpart in Eq. (4-22), and

we simply end up with


V, = V,, '" + O(p/R3), p F, (4-23)

where the remainder term O(p/R3) vanishes in the limit that p approaches F and

gives no contribution to the self-force.

For the rest of this C'! Ilpter, QR replaces ', '' for an explicit computation of

the self-force, and the alternative split of Qret is adopted, namely

,ret s + R (4-24)


where bs is termed the Singular Source field, and CR the Regular Remainder field.

We determine an analytical approximation of bs via a multiple expansion, then

subtract this from the numerical solution of ,, for the ultimate calculation of the

self-force.

4.2 Mode-sum Decomposition and Regularization Parameters

The self-force problems having more general conditions, such as a strong

gravitational field in the background and particle velocities beyond the non-

relativistic limit, are not approachable fully analytically via direct calculation of

the Green's function in the tail part. Barack and Ori [9] sI----. -1 .1 a method to

analyze such problems when the background spacetime is spherically symmetric









by combining both analytical and numerical computations. In their analysis, the

self-force may be considered to be calculated from

self lim [Fet() di(p)], (4-25)
p-p'

where p' is the event on F where the self-force is to be determined and p is an

event in the neighborhood of p', and Fa(p) is related to Q(p) via Eq. (4-1). For

use of this equation, both Fret(p) and F"ir(p) would be expanded into multiple

f-modes, i.e y:, F7 (p) and Z:, F r' (p), respectively, where ire (p) is determined

numerically and Ff (p) determined analytically. In order to determine ~j (p), we

solve Eq. (4-3) using spherical harmonic expansions. The source Q in the equation

is expanded into spherical harmonics, and similarly the field Qret is expanded

', r ')(r, t), (4-26)


where (r, t) is found numerically. The individual components '(r, t) in this

expansion are finite at the location of the particle even though their sum, the right

hand side of Eq. (4-26) is singular. Then we have

qret q V iretv ) (4-27)
rn

which is also finite. The remaining part ff(p) is determined by a local analysis

of the Green's function in the neighborhood of the particle's world-line. Ref. [9]

provides

lim j-- (p) (i A B + + + O(f-2), (4-28)
P P' \ 2 / +
where Aa, Ba and Ca are constants and are generically referred to as R il., rization

Parameters. The remainder is defined as

D [- lim f (p) + 2)A B --e- (4-29)
So-P' 20 f -
=0








which is shown to vanish [9]. Then the self-force is ultimately calculated as

.Fself l- [iimi _ret(P) (C) A (4-30)
e (p)- + A B D (4-30)
o 2 -P' a + -

Our approach closely follows Barack and Ori [9], but there is some difference in
the regularization scheme due to our different split of Qret as described in
Eq. (4-24). From our perspective, via Eq. (4-1) the self-force can be explicitly
evaluated from

pelf li [. et(p)- _F(p)] =-T(pI)

q lim Va('ret- s) qVaR, (4-31)

where similarly to the above, we expand both et(p) and FS(p) into multiple f-
modes e-,T Pt (p) and e-e^S,(p), respectively, with rt(p) determined numerically
and F~a(p) determined analytically. This implies that our self-force is




r
.Fself lir [Pm(p)-_ Sa(P)] yj t

q alim V (,, l q liVa (4-32)
p____tp/ ,, 7
m m
evaluated at the location of the particle. Here the individual fm components
'rt and are finite at the location of the particle even though their sums

are both singular. The f-mode derivatives F =- qVa Zm(,' itm) and Fsa

qVa (' "' YAm) are also finite at the point of the particle, and we take the
difference between the two, which is T, = qVa Em QmYm, then take the sum
of this quantity over which produces a convergent value for the self-force.
Our computation of the retarded field part is identical to that of Barack and
Ori, but our f mode-decomposition of the singular field part, i.e. Fsa is slightly
different from their J7r. We describe JTsa in the coincidence limit p -i p' via the
regularization parameters











( 1 A 2 2Da
lim as + A + L + B, + C) + 0(-4), (4-33)
p,-p' 2 (2f- 1)(2f + 3)

where the first two terms look just identical to those in Eq. (4-28), but the third

term for Ca looks different from its counterpart. Our regularization parameters are

classified in terms of singularity and differentiability of FS, in the limit p p',

namely into e-2-order, e-l-order, co-order, el-order terms, etc. (see Section 4.4),

and all the f-dependences associated with them, as seen in Eq. (4-33), naturally re-

sult from the multiple decomposition of JS, via Legendre polynomial expansions.

We will see later in Section 4.4 that our co-order term has no clear dependence

on But in Barack and Ori [9] L + 1 is introduced as a perturbation factor

and limp_,p/ J~~F is expanded as a power series in L, in which the third term gains

L in its denominator as shown in Eq. (4-28). This discrepancy between the two

approaches, however, is resolved by the fact that Ca vanishes alv--iv. We will prove

this in Section 4.4. Also, one should note that our last parameter Da is defined dif-

ferently from D' in Eq. (4-29) (note the difference in notation). Our Da originates

from the non-singular but non-differentiable behaviors of the field in the neighbor-

hood of the world-line of the particle. Again, its coefficient -2-/2/(2L 1)(2L + 3)

results from the Legendre polynomial multiple expansions. The use of this param-

eter in the mode sum calculation will result in more rapid convergence and more

accurate final outcome.

In Section 4.4 we present in detail the derivations of all these regularization

parameters. The results are summarized as below:


At sgn(A) 2 A (434)

q2 E
A, -sgn(A) fA (4-35)









A, = 0,


92
Bt = -E0
To
O


F3/2 3F5/2
A3/2 2A5/2 '


I F/2 (f 22) F/2
A1/2 2fA3/2


F/2 F3/2
S1/2


3i2 F5/2
2fA5/2 '


3( 25/2 3/2
2A3/2 '


Ct = Cr = Co 0,


/2F_ (f 1)F1/2
f)A1/2F -1/2 + 2
2A1/2


-1)E2 + 3f(f
8fA5/2


4)] F5/2


5 [-2(f + 3).2 +6(1 f)2 + 3f2]F7/2
8fA7/2


(1 f) (,4 E4)F-1/2
22 f 1/2
1).4 f2 + (1 f)E4 + 3fE2] F12


4f2 3/2


[(f + 9)r4 + (-(6f + 4)E2 + f(f + 4)) i2 + 5(f
8f2A5/2


35 289/2
+ 8fA9/2 '


18)E2 3f(3f + 2)) i4


+3 (3(1 f)E4 + f(f + 4)E2 + f3) i2 + 6f(f


5i2 [-(f + 3)4 + ((6


2f)E2 + 3f2) i2 + 3(f
4f3A9/2


1)E4 3E2] F5/2
8f3a 7/2
- 1)E4- 3f2E2] F7/2


35i4F9/2
8fA9/2


(4-42)


q2
r 0
To


i2
J


(4-37)


(4-38)



(4-39)

(4-40)


92
D

+[(9


F3/2
4A3/2


q2
0r I


(4-41)


+ [(9- f)i6 + ((10f


1)E4 f2E2] F3/2


(4-36)


f)2 + 9(f









2 j A3/2[1/2 2 + (1 f)E2 l /2
7/2- 2 4fA1/2
[(f + 9)2 (13f + 9)E2 + 3f(f + 8)] F3/2
8fA3/2
+ [(9 f)r4 + ((10f 18)E2 f(5f + 8)) r2

+9(1 f)E4 3f(13f 8)E2 + 21f3] F5/
8f2A5/2
5 [-(2f + 6)>4 + ((12 4f)E2 + 12f2) P2 + 6(f 1)E4 3f2E2] F7/2

35- t /2F/2 (4-43)
8A 7/2 I

Ao = Boe Co = Do = 0, (4-44)

where A r r- and E -ut = (1 2M/or) (dt/d'r) ('r: proper time)
and J u =- ro (df/dr)o are the conserved energy and angular momentum,
respectively, and = u" = (dr/dr)o, f = (1- 2M/ro), A (1 + J2/r2). The
subscript o denotes evaluation at the location of the particle. Also, shorthand for
the hypergeometric function is Fp 2F1 (p, 2; 1; J2/(r + J2)) (see Appendix A
for more details about the hypergeometric functions and the representations of the
regularization parameters in terms of them).
4.3 Description of Singular Field and THZ Coordinates
4.3.1 Introduction of THZ Coordinates

Intuitively, one may expect that the singular source field bs will behave like
a Coulomb scalar field due to its analogy to Dirac's mean of the advanced and
retarded fields in flat spacetime [2]. In fact, this intuition can be supported via
some local analysis of is. If bs resembles the Coulomb q/r piece of the scalar
potential near the particle with scalar charge q, where r is the distance between a
source point p' and a nearby field point p, we can think of bs as the field measured
by a local observer sitting on the particle, to whom the background geometry in the
vicinity of his location looks flat. The description of bs will then be advantageously
simple in this observer's frame of reference, and we are motivated to use some









S7....i/; inertial or normal coordinate system for this purpose. A normal coordinate

system can alv--, be found where the metric and its first derivatives match

the Minkowski metric on the particle's world-line F, and the coordinate time T

measures the proper time [19]. Normal coordinates for a geodesic, however, are

not unique and have an ambiguity at O(p3), where p is the proper distance from p

to F measured along the spatial geodesic which is orthogonal to F. For example,

differences of O(p3) distinguish Riemann normal from Fermi normal coordinates

[19]. For our purposes a normal coordinate system introduced by Thorne and

Hartle [20] and later extended by Zhang [21] (henceforth, referred to as THZ

normal coordinate system) is particularly advantageous. It will be shown later in

Subsection 4.3.2 that in this coordinate system the scalar wave equation takes a

simple form and that as a result we obtain


S= q/p + (p3/R4), (4-45)

where R represents a length scale of the background geometry. The approximation

in Eq. (4-45) is accurate enough for self-force regularization because

Va,) = Va(q/p) + O(p2/R4), (4-46)

and the O(p2/74) remainder vanishes in the coincidence limit p i p'.

The THZ coordinates XA = (T, X Z) associated with a given geodesic F

have the following features [21]:

(i) Locally inertial and Cartesian; more specifically, on F, gAB = rAB and

aC9AB = 0. And T measures the proper time along the geodesic F, and
X = = Z =0 on F. Also, the metric is expandable about F in powers of

p /2 + 2 + Z2 in a particular form like


gAB = TAB + Pf x (homogeneous polynomials in XI of degree q)


(4-47)









with p > 0 and p+ q > 2.

(ii) The coordinates satisfy the de Donder gauge condition


OAQAB = O(Xn),

where gAB /gAB

The metric perturbation in THZ coordinates is described as [18]


gAB


TIAB + HAB

rIAB + 2HAB + 3HAB + O(p4/R4), p/R 0,


with


3HABdXAdXB


-SjXI XJ(dT2 + 6KLdXKdXL)

+ KPQ3I XPXIdTdXK
3
-20 XXJXK 2PIX dTdXK


+ XIJPQeQK PK X 1 pi2 BQ.XP dX'dXJ

(4-50)


-SIJKXIXXK (dT2 + 6KLdXKdXL)
3
2
+ KPQBQIJXPXI JdTdXK
3(p/ dXdXJ,
+O(p4/R4 IJjdXdXj,


where rlAB is the flat Minkowski metric, CIJK is the flat space Levi-Civita tensor,

p = (X2 + 2 + Z2)1/2, and the indices I, J, K, L, P and Q are spatial and
raised and lowered with the three dimensional flat space metric 61j while the

index 0 denotes the time component. The external multiple moments are spatial,

symmetric, tracefree tensors and are defined in terms of the Riemann tensor


(4-48)


2HABdXAdXB


(4-49)


and


(4-51)









evaluated on F as

EJ RoloJ, (4-52)

B3 = iPQRpQjo, (4-53)
2

SIJK [VKRoIo STj (4-54)

IJK IPQVKRPQJO]STF, (4-55)

where STF means to take the symmetric, tracefree part with respect to the

spatial indices I, J, ... and the dot denotes differentiation of the multiple

moment with respect to T along F. Dimensionally, S j ~ BIJ 0(1/R2) and

SIJK U31JK SIJ J ~ 0(1/Rt3). The fact that all of the external multiple
moments are tracefree comes from the assumption that the background geometry is

a vacuum solution of the Einstein equations.

The THZ coordinates are a special kind of harmonic (or de Donder) coordi-

nates. We may express the perturbed field by defining

HAB AB AB, (4-56)

where gAB ggAB. A coordinate system is harmonic if and only if

AHAB = 0. (4-57)

Zhang [21] provides an expansion of gAB for an arbitrary solution of the vacuum

Einstein equations in THZ coordinates, in his equation (3.26). In the leading lower

order terms, the metric perturbation HAB in this expansion is described as [18]


HAB 2 AB + 3HAB + O(p4/R4),


(4-58)









where


2HO -2S1jXIXJ
2HOK 2 10 JIJ K 2K I 2]
2t = PKpQBQIXpXI+ 21
3 21 5

2HIJ = X(;' J)PQQKXpXK PQ(IJ) Qxp2 (4-59)

and


3Hoo0 lIJKXIXJXK
3
3HOK KPQBQIjXp XJ
3
3HIJ = O(p4/4). (4-60)

The metric perturbation HAB is the trace reversed version of HAB at linear

order,
1
HAB HAB- 9ABHCc, (4-61)
2
and the expansion shown in Eqs. (4-49)-(4-51) precisely corresponds to Zhang's

[21], the first leading terms of which are expressed in Eqs. (4-58)-(4-60).
4.3.2 Approximation for the Singular Field in THZ Coordinates

It was seen at the beginning of the previous Subsection that the motivation

for the use of THZ coordinates is to obtain a simpler approximation of the singular

source field, as represented by Eq. (4-45). The result in this equation can be

derived via Eq. (4-17). We develop local expansions in the THZ coordinates for the

elements u(p,p'), & and v(p,p') on the right hand side of the equation, and combine

them to give an approximate expression for bs.

First, for a vacuum spacetime (RAB 0) which is nearly flat, according to

Thorne and Kovhcs [22] we have


u(p,p') = 1 + (p4/ 4).


(4-62)









When the source point p' is on F, Synge's world function a(p,p') is particularly

easy to evaluate in THZ coordinates for p close to p'. The world function is shown

by Thorne and Kovhcs [22] to be


a(p,p')= 1xAB (TAB + HABdA + O(p6/T4), (4-63)

where XA is the THZ coordinate representation of the field point p while the source

point p' is represented by (T', 0, 0, 0) [18]. The integration of the coordinate along a

straight path is given by

(A() A(XXA '6A), (4-64)

where A runs from 0 to 1.

Working through only the lower order expansions of the perturbed field,

namely HAB 2HAB + 3HAB + O(p4/R4), the integral of HAB along the straight

path C is evaluated with the help of Eq. (4-64) to be [18]



Hoo /HoodA
Jc
SKL + SKLM K L ] dA + O(P/Z4)

SSKLXKXL -- KLMXKXLX + O(p44/R), (4-65)
3 12


Hoi HoidA

I 2 eKP8 L KL 1_ KLoK L

21 3 J
+ 4 KI% K + IKP8 LM LM d A+ (p/4)
2 5
SIKP8 L KX L KLX XKLX
9 42
1 1
+ tSKIXK 2 + KPi LMX KXLX M + O(p4/R4) (4-66)
21 12









and

iJ = HijdA
Jc

= L-IJS K L 6IJKLMK L
+ 5 KP L K 1KP JK dA + (p4 /4)

6IJSKL X 6IJSKLMXK XLXM
3 12
+ 5IKpLPLXKX LxJ t IKPP JXK 2 + O(4/R4). (4-67)
84 84

In terms of 'HAB, Synge's world function is expressed as [18]

1 1 1
7(p, p') ABrlAB + 1(7 _-)2YHo + (7 T')X HIO +IJ
2 2 2
+O(p6/R4)

2 (1 Roo) [(r' 7 + X'H0o)2 _- IXJ(r + 'j)/(1 oo)]

+O(p6/R4). (4-68)

With the source point p' on F, the second term in the square brackets above
can be modified with the help of Eqs. (4-65) and (4-67),

X'X(rqi, +'Huj) = p2( +' oo) + O(p6/R4), (4-69)

where

p2 XI XJj. (4-70)

Substituting Eq. (4-69) into Eq. (4-68) and factorizing o, we obtain [18]

S(p,p') 2(1 Roo) [- 7 + X'jo- p(1 + Hoo)]

x [T' T + X'-Io + p(l + Hoo)] + O(p6/R4). (4-71)









At the retarded time, p' is on the past null cone emanating from p, where

a(p,p') = 0, and the first factor in the square brackets in Eq. (4-71) is


T' T + XI"Io p(l + 'oo) ~ p


(4-72)


due to the fact that T' T < 0 and IT' TI ~ p, and the second must be

T' T + X'N0o + p(l + Hoo) p-1 x O(p6/j4) O(p5/R4)


(4-73)


to cancel the term O(p6/TR4) in Eq. (4-71) such that a(p,p') = 0 precisely. Then,

the differentiation of Eq. (4-71) with respect to T', evaluated at the retarded time

is dominated by the differentiation of the second factor,

da(p,p')] -( oo) [7"- 7 + XIio p(l + oo)] + O(p6/75)
dT- I ret ret


1
2(1 Noo) [-2p( + Noo) + O(p5/R4)]
p [1 + O(p4/4)] ,


(4-74)


where the second equality follows from the fact that T -p for the retarded

time, and the third equality from Eq. (4-65) [18].

Similarly, at the advanced time, p' lies on the future null cone of p, where

a(p,p') = 0, and the first and second factors in the square brackets in Eq. (4-71)
now reverse their roles,


T' T + X',Ho p(l + ,oo) ~ O(p5/R4)


(4-75)


and


T' T + XI7-po + p(1 + -oo) ~ p,


(4-76)


due to the facts that T' T > 0 and 7T' TI ~ p and that r(p,p') = 0. Then, the

differentiation of Eq. (4-71) with respect to 7', evaluated at the advanced time is









now dominated by the differentiation of the first factor,

d(p, p -) (1- 'oo) ) [T' + X"Hno o+p(lt + oo)] + O(p6/R5)
dL adv 2adv

(1 -oo) [2p(1 + Roo) + O(p5/i4)]
p [1 + O(p4/R4)], (4-77)

where the second equality results from the fact that T' T p for the advanced
time.

Dewitt and Brehme show that in general

1
v(p,p') = R(p') + O(p/n3), p F. (4-78)
12

However, in vacuum spacetime, where R = 0,


v(p,p') = O(p2/n4). (4-79)

When integrated over the proper time, the dominant contribution from this term is

O(p3/R4) in the coincidence limit p F.
Then substituting all the results in Eqs. (4-62), (4-74), (4-77) and (4-78) into
Eq. (4-17), we eventually obtain [18]

s + O(p3/R4). Q. E.D. (4-80)
P

From Eqs. (4-49)-(4-51), one notes that our expressions of THZ coordinates

are well defined up to the addition of a term O(p5/R4), which corresponds to a
term O(p4/R4) in the metric perturbation. The change in p due to the addition

of such a term is O(p3/R4), which would be consistent with the order term in
Eq. (4-80). The differentiability of the order term is of interest, and a term of

O(p3/R4) is C2 in the limit p --- 0. From the fact that R = ret _- s, where
'R is a homogeneous solution of the scalar wave equation, we find that Eq. (4-80)

clarifies the relationship between the accuracy of an approximation for bs and the









differentiability of the subsequent approximation for QR, and the self-force 0a R: if
the approximation for bs is in error by a C" function, then the approximation for

bR is no more differentiable than C" and the approximation for 0tQR is no more
differentiable than C"-1. This concern of differentiability is associated with our
last regularization parameters, Da-terms. According to Eq. (4-33), Da-terms are

determined by the el-order terms of 0,s8, and correspond to the accuracy ~ p2/.R3
in is. This allows errors in bs by O(p3/R4), which is C2 in the limit p -- 0. Then,
our self-force 0dR is no more differentiable than C1.

It would be instructive to give an intuitive interpretation of Eq. (4-80) using
the features of THZ coordinates. The scalar wave operator in THZ coordinates is

[18]

VA A a = A (ABaB ) A (HABaB)

f= TQABOAOB b HIJOIj 2HIO(10io)

-HOO, ,1', (4-81)

where the second equality follows from the de Donder gauge condition, Eq. (4-57).

When ) is replaced by q/p in Eq. (4-81) [18],


gV A A(q/p) -4q6(3)(X) + (p/R4), p/R 0, (4-82)

for which we used Eqs. (4-58)-(4-60) and the fact that p is independent of T. A

C2 correction to q/p, of O(p3/R4), would remove the order term on the right hand
side of Eq. (4-82) and we are led to the conclusion that s = q/p + O(p3/R4) is
an inhomogeneous solution of the scalar field wave equation [18]. The error in the

approximation of bs by q/p is C2.
4.3.3 The Determination of THZ Coordinates

The singular field is approximated in a simple form via THZ coordinates as

in Eq. (4-80). In our self-force problem, this bs depends on the geodesic of the









particle as well as on the geometry of the background spacetime. Thus, in order

to derive the regularization parameters from the multiple components of Va8s,

where a denotes the coordinates of the background geometry, one requires that p

in Eq. (4-80) be expressed in terms of the coordinates of the background geometry.

For this purpose, given the definition p2 X2 + y2 + Z2 in the THZ coordinates,

we need clarify the relationship between the background coordinates (t, r, 0, 4) and

the THZ coordinates (T, X, Y, Z) associated with an event p' on the world-line

F. In this Subsection we provide the expressions of the THZ coordinates in the

vicinity of the source p' moving on a general orbit F, in terms of expansions of the

Schwarzschild background coordinates up to the quartic order. The procedure to

complete this task can be summarized in the following steps:

(i) Find initial inertial coordinates XA (A 0, 1,2, 3) in the neighborhood of

the event p' on F in terms of Taylor expansions of the background coordi-

nates Xa = (t, r, 0, 4) about x', where x' represents p' in the background

coordinates (henceforth, the subscript o denotes the evaluation at p'). This

coordinate frame is static in the sense that the event p' is not in motion along

F yet.

(ii) Construct Fermi normal coordinates, which have vanishing ('!i 1-i.!! !

symbols along F. With the metric components being expandable about F in

powers of proper distance to F for all time, Fermi normal coordinates provide

a standardized way in which freely falling observer can report observations

and local experiments [23].

(iii) Determine the transformation from Fermi normal coordinates to THZ

coordinates and finally combine this with the results of Steps (i) and (ii).

Step (i).

We build initial inertial coordinates XA via the expansions of the Schwarzschild

coordinates x" = (t, r, 0, 4) about xt. Weinberg's [24] Eq. (3.2.12) shows that











XA + MA(Xa ) + MAa _
'A 2 a c -lo ()xb x-)(xC- 4), (4-83)

where we may choose XA 0 and

MAa diag [Mt,, Mxr, M Mz ]

= diag [(1 2M/ro)1/2, (1 2M/ro)-1/2 rosino, -ro] (4-84)

for convenience as this choice re-centers and re-scales the Schwarzschild coordinates

to = (1 2M/ro)1/2 (t to), X = (1 2M/ro)-1/2 (r- ro), To sin 0o Oo),

Z = -ro( 0o). Taking advantage of the spherical symmetry of the background,

we may take only the equatorial plane 0o = r/2 for this transformation. Then, for

Eq. (4-83) the non-zero C'!hi -1' !!. symbols evaluated at xo are

M f M M
tr o r2 to 2 Io 2 o 0o,

Fr, -fro, FOo (4-85)
o ro

where f 1 2M ) These coordinates are static and we have no information

about the particle's motion.

Following Ref. [24], the metric expansions in these coordinates can be deter-

mined via
axa aXb
gAB = gab (4-86)
8XA OX B
where the background metric gab can be expanded about xa in the Taylor series

1
gab = gab + gab,c o (xc X +2 g ab,cd j (xc- X )(xd d)

+6 9gab,cde 0 ( )(d )(e X) + 0[(x )4]. (4-87)









To carry out the calculations with Eqs. (4-86) and (4-87), we need invert XA(xa) in

Eq. (4-83) by iteration to

Xa = xa + NaAA 1 p aNPA B

1
+ t o P oNqANrBN8N^AXk

-t (4 a 0 F |o F' o N q AN'A' rB^ D

+ o o v o NrANsBNuCNvD) XABCXD

+0(X5), (4-88)

where NaA is the inverse of MAa such that MA aN aB AB and MAaNbA b= a.

Then, using Eqs. (4-86)-(4-88) together with (4-84) and (4-85), we obtain


gAB = rAB + HABCD

S]AB + HABCDXCXD + HABCDEXCXDXE + O(X4/R4) (4-89)

or in the contravariant form

AB rAB HAB

AB rAB ABcDXC D- ABcDEXCXDXE + (X44), (4-90)


where the non-zero components HABCD = H(AB)(CD) and HABCDE = H(AB)(CDE)

turn out to be

S M2 1 (2M 3M2" H M2
Hoooo = ~m, Hoolln = Hoto( 3~2i\_M
fr4 H fr r fr
0202 M M2 1 /2M M2
ft0202 ft303 M Hiloo m2, Htl111 t (2M '

f -1 M
H1122 = H1133 H1212 H1313 2- 2 '
r 1 2 \r r3
f 1 f f
H2222 H2233 2 H2323 fH3333 (4-91)
r2 r 2r. o
Oo o o









and


0001 3f3 2M2 3M3) H00111 1 (2M 6M2 +51M3
Hoooo1 3 f3/2 5r r ]6 Yoom = 3/2 4 ~5 ~6 '
OO0 \ 00 /o 0o o 0 '
S1 2M 3M2\ ) M3
fH00122 H- 00133 -f1/2 H501000 f l 3/2r c'



1Oll 33f32 2
/1/2 ro 3 3
HO02012 HO3013 6 fl/T2 p4 ~ ~^ H "11001 3 f3/ p5 p6~+ ^ '


1


(2M


6M2 3M3


11111 f32\r r r

H11122 = H11133 3f 1/2 r 4
0 0 0
1 M M2
H12002 f H12112

fl/2 1 M
fH12222 ft322 ; H) 12233

1 M M21
H13003 6 f 1/2 ( 4 /5) H13113

fl/2 1 M)
H13223 --, H 13333
6 H r3 r
2 f 1/2 1 3M)
H22122 = H33133 3 3 4'
0 0
Sfl/2 1 3M)
H23123 = 0
3 r r


'


+15M2

1 4 11


fl/2 (1 M



3r
6 r ri
t1 11


fl/2 1 M
2 r3 ri
2 0 1/2
, H22133 23 r
3 r


M
4+
0



M
4+
0

) I


9M2
5M '



9M2
S5 '


(4-92)


The components HABCD and HABCDE serve as building blocks to evaluate the

quantities BAC, D ABCD, C,DE and RABCD, E at the location of the particle


pA
BC, D
0
ABCD
o
^A
BC, DE

RABCD, E
10


HABCD + HACBD HBCAD,


tBCA D tA CBD fiBDAC + lA DBC,

3 (H BCDE + A CBDE HBCADE) ,

3 (HBCADE HACBDE HBD A CE + HADBCE)


(4-93)

(4-94)

(4-95)

(4-96)


H,,,,,


)









and these quantities are essential for transforming the initial coordinates XA into

Fermi normal coordinates and then finally into THZ coordinates through Steps (ii)

and (iii).

Step (ii).

To construct Fermi normal coordinates out of the initial coordinates XA,
first we evolve the particle's motion along F from the initial point XA = 0,

which corresponds to x' in the Schwarzschild coordinates. Since F is a geodesic

of the particle's motion, its tangent vector iA, the four-velocity of the particle is

transported parallel to itself along F

BVB A = 0. (4-97)

We call F the timelike geodesic, and the time-axis of an observer's frame that is co-

moving with the particle is tangent to this geodesic. While an observer is traveling

with the particle along F, his space triad remains orthogonal to F-parallel

transport preserves orthogonality to F [25], i.e.


UBVBB ) = 0, (4-98)

where iA (I = 1, 2, 3) are basis vectors for the space triad, spanning the hyper-

surface orthogonal to F. Along each direction of ,the physical measurement

made by the observer should not be affected by where it is made, thus each of

n A should be transported parallel to itself. At the same time, each of A) should

alv--,- remain orthogonal to the others Aj) (J $ I). Then, altogether


(J) VB() 0, (4-99)

which gives three spacelike geodesics, F(I) (I = 1,2, 3).

The set of vectors { A, if), fA(), nf (} above form an orthonormal basis for

the co-moving observer's frame. Now having this basis we may construct a family









of geodesics XA(s, A1, A2, A3), which will be inverted later to give Fermi normal

coordinates: here s is a parameter for a temporal measure along F and becomes

the time coordinate TFN via the inversion, and A' (I = 1, 2, 3) are parameters

for spatial measures along F(I) and become the spatial coordinates XIN via the

inversion. By combining the integral solutions of the geodesic equations (4-97)-

(4-99) we obtain


XA(s,Al,A2,A3) dsA ds I + dA' ds
r= ( )s )

+dA dA (4-100)
I=1 () J=-1

where A = (A1, A2, A3), and the subscripts outside () mean that these variables

are held fixed while the partial differentiations are performed with respect to the

others.

To evaluate the above integral, one need find proper expansions of each

integrand in terms of s and A. From Eqs. (4-97)-(4-99) we have


(8 ) -0 BC X-

BC, D u + 2 BC, DE BCD




a(/ ) (-BCU(I)o
8( X=o

iBC, D (I)U BC, DE (I)B Ci D E
S(1

+0(_k3/R4), (4-102)









(nI) pA B C
AJ B (I) (J) s=
dA h C _D + 1A )nC)X 8
SBC, D (I) J) 2 BC, DE o(I)(J) X s 0

+0(X3/4), (4-103)

where ABc is expanded around A = 0, and its expansion coefficients iAB, D and
0
iAC DE are computed via Eqs. (4-93) and (4-95) along with (4-91) and (4-92).
The first order approximation for XA(s, A1, A2, A3) near the initial point A = 0 is

axA axA
XA(s, A1, 2, A3) s + A' + [(so )2/1R]
0 0
s ~,)I + O[(s, )2/R], (4-104)

where A, A(1) n(2), n (3) are the orthonormal basis vectors evaluated at the
location of the particle, and the summation is assumed over the repeated index
I = 1, 2, 3 (hereafter, we omit the summation sign and assume the summation
convention for the up-and-down repeated spatial indices I, J, K, L,... = 1, 2, 3).
Substituting this into Eqs. (4-101)-(4-103), we obtain the quadratic approximations
for the integrands

( IU \ A j%CjDS/ t1 4 OiiCii
= X -0 BC, D C, DE 0o 0 0 0
+O(S3/R4), (4-105)


((i) A iB 1CfD t A iB iCDfE
as AO BC, D o(lJ) 0o 2 BC,DE o 0 (J)o0 U0

+O(S3/R4), (4-106)


(a ) ^A B (Ds + D K)
Oa j BC, D 'o(I) o(J) o S (K)
S0 (s +A D K) (s + o(L)A)
O sC,DE () (J) 0 (K) AK 0 (L) )
[ 3/]. (4 7
+o[(s, )3 /4]. (4-107)









Our aim here is to find the quartic order expansions for Fermi normal coordinates,

which are derived from the inverse transformations of the quartic order expansions

of XA in s and A. XA are determined to the quartic order in s and A when

we evaluate the integral in Eq. (4-100) along with Eqs. (4-105)-(4-107). The

contributions from the order terms O(s3/ 4) or O[(s, A)3/)" 4] can be disregarded

since through the integration via Eq. (4-100) these become higher than quartic.

Substituting Eqs. (4-105)-(4-107) into Eq. (4-100) and performing the integral,
we obtain


XA(s, 1, A 2, 3)


AS+i nA +K
0 o(K)

SBC,D Uo CD + 3 0C 0D 2 ( BK)
+3Djs (^B K) L B ( K (C L (D M

-24 C, DE 0 C D 00 fs4 C+ 4D 10 3 ()ABK)

+6uDuE 2 (K) K (L) A


+(4 B K) L)( (L D ) () E
+ [(o(K)A ) (OL) ) L (M)M () (N)

+0[(s, A)5/R4]. (4-108)


The parameter s for a temporal measure along F becomes the time coordinate

TFN and the parameters A' for spatial measures along F(I) become the spatial

coordinates X1N (I = 1, 2, 3) when Eq. (4-108) is inverted and solved for s and A'.








To solve for s, take an inner product of both sides of Eq. (4-108) with -i-oA,
exploiting the fact that A UoA = -1 and that noK)UoA 0. Then

TEN s +0 {XA [+BjCjDS3 + 3ACDS2 (AAK)
TFN -S -oA A BC, D6 BC 3DC 0D 2 0 (K)
1 0o
+3S (K)i K) (nLAL) + (<)A) (
+ C DE o CBCDE 4 C + 4 fD 3 ( AK)
24 BC,0 00 0 0O 0(K) A
+6DifE 2 (K)UB K (' L)

+4(s (< K) L) (M)A
+4^O s [n^O(K)A> ) 7() on(M)A )
I l^B L0 -D E N^ I
+ (K) ) K(o(L)A) L(n AM) (M (N)A) }

+0[(s, A)5/74]. (4-109)

To solve for A', take an inner product of both sides of Eq. (4-108) with nl ,
exploiting the fact that uAnA =- 0 and that A (I^A = K. Then
o oA oAo( K. Then

F N A CoA Do BC D 3 0CD 2 0 aB(K)
+3jS (K K) +3 1 A(K) ABCD u(L) A uE3(K) A BAK>(L) A m) Am)

2 C,DE 0B C D 4 uCu D E 3 B(K)
+1 BC,DE o oKuoo +4uouoo (o)A)
6DjE 2 (B K ) C L
+6u0 0 ((K)A L) (noL (M)A
+4fB K)AC L )AD )AM)

+ (K)A ( L)A) (Ln AM) (
+0[(s, A)5/74]. (4-110)

Similarly, we may invert Eq. (4-104) to solve for s and A' by contracting both its
sides with -kUA and n I

s -oA A + O(X2/ ), (4-111)


A' =< (IXA + 0(x(2/R).


(4-112)









Substituting these linear approximations for s and A' on the right hand sides

of Eqs. (4-109) and (4-110), we finally obtain the expressions of Fermi normal

coordinates, written in terms of the initial coordinates XA, precisely up to the

quartic order



TFN = -UA [XA + ABCDBCXD + A BCDEXXXCDE + O(X5/R4),
(4-113)

XN A'I L A + ABCDX BC D + (A BCDEX B_" X + o(X5/R4),

(4-114)

where


ABCD PQ, R V B TT RD + 37TQB rRchPD + 376RBhPChQD
b o
+hPBhQchRD) (4-115)


BCDE PQ,RS (7 PB7Q CRDTSE + 47r QB rRr SDhPE
ABCDE 4 P
+6rRB rSchPDhQE + 47RBhPchQDhRE

+hPBhQchRDhSE) (4-116)

with 7AB being the time-projection tensor and hAB being the space-projection

tensor, which are defined as

7AB -U oo (4-117)

hAB A (I) AB Au (4-118)
ho(I)noB A o 0oB,

respectively.1



1 The identity provided in Eq. (4-118) is obvious from the local tetrad, or lo-
cal vierbein: vAP) {uA, uj, n2, n ) ) where P e (0,1, 2, 3) is the label for









The transformation via Eqs. (4-113) and (4-114) reproduces the desired
geometry of Fermi normal coordinates. We examine this in Appendix B.

Step (iii).
Out of any I .i.. ,l Inertial Cartesian coordinates, we can develop the three
geodesic equations as described by Eqs. (4-97)-(4-99). Thus the coordinates XA in
Step (ii) may be replaced by another locally inertial coordinates XA to make initial
coordinates. Then by combining the integral solutions of the geodesic equations
(4-97)-(4-99) we now construct a new family of geodesics XA(s, A1, A2, A3)


XA(, ', A2, 3) ds ds us + I dA' I1 ds ( 0
I a9s 1-1 M as ) X_
3 1 2 3 A I) \
+ dA dA a (4-119)
I 1 (M) J=1 ()

where s becomes the time coordinate TFN and A' (I = 1,2, 3) become the spatial
coordinates X1N when the equation is inverted.



each vector of the tetrad. These vectors constitute a local orthonormal frame at
each point along the timelike world-line F of a particle, in which the timelike vec-
tor uA, the four-velocity of the particle is tangent to F and gives the direction for
the time-axis, and the spacelike vectors n() (I 1,2, 3) serve as the basis vec-
tors for the space triad. Then it follows that (i) gABV p)V = rpQ and that (ii)
gABV (P) rl Q and thQ)ii
IPQV Ap V)B AB. Splitting the tetrad VA into uA and n), the relation (ii)
can be rewritten as -uA B + I n1AI) (I)B gAB, which proves the identity in
Eq. (4-118).









Following the same procedure as in Eqs. (4-101)-(4-108), Eq. (4-119) develops

into


XA (TFN, FN)


A A K)XK
U TEN + "o(KXFN
1 FA [D B C D r3 i I C Dm2 XK
- 6 ABC, D 0oT#N +' .) 0TNXFKN

-+" C D 7cK xL + B C D xK xL M
+3,, -) )no(L)Uo TFN FN FN + o(K) o(L) o(M) FN FN FN
1 A BCDE4
24 BC,DE oLo o EFN .+ ), D o C D ETNXFKN

+6ro(K)nO(L) Uo FNXFNXFN

+ ,R -)rC D (E TFNX K LNX M
B C D E rK rL rM rN
+o(K) o(L) o(M) o(N) XFN FN FN FN

+O(X#N/T4), (4-120)


where the subscript o denotes the evaluation at the location of the particle, and the

quantities uA, nA A FcD o and FAC,DE o are evaluated in the coordinates XA. If

we identify XA with THZ normal coordinates, then this equation exactly tells us

how Fermi normal coordinates transform into THZ normal coordinates.

The linear part of the above transformation implies the inverse-Lorentz boost

and is responsible for the relationship between the metrics of the two geometries at

the location of the particle. The two metrics are in fact both Minkowskian there,


(4-121)


9(THZ)ABo g- (FN)AB o -7AB.


According to Ref. [24], this relationship must be satisfied via

A'B' AB OxA' OXB'
9(THZ) (FN) 8XAN AXBN.
FN EBN


(4-122)









Then the inverse-Lorentz boost must be the identity transformation,

oA T A-A "A, (4-123)
A TFN o

A -A AA XA 6Ai, (4-124)
FN o

and the whole transformation in Eq. (4-120) is characterized by cubic or higher

order corrections between the two coordinate expressions. Also, we can specify

A A (4-125)
IBC,D o =HABCD + HACBD HBCD, (-125)

BFC,DE o = 3 (HABCDE + HACBDE HBCDE) (4-126)

using the components of the metric perturbations taken from Eqs. (4-50) and

(4-51). Then, with iU, nI) C, D and FBDE o specified, the transformation in

Eq. (4-120) is completely determined to be



S= TN KLXXF NpN + O(X N/R4), (4-127)
S FN 1N 68 KL FN N
1 1 K L
X XN SCKXFNP~N + 'KLXFNX FNN
6 3
1 1
SKXFNPFNTFN + t'KLXFNXFNXFNTFN
6 3
1 1
SKLX KNX K N 2 KLMX NXN XM X N
~ KLFN FNPFN 1 i FN FN FN FN
24 2
-2MK MLXKFNX pN + O(X5 N 4), (4-128)
63 MK LFNFNPFN FN

where T TTHZ, X' XT HZ and PFN XFN + YN + ZFN. In Appendix C it is

verified that the coordinate transformation via Eqs. (4-127) and (4-128) properly

converts the metric of Fermi normal geometry into that of THZ normal geometry

with the help of some properties on the Riemann tensors for vacuum spacetime.

Finally, in order to express the THZ coordinates XA in terms of the back-

ground coordinates x" = (t, r, 0, 0), we combine Eqs. (4-113) and (4-114) with

(4-127) and (4-128) along with (4-83). The final result is











T -UoA [XA + KBCDXBXkX_ + AABCDEXBXkXkDE] +O(X5/R4), (4-129)

X= o(A + ABCDXB CX + D ABCDE B CD E + O(X5/ -tTR4) (4-130)

with
XA = MA(a x- x) + MAa lo (xb )(x x,), (4-131)



A1 ] A6 PR o ( QrPBQCRD + 37 QB rRChPD
K BCD 6 PQ, B

+37 RBhPchQD + hPBhQchRD), (4-132)


A BCDE A PQ,RS (rPBTQ RDTSE + 47QBTRCTSDhPE
24 o 0
67RBsrSchPDhQ E + 47TSBhPhQDhRE + hPhQchRDhSE)

1 APQR,S 7QShPBhRchDE, (4-133)
168 o


M ABCD -1 Ap, R ( PBQ C RD + 37TQB7RC hPD
ABCD-6 o
+3wRBhPch.QD + hPBhQ ChRD)

EPQR (t6AEUPQhRBhCD + t B QEhp D (4-134)



VABCDE PQ, RS 7 B CT T +47TQB7 C

67 RBT7S ChPpDhQ + 47rSBhPchQDhRE + hPahQchRDhSE)

F- PQR, S o ( FAF PQSBhRChDE + ABQF ShPDhRE
1b 1
+ AFT PQhRBhSChDE + b--6 B FhPChRDhSE
24 24
+ 6AF7 RShPBh DE (4-135)
63








where A, E and P, S = 0, 1,2, 3, and I = 1,2, 3. For the Schwarzschild
spacetime as the background, we may take the equatorial plane 0o = r/2 and
describe the source point in Eq. (4-131) as

x (to, ro, o) (4-136)

We also have
MA diag [fl/2 f-1/2, r ] (4-137)

together with all non-zero F% o for the background taken from Eq. (4-85). Describ-
ing the four-velocities,

i f-1/2E, f-1/2 -, (4-138)
( ro
(1)A -12 1+ f 0 (4-139)
0 t + fl2E+f' ro(E+f/2)' l
(2)A 1+ 0)} (4-140)
) ro o (E + f1/2)' + 2 (f-/2E + 1) 4

(3)A = (0, 0, 0, 1), (4-141)

where f 2M), and E -ut (1 2M/r) (dt/dr)o (T: proper time)
and J uu = ro (d/dr)o are the conserved energy and angular momentum
in the background, respectively, and i u" = (dr/dr)o. These are also used
to compute 7tAB and hAB via Eqs. (4-117) and (4-118). Also, one can evaluate

BC,D BCD C,DE ad ^a BCDE using Eqs. (4-93)-(4-96) along with
(4-91) and (4-92).
4.3.4 Determination of the Singular Field
We determined the THZ coordinates in terms of the background coordinates
in the previous Subsection. Using these results, we are now able to specify the
singular source field bs in terms of the background coordinates. In the approxima-
tion of bs as represented by Eq. (4-45), p is defined as the proper distance from
p to F measured along the spatial geodesic which is orthogonal to F. In the THZ









coordinates, its square is expressed as

p2 XXI x 2 + y2 + Z2. (4-142)

Substituting Eqs. (4-130) along with (4-132)-(4-135) into this equation and

simplifying the algebra, we obtain

P2 -hABA A + lhAE iABC, D BXC D E TAC ABCD BDE XE
3 o 3 o
+ hAF ABC,DE B C D E F _- TAC ABCD, E BXDXEXFX
12 o 12 o
1 TACT EF ABCD,E B D F G G
4 o
+ 7ACR7 E7GH -ABCD,E XBXDXFXGXH + 0(6/R4), (4-143)
6 0

where XA represents the initial normal coordinates and is constructed from the

background coordinates x" = (t, r, 0, 0) via Eqs. (4-131) together with (4-85),

(4-136) and (4-137). Also, 7AB and hAB are computed via Eqs. (4-117) and (4-118)

along with (4-138), and ABC, D RABCD BC, DE and RABCD, E are evaluated

using Eqs. (4-93)-(4-96) along with (4-91) and (4-92). The actual expression of p2

in the background coordinates x" = (t, r, 0, 0) would be very lengthy to the order

specified in Eq. (4-143), and here we specify it only to the cubic order in order to









describe the main features of bs for the next Section

2EF
P2 (E2 f)(t t) 2 (t to)(r- o)- 2EJ(t-
f
+(+ ) (r ro)2+ 2 ( 7)2 2(
f f 2 f
+(r2 + J2)( 2 (t-to)3
00
To
M ( 2E2 r 2 MJr
+ + 2 + t- o)2(r o + -o (t
MEi 2(ro M)E
f2 2 (t to r _r0)2 fr (t t)(r
J '0 J'0
+roE(t to) (0- 2) + roE (t to)( o)2
M ( 2 (2ro-2)r 5M)Jr
22o 3 +12 2 r0
(r 2 0f2 2 ( 2

f 2f r
7+ r2 (+-r ( t r2 + 2J

-roJ ( ) ( o)- oJr ( o )3

+P1v(t, 0, 0) + Pv(t, r, 0, 0) + O[(x Xo)6,


o)(0 o)


to)2( o)

ro)( (- o)


2


r(r
2 /
0


0o)

ro)(0 o )2



(4-144)


where Piv(t, r, 0, 0) and Pv(t, r, 0, 0) represent the quartic and quintic order terms,

respectively. All these terms can be specified using MAPLE and GRTENSOR.
It is important to note that only the minimum information about the back-
ground spacetime is required for constructing XA in order to determine p2 to any
high order we desire. In other words, we specify XA only to the quadratic order

as in Eq. (4-131) for its use in Eq. (4-143), and the specification of XA to cubic or
higher order would not make any difference in p2

4.4 Determination of Regularization Parameters

In Section 4.3, we saw that an approximation of bs is particularly simple in
THZ normal coordinates,


s q/p + O(p3/R),


(4-145)









where p2 = X2 + Y2 + Z2 and 7 represents a length scale of the background

geometry. Following Refs. [17] and [18], the regularization parameters can be

determined from evaluating the multiple components of


Va,) = V,(q/p) + O(p2/R4), (4-146)


where a labels the Schwarzschild background coordinates x" = (t, r, 0, 0). The

remainder O(p2/R3) in the above approximation is disregarded since it gives no

contribution to V,s as we take the "coincidence limit", x x' where x"

denotes a point in the vicinity of the particle and x' the location of the particle in

the Schwarzschild background.

In evaluating the multiple components of Vabs via Eqs. (4-146) and (4-144),

singularities are expected to occur with certain terms. To help identify those

singularities, we introduce an order parameter c which is to be set to unity at the

end of the calculation: we attach e to each n th order part of p2 in Eq. (4-144)

and re-express p2 as


p2 2, II + 3,PI + c4Piv + C5PV + 0(6), (4-147)

where PII, PIII, PIV and Pv represent the quadratic, cubic, quartic and quintic

order parts of p2, respectively.

We express 0, (1/p) in a Laurent series expansion, and every denominator

of this expansion takes the form of P3/2 (n = 3, 5, 7, 9, ... ). Thus, P IIl 1 i, an

important role in the multiple decomposition, but the quadratic part PII, directly

taken from Eq. (4-144), is not yet fully ready for this task. First, 4 0o must be

decoupled from r ro so that each appears only as an independent complete square.

Coupling between t to and 4 (o does not create difficulty in the decomposition.









Thus, we reshape the quadratic part of Eq. (4-144) into


9Fr2
(E2 f)(t_ f 2_ 2 ot (t
f (ro + J2)

f2 (r + J2)
0-W^j -


- t)A 2EJ(t to)(

'/)2 +r2( )2


with
JA
7 #o -(4-149)
f (r + J2) (4-49)
where A = r ro, and an identity ,2 = E2 (1 + J2/r ) is used for simplifying

the coefficient of A2. Here, taking the coincidence limit A -+ 0, we have t' -+ Oo.

This same idea is found in Mino, Nakano, and Sasaki [26]. Also, for the multiple

decomposition the quadratic part must be analytic and smooth over the entire

two-sphere, and we write


2Err2
(E2 -f)(t )2 2E (t o)A-
f (ro + 2)
E2 2r2
2 A2 + (r2 + J2) sin2 0sin2(o
+ [(x- xo)4].
+O[(x X0)4].


2E.J(t t) sin 0 sin( 0')

0') + ro cos2


(4-150)


Here we have used the elementary approximations = sin(O 0') + O[( 0')3]

and 1 sinO + 0[(0- 7/2)2].

To aid in the multiple decomposition we rotate the usual Schwarzschild

coordinates by following the approach of Barack and Ori [27] such that the

coordinate location of the particle is moved from the equatorial plane 0 = r/2

to the new polar axis. The new angles O and 4 defined in terms of the usual

Schwarzschild angles are


sin 0 cos(0 0')

sin 0 sin(O 0')

cos 0


cos 0

sin O cos 4

sin O sin 4.


(4-148)

(4-148)


(4-151)









Also, under this coordinate rotation, a spherical harmonic Yem(0, 0) becomes


Ym(,) rnmm, '( ), (4-152)


where the coefficients aim, depend on the rotation (0, 4) (6, E) as well as on

, m and m', and the index f is preserved under the rotation [28]. As recognized

in Ref. [27], there is a great advantage of using the rotated angles (6, 4): after

expanding a,(q/p) into a sum of spherical harmonic components, we take the

coincidence limit A -- 0, -- 0. Then, finally only the m = 0 components

contribute to the self-force at = 0 because Ym(0, ) = 0 for m / 0. Thus, the

regularization parameters of Eq. (4-33) are just (, m = 0) spherical harmonic

components of a,(q/p) evaluated at x0.

Now, with these rotated angles, PuI is re-expressed as

o2Epyr2
P =I (E2 f) (t_ to)2 2E (t o)A 2EJ(t to) sin 0 cos 4
f (r + J2)
+ -2 o 9 2 J2 2

r E2 2
+2(J)(1 n )


x r2 2 + 1 -cosO + O[(x )4], (4-153)
2f2(r2 + J2)2 ( 2 )

where the elementary approximation sin2 ( = 2(1 cos () + 0(94) is used. We may

now define

2Err2
p2 (E2 f)(t 2 )2 2E ( to)A- 2EJ(t to) sin cos
+2 2 of ( + J2)
o0 +
( J2)( 2 1 sin2 i)

r E2A2
x 0f-(r-- r2EA 2 + 1 cos (4-154)

In particular, when fixing t t we defin
In particular, when fixing t to, we define


p2 p2 t = 2(r + 2)X (2 + COS )


(4-155)









with
J2 sin2 (
X 1 ----- (4-156)
r + J2
and
r E22A2
62 = (4-157)
2f2(r + J2)2
Now we rewrite Eq. (4-147) by replacing the original quadratic part Pii with
/2

p2 p 22 + C3 I + 4PIV + 5V + 0(C6), (4-158)

where Piv now includes the additional quartic order terms that have resulted from

the replacement of PII by p2 through Eqs. (4-150) and (4-153). A Laurent series

expansion of 9 (1/p)|t=to is then

S1 a(2)t to -2 1 a IIlt=to 3 [aa (P2) IIItto -1
p t=to 2 2 p3 4 p5
1 a0aPiv t-t 3 [a+ (P2) IV t=t + (aPIIlI) -PII t=to
S2 p4 p5
15 [aa (p2)] I to
16 p0
1 .Pvt t=to
1 2 p
3 [a. (p2)] Pvl tto + (aaPIII) PIV t=to + (aIv) PIIIt= to
+4
4 p5
15 2 [a (p2)] PPiv lto + (aPjIII) 2II t
16 p
35 [ (p2)] 3 I tto
+I35 p1 + O(C2). (4-159)
32 p0

After the derivatives in Eq. (4-159) are taken, the dependence upon 0, 4 and

r may be removed in favor of po, X and 6 by use of Eqs. (4-155)-(4-157). Then

the three steps of (i) a Legendre polynomial expansion for the dependence,

while r and 4 are held fixed, followed by (ii) an integration over 4, while r is held

fixed, and finally (iii) taking the limit 6 -+ 0, together provide the regularization









parameters. The techniques involved in the Legendre polynomial expansions and
the integration over ) are described in detail in Appendices C and D of Ref. [18].

Below in Subsections 4.4.1-4.4.4, we present the key steps of calculating the

regularization parameters Aa, Ba, Ca and Da in Eqs. (4-34)-(4-44).
4.4.1 A,-terms

We take the c-2 term from Eq. (4-159) and define

Qa[e-2] q t to (4-160)
2 po

We may express this in a generic form
k -A k 1-0 o k- Ip \P
Q,[ _2] a k p(,)A- -o) (0(- -)
0Qa 210>k3 2 (4-161)
k=0,1 p=0 90

where A r ro, and akp(a) is the coefficient of each individual term that depends

on k and p as well as on the component index a, with a dimension Rk for a = t, r

and R~k+1 for a = 0, The behavior of Qa[e-2], according to the powers of each

factor in Eq. (4-161), is

Qa -2] p-3A -k o-k p (- ) P, (4-162)

where s = k for a = t, r and s k + 1 for a = 0, 0. We recall from Eqs.

(4-148) and (4-149) that the first of the steps to lead to p2 is replacing 0 Q o

by (0 0') JrA/f(rQ + J2) to eliminate the coupling term A (0 to) in PII.
This makes a sum of independent square forms of each of A and 0 ', which is

a necessary step to induce the Legendre polynomial expansions later. Thus, to be

consistent with this modification made for po, the remaining 0 &o in Eq. (4-162)








should be also replaced by ( ') JtA//f (r + J2). Then,
J k-p
(- Q)k-P __') 7 _2)
k-p
= ACp ( k-p-i ( )Ai k-p-i/Z--i (4-163)
i=0
S (sin 0)' (cos 4) Ak --i /P k--i + O[(x Xo)k-p+2], (4-164)

where a binomial expansion over the index i = 0, k p is assumed with the
coefficient ckpi 1/Rk-p-i in Eq. (4-163), and in Eq. (4-164) ( 0')' is replaced
by [sin(O o')]1 + O[(4 #')i+2] the term O[(x xo)k-p+2] at the end results
from this O[(0 ,')i+2], then the coordinates are rotated using the definition of
new angles by Eq. (4-151). Also, by Eq. (4-151) again

( 5 (sin O)P (sin )"p + O[(x- xo)p+2]. (4-165)

Using Eqs. (4-164) and (4-165), the behavior of Qa[c-2] in Eq. (4-162) now looks
like

Qa[e-2] 3A1-p-i (sin Q)p+i (sin ))P (cos )i Z, (4-166)

where s = p+i for a = t, r and s = p+i+1 for a = 0, and any contributions from
O[(x- xo)k-p+2] in Eq. (4-164) and from O[(x- xo)p+2] in Eq. (4-165) have been
disregarded: by putting these pieces into Eq. (4-162) we simply obtain eo terms or
0(C2), which would correspond to C,-terms or (0-4) in Eq. (4-33), and later in
this Section it is proved that they ahv--, vanish. Qa[c-2] in Eq. (4-166) then can be
categorized into the following cases:
(i) i 1
The integrand for the integration-averaging process over K) is F() -
(sin )"P (cos )i = (sin )" (cos )), and it has the properties that F(4) + r)









-F()) for p = 0 and that F(4 + r/2) = -F()) for p = 1. Thus


(Qe-2]) =0, (4-167)

(ii) i 0
The integrand F(4) (sin )") has the properties that F() = 1 for p = 0

and that F(4) + r) = -F(4) for p = 1. Thus, the only non-zero contribution

to Qa[C-2] comes from the case p = 0, i.e.

Qa [-2] P 3ARs. (4-168)

The significance of this analysis is that the non-vanishing A,-terms should al-

ov-iv take the form of Eq. (4-168) and therefore in calculating the regularization

parameters we need to sort out only this kind.

Then, we proceed with our calculations of the regularization parameters one

component at a time.

At-term:.

First we complete the expression for Qt[c-2] by recalling Eqs. (4-154) and

(4-155)


Qt [C-2] a(p2)
q2 [2 c + j2) x (2 + t cosO )]-32 (2E +2Js cos )
2 [(r + J2 21-2 + 2EJ sin O cos
q2E r2 A-3/2
X (2 + COS ) -3/2
2V2f (r2 + J2)5/2
q2EJX-3/2 cos -1/
q 2E E C 2)/ a (62 + 1- cos)-1/2 (4-169)
(r + J2)32 3/2

where a/0ae means that A is held constant while the differentiation is per-

formed with respect to 0.








According to Appendix D of Ref. [18], we have the following Legendre polyno-
mial expansions of (62 + 1 cos )--1/2: for p > 1


(62 + CS ) -"-1 2p-12p 1) [1 + O(M)] P(cos ), 6 0, (4-170)
t o
and for p = 0

(62 COSO) -12 = [2+ O(M)] P(cos), ---0. (4-171)
t 0
Then, by Eq. (4-170) for p = 1, and by Eqs. (4-171) and (4-157), in the limit 6 0
(equivalently A 0) Eq. (4-169) becomes



lim Qt[ -2 sgn(A) q2r C P COS
Ao 2 2 + J )/2=0 2Y
q2EJX-3/2 cos 4 D P(cos ). (4-172)
(r2 + J2)3/2 a o

Then, we integrate limao Qt[c-2] over 4 and divide it by 27r (henceforth, we
denote this process by the angle brackets "()")

olim Qt[-2]) sgn(A) q2 yro (X-1) P(COS ), (4-173)
(r2 + 2 2 3J 0 2

where we exploit the fact that (-3/2 cos )} = 0 to get rid of the second term in
Eq. (4-172).2 Appendix C of Ref. [18] provides (X-1) 211 (1, 1; 1; a) F

(1 a)-1/2, where a J2 (r2 + J2). Substituting this into Eq. (4-173), the
regularization parameter At is the coefficient of the sum on the right hand side in

a
2 Or alternatively, one can use the argument a- P(cos 6) = 0 as 8 -+ 0, to
show that this part does not survive at the end.