PAGE 4 FirstandforemostIwouldliketothankmyresearchadvisorProfessorStevenDetweiler,forhisconstantencouragementandguidancethroughouttheentirecourseofmyresearchwork.IwouldalsoliketothankProfessorBernardWhitingandProfessorRichardWoodardforvaluablediscussions.IamhonoredandgratefultoProfessorJamesFry,ProfessorDavidReitze,andProfessorAtaSarajediniforservingonmysupervisorycommittee. iv PAGE 5 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 PAGE 6 ............................ 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 PAGE 7 ............................ 160 vii PAGE 8 Figure page 4{1Self-forceofascalareldintheSchwarzschildspacetime ......... 112 viii PAGE 9 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 PAGE 10 Inthisdissertationwedescribesystematicmethodsforndingmultipoledecompositionsofthesingularsourceeldsforbothcases.Thisimportantstepleadstothecalculationoftheself-forceonascalar-chargedparticleorapointmassorbitingaSchwarzschildblackhole. x PAGE 11 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 PAGE 12 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 PAGE 13 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 PAGE 14 followanalysesbyDetweilerandWhiting[ 11 ]tondthemethodsforcalculatingtheregularizationparameters. PAGE 15 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 PAGE 16 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 PAGE 17 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 PAGE 18 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 PAGE 19 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) PAGE 20 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. PAGE 21 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)). PAGE 22 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. PAGE 23 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) PAGE 24 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) PAGE 26 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) PAGE 28 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) PAGE 29 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) PAGE 30 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 ). PAGE 31 . 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 PAGE 32 wheretheintegrationisperformedovertheregionbetweenanytwospacelikehypersurfaces.Withvariationstakeninthedynamicalvariablesza0andAawhichvanishonthesehypersurfaces,theactionsuersthevariation providedthatistakentobethepropertimeoftheparticle(andwillhenceforthbeassumed)suchthat Applicationofthisactionprincipleyieldsthedynamicalequations Bythefactthat 2RbaacFcb+RbabcFac=0;(2-108) onecanshowviaEq.( 2-107 )thatthecurrentdensityisconserved. Conservationofthestress-energytensor Thestress-energytensorofthesystemisgivenby PAGE 33 where 4gabFcdFcd: Thedivergencesofthesetensorsarefoundtobe 2gab(Fcd;b+Fdb;c+Fbc;d)Fcd=FabJb: Combiningtheseresults,oneobtainstheconservationlaw Vectorpotentialsandelectromagneticelds IntheLorenzgauge theelectromagneticeldequation( 2-107 )mayberewrittenasaninhomogeneousvectorwaveequation Particularsolutionsofthisequationaregivenby PAGE 34 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 PAGE 35 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. PAGE 36 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. PAGE 37 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) PAGE 38 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, PAGE 39 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: PAGE 40 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(): PAGE 41 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. PAGE 42 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. PAGE 43 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 PAGE 44 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 PAGE 45 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 PAGE 46 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. PAGE 47 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 PAGE 48 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 PAGE 49 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 PAGE 50 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 PAGE 51 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 PAGE 52 combined,theradiationdampingforcecancelsouttheantidampingforce,sotheequationofmotionisconservative,anditagreeswiththeappropriatelimitofthestandardpost-Newtonianequationofmotion. PAGE 53 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 PAGE 54 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 PAGE 55 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. PAGE 56 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 PAGE 57 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 PAGE 58 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. PAGE 59 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, PAGE 60 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 PAGE 61 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 PAGE 62 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) PAGE 63 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 PAGE 64 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) PAGE 65 23=2;(4-39) PAGE 66 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 PAGE 67 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 PAGE 68 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 PAGE 69 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) PAGE 70 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 PAGE 71 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) PAGE 72 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): PAGE 73 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 PAGE 74 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 PAGE 75 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 PAGE 76 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 PAGE 77 ^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]: PAGE 78 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; PAGE 79 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; PAGE 80 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 PAGE 81 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) PAGE 82 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]: PAGE 83 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. PAGE 84 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 PAGE 85 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 PAGE 86 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 ). PAGE 87 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 PAGE 88 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 PAGE 89 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; PAGE 90 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 PAGE 91 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 PAGE 92 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, PAGE 93 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. PAGE 94 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: PAGE 95 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) PAGE 96 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 PAGE 97 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 ) PAGE 98 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(+)= PAGE 99 (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. PAGE 100 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. PAGE 101 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 PAGE 102 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) PAGE 103 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 PAGE 104 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. PAGE 105 (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) PAGE 106 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 PAGE 107 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) PAGE 108 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 PAGE 109 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 PAGE 110 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 PAGE 111 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 PAGE 112 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 ). PAGE 113 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=;. PAGE 114 (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=;. PAGE 115 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 PAGE 116 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), PAGE 117 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) PAGE 118 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. . PAGE 119 . 2(f1)1+J2=r2o3=2F1=2+[(f1)_r2+(1f)E2f]F1=2 PAGE 120 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 PAGE 121 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 ]. PAGE 122 112 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 ]. PAGE 123 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 PAGE 124 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 PAGE 125 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 PAGE 126 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 PAGE 127 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 PAGE 128 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 PAGE 129 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 ). PAGE 130 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 ] PAGE 131 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) PAGE 132 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 PAGE 133 [ 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 PAGE 134 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 PAGE 135 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: PAGE 136 (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 PAGE 137 (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 PAGE 138 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 PAGE 139 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 PAGE 140 where and Vectors:fhSt;hSt;hSr;hSrg FromEq.( 5-52 )wedene and wherethelabelcrunsovertandr,andthelabel^arunsoverand. Takingthecasewithc=tand^a=forexample,wehave PAGE 141 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(;)#: PAGE 142 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 PAGE 143 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 PAGE 144 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 PAGE 145 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 PAGE 146 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) PAGE 147 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 PAGE 148 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 PAGE 149 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) PAGE 150 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) PAGE 151 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, PAGE 152 ~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 PAGE 154 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 PAGE 155 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. PAGE 156 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 PAGE 157 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 PAGE 158 whichareidenticaltotheresultsofBarackandOri[ 27 ]. PAGE 159 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 PAGE 160 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 PAGE 161 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 PAGE 162 ( 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); PAGE 163 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 ]. PAGE 164 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 PAGE 165 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, PAGE 166 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