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The Dynamics of Point Particles around Black Holes

Permanent Link: http://ufdc.ufl.edu/UFE0024687/00001

Material Information

Title: The Dynamics of Point Particles around Black Holes
Physical Description: 1 online resource (177 p.)
Language: english
Creator: Vega, Michael
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: force, general, gravitational, radiation, reaction, relativity, self, waves
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A point particle moving in a curved spacetime gives rise to fields that in turn affect its motion. One conveniently thinks of this interplay as the response of the particle to its {\sl self-force}. To date, models of point particle motion in the vicinity of black holes have ignored parts of this self-force because it is such a challenge to calculate. This work is part of a larger effort to develop systematic tools for the efficient calculation of such self-forces. This development is made with the aim of accurately simulating the inspiraling motion of compact objects onto supermassive black holes (also known as extreme-mass-ratio binary inspirals, or EMRIs), and of obtaining good predictions of the gravitational waves they emit. EMRIs are the main targets for the proposed space-based gravitational wave detector, the Laser Interferometer Space Antenna (LISA). For the mission to succeed, accurate templates of the gravitational waves it will pick up are necessary. This work is an attempt to address this need. The main contribution of this dissertation is the design and testing of a novel method for simultaneously calculating self-forces and radiation fluxes due point particle sources using (3+1) codes. Concrete calculations of self-forces for particles in strong-field gravity have only previously been done through mode sum approaches, which, while having been critical to the development of the subject, appears inconvenient for the eventual goal of using a calculated self-force to update particle trajectories. The new method avoids a mode decomposition entirely, and instead properly replaces the distributional source of the curved spacetime wave equation by an effective regular source. The resulting regular solution of the wave equation, under appropriate boundary conditions, results in the physical retarded field when evaluated in the wavezone, while its gradient at the location of the particle gives the full self-force. This prescription is founded on the possibility of properly smearing out or regularizing delta function sources using an elegant decomposition of point source retarded fields, introduced by Detweiler and Whiting. Concrete implementations of the method are presented here, focusing exclusively on the ideal test case of a scalar point charge in a circular orbit around a Schwarzschild black hole. For a quick proof-of-principle, the method is first implemented in time-domain, using a 4th-order (1+1) algorithm for evolving the wave equation. This was used to calculate the self-force and the retarded field in the wave zone. To assess the quality of the numerical results, they were compared with the results of highly accurate frequency-domain calculations found in the literature. Encouraging agreement to within $\lesssim 1\%$ is achieved. This work also presents the first successful self-force calculations performed with (3+1) codes. For this task, two independent (3+1) codes (finite difference and pseudospectral) developed originally for full-fledged numerical relativity applications were adapted to implement our new technique. Again, good agreement of $\lesssim 1\%$ error in the self-force and fluxes is achieved. These results open the door towards employing the well-developed machinery of numerical relativity in tackling the extreme-mass-ratio regime of black hole binaries, and consequently pave the way towards long sought self-force waveforms for EMRIs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Michael Vega.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Detweiler, Steven L.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024687:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024687/00001

Material Information

Title: The Dynamics of Point Particles around Black Holes
Physical Description: 1 online resource (177 p.)
Language: english
Creator: Vega, Michael
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: force, general, gravitational, radiation, reaction, relativity, self, waves
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A point particle moving in a curved spacetime gives rise to fields that in turn affect its motion. One conveniently thinks of this interplay as the response of the particle to its {\sl self-force}. To date, models of point particle motion in the vicinity of black holes have ignored parts of this self-force because it is such a challenge to calculate. This work is part of a larger effort to develop systematic tools for the efficient calculation of such self-forces. This development is made with the aim of accurately simulating the inspiraling motion of compact objects onto supermassive black holes (also known as extreme-mass-ratio binary inspirals, or EMRIs), and of obtaining good predictions of the gravitational waves they emit. EMRIs are the main targets for the proposed space-based gravitational wave detector, the Laser Interferometer Space Antenna (LISA). For the mission to succeed, accurate templates of the gravitational waves it will pick up are necessary. This work is an attempt to address this need. The main contribution of this dissertation is the design and testing of a novel method for simultaneously calculating self-forces and radiation fluxes due point particle sources using (3+1) codes. Concrete calculations of self-forces for particles in strong-field gravity have only previously been done through mode sum approaches, which, while having been critical to the development of the subject, appears inconvenient for the eventual goal of using a calculated self-force to update particle trajectories. The new method avoids a mode decomposition entirely, and instead properly replaces the distributional source of the curved spacetime wave equation by an effective regular source. The resulting regular solution of the wave equation, under appropriate boundary conditions, results in the physical retarded field when evaluated in the wavezone, while its gradient at the location of the particle gives the full self-force. This prescription is founded on the possibility of properly smearing out or regularizing delta function sources using an elegant decomposition of point source retarded fields, introduced by Detweiler and Whiting. Concrete implementations of the method are presented here, focusing exclusively on the ideal test case of a scalar point charge in a circular orbit around a Schwarzschild black hole. For a quick proof-of-principle, the method is first implemented in time-domain, using a 4th-order (1+1) algorithm for evolving the wave equation. This was used to calculate the self-force and the retarded field in the wave zone. To assess the quality of the numerical results, they were compared with the results of highly accurate frequency-domain calculations found in the literature. Encouraging agreement to within $\lesssim 1\%$ is achieved. This work also presents the first successful self-force calculations performed with (3+1) codes. For this task, two independent (3+1) codes (finite difference and pseudospectral) developed originally for full-fledged numerical relativity applications were adapted to implement our new technique. Again, good agreement of $\lesssim 1\%$ error in the self-force and fluxes is achieved. These results open the door towards employing the well-developed machinery of numerical relativity in tackling the extreme-mass-ratio regime of black hole binaries, and consequently pave the way towards long sought self-force waveforms for EMRIs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Michael Vega.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Detweiler, Steven L.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024687:00001


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THEDYNAMICSOFPOINTPARTICLESAROUNDBLACKHOLES By MICHAELFRANCISIANG.VEGAII ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c r 2009MichaelFrancisIanG.VegaII 2

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ToMamaandPapa,forgettingmestarted AndtoMonetteandbabyJanna,forgivingmethestrengthton ish 3

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ACKNOWLEDGMENTS Myrstthanksgoestomyparents,MichaelandMa.CeciliaVeg a,whohavealways encouragedmetopursuemyinterests.Theydeservemorethan ksthanIcanpossibly give.Thanksalsogoestomyyoungersiblings,AnnaKathrina ,JohnThomas,andAngela Paula,whohaveallgrownuptobecomeresponsibleandintell igentadults,andhavemade meaveryproud kuya ,inspiteofmydecade-longabsencefromhome. IamimmeasurablygratefultoSteveDetweiler,whohasdones omuchmorethanact asmyPhDsupervisor.Hisboundlessoptimism,unshakablewo rkethic,refreshingcandor, sinceregenerosity,anduniqueinsightonphysicsandlifei ngeneral,willforeverremain idealstowhichIshallaspire.IalsowishtothankSandyFish er,Steve'swife,whowas abigsourceofencouragementthroughmyyearsinGainesvill e.Myfondestmemoriesof Gainesvillewillalwaysincludethem. TotherestofmyesteemedPhDcommittee:BernardWhiting,Ji mFry,David Tanner,andJonathanTan,Iwishtoexpressmygratitudefort heirpatienceandgenerous commitmenttomakemeabetterscientist.Specialthanksgoe stoJimFry,whosebrilliant teachingandinfectiousexcitementforallthingsphysicsh adearlyonconvincedmethatI wasintherightplaceforgradschool,andtoBernardWhiting ,whoserigorousincisiveness andunyieldingquestforclarityaresightstobehold. IoweagreatdealtoJerroldGarcia,mybestprofessorattheA teneodeManila University,whoseexampleandcondenceinmeprovidedtheo riginalinspirationbehind mypursuitofadoctoratedegree,andwhocontinuestobeoneo fmysurestguidesinlife. TofellowpeonsoftheUFphysicsgraduateprogramwhosefrie ndshipshavebecome mostvaluabletome:DeepakKar,LarryPrice,KarthikShanka r,RanjaniNarayanan, ScottLittle,EmreKahya,Dong-HoonKim,WayneBomstad,Jin myungChoi,and Byoung-HeeMoon,IwishtogivemythanksformakingGainesvi llealotmoreinteresting, intellectuallyorotherwise. 4

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Gainesvillequicklybecameasecondhometomebecauseofthe extraordinarily vibrantcastofFilipinograduatestudentswhostudiedatUF .Iwishtoespeciallythank SuzetteandEdPabit,MelvinMeana,JongMalabed,MikeVelor o,JohnYap,Joyand EmilCagmat(+Koji),JhoannaandDodgeBaluya,JemyandFair Amos,theGatafamily, theJavelosafamily,theBulanonfamily,MachelMalay,Jani ceYoung-Silverman,Cris Dancel,JoeyOrajay,andtheDirainfamily. Tofriendswhosharedinmy\physicsexile"andkeptthemselv esatphonecallreach: LitodelaRamaatUrbana-Champaign,AdrianSerohijosatCha pelHill,andReinabelle ReyesatPrinceton,Iwishtoexpressmygratitudefortheirc onstantwillingnesstoshare andlisten. DarleneLatimer,YvonneDixon,ChrisScanlon,NateWilliam s,DavidHansen, BrentNelson,andBillieHermasendeservecountlessthanks forhelpingmewithmanyof theessentialworkadayaspectsoflifeasagraduatestudent .Theymademystayatthe physicsbuildingunbelievablycomfortable. IalsowishtoacknowledgetheNutterFellowshipoftheColle geofLiberalArtsand Sciences,andtheHarrisFellowshipoftheInstituteforFun damentalTheory,forproviding thenecessarynancialsupportthroughoutthewritingofth isdissertation. Finally,Ithankthelovesofmylife,MarienetteMorales-Ve gaandMalinJannaVega, whoarethefoundationandstrengthbehindallthatIdo.Ihop etheyknowhowallthis wouldbemeaninglesswithoutthem. Invariably,theactofwritingacknowledgementscomeswith afailuretoacknowledge everyonewhodeservesit.Mysincerestapoogiesgotoanyone Imayhaveforgotten. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1Self-ForceandGravitationalWaveAstronomy ................ 13 1.1.1RadiationReactioninElectromagnetism ............... 15 1.1.2TheExtreme-Mass-RatioTwo-BodyProbleminGeneralR elativity 16 1.2AstrophysicalContext:LISAandGWsfromEMRIs ............ 19 1.3EMRIModelingandWaveforms ........................ 22 1.3.1PerturbationTheoryinGeneralRelativity .............. 23 1.3.2First-OrderPerturbationTheoryonaSchwarzschildB ackground .. 24 1.4AchievementsofthisWork ........................... 30 1.5OutlineofthisManuscript ........................... 31 2IDEASINRADIATIONREACTIONANDSELF-FORCE ............ 34 2.1MotionofaChargedParticleinFlatSpacetime ............... 34 2.2MotionofaChargedParticleinCurvedSpacetime ............. 44 2.3MiSaTaQuWaEquation ............................ 53 2.3.1AxiomaticApproach .......................... 54 2.3.2ConservationofEnergy-Momentum .................. 56 2.3.3MatchedAsymptoticExpansions .................... 57 2.4Green'sFunctionsfortheWaveEquationinCurvedSpacet ime ....... 58 2.5Self-ForcefromaGreen'sFunctionDecomposition .............. 63 3REGULARIZEDPOINTSOURCESFORSELF-FORCECALCULATIONS .. 75 3.1GeneralPrescription .............................. 75 3.2ApproximatingtheLocalSingularField ................... 80 4NUMERICALEXPERIMENTINTIME-DOMAIN ................ 88 4.1ScalarFieldsinaSchwarzschildGeometry .................. 89 4.2EvolutionAlgorithm .............................. 91 4.3InitialDataandBoundaryConditions .................... 99 4.4Self-ForceCalculation ............................. 99 4.5CodeDiagnostics ................................ 100 4.5.1Convergence ............................... 100 6

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4.5.2Highl Fall-O .............................. 101 4.5.3DependenceontheWindowFunction ................. 103 4.6Results ...................................... 105 4.6.1RecoveringtheRetardedField ..................... 105 4.6.2ComputedSelf-Force .......................... 106 4.7Discussion .................................... 107 5SELFFORCEWITH(3+1)EVOLUTIONCODES ................ 113 5.1Preliminaries .................................. 114 5.1.1Kerr-SchildCoordinates ........................ 115 5.1.2Self-ForceforPerpetualCircularOrbitsofSchwarzs child ...... 117 5.1.3How F t andtheEnergyFluxareRelated ............... 118 5.1.4ScalarEnergyFluxinKerr-SchildCoordinates ............ 122 5.1.5EectiveSourceandtheNewWindowFunction ........... 124 5.2Descriptionofthe(3+1)Codes ........................ 127 5.2.1Multi-BlockCode ............................ 127 5.2.2PseudospectralCode .......................... 130 5.3Results ...................................... 133 5.3.1 F t andtheEnergyFlux ......................... 133 5.3.2TheConservativePiece, F r ....................... 139 5.3.3Discussion ................................ 141 6SUMMARYANDOUTLOOK ............................ 147 6.1ReviewofMainResultsandFutureWork .................. 147 APPENDIX ATHORNE-HARTLE-ZHANGCOORDINATES .................. 150 A.1ConstructionoftheTHZCoordinates ..................... 150 A.1.1SpatialComponentsoftheGaugeVector ............... 151 A.1.2 t -ComponentoftheGaugeVector ................... 155 A.2THZCoordinatesforaCircularGeodesicaroundSchwarzs child ...... 156 BSPHERICALHARMONICSANDTHEREGGE-WHEELERGAUGE ..... 159 CPERTURBATIONTHEORYINGENERALRELATIVITY ........... 165 C.1Twometricsonamanifold ........................... 165 C.1.1 R abc d and R ab .............................. 167 C.1.2 G ab R ab 1 2 R ............................. 169 REFERENCES ....................................... 171 BIOGRAPHICALSKETCH ................................ 175 7

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LISTOFTABLES Table page 4-1Summaryofself-forceresultsfor R =10 M and R =12 M ............ 107 5-1Timelags. ....................................... 139 5-2Summaryof(3+1)self-forceresults. ........................ 144 8

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LISTOFFIGURES Figure page 1-1PlannedorbitforLISA. ............................... 21 2-1SupportofvariousGreen'sfunctionsforthewaveequati onincurvedspacetime. 64 3-1Equatorialproleofthetime-domaineectivesource, S e ............. 80 3-2Zoomed-inequatorialproleofthetime-domaineectiv esource ......... 81 4-1Staggered(characteristic)gridwithunitcellfor(1+1 )evolutionintime-domain 92 4-2Unitcellofthe(1+1)characteristicgrid ...................... 95 4-3Convergenceattheparticlelocationandinthewavezone ............. 101 4-4( r r R ) l versus l ................................... 103 4-5( r t R ) l versus l ................................... 104 4-6Fractionalchangesin R 22 asaresultofusingdierentwindowfunctions. .... 105 4-7Comparisonoftime-domainandfrequency-domainresult sfor f 22 ( r ). ...... 106 4-8Relativeerrorbetweentime-domainandfrequency-doma inresultsfor f 22 ( r ). .. 107 4-9Equilibriumtime. ................................... 108 4-10( r r R ) m versus m .................................. 111 5-1Equatorialproleofthe(3+1)eectivesource ................... 127 5-2 S e zoomedinatthelocationofthecharge ..................... 128 5-3Contourplotof(3+1) S e atthelocationofthecharge .............. 128 5-4 F t computedatdierentangularresolutionsofthemulti-bloc kcode ....... 135 5-5 F t computedatdierentangularresolutionsoftheSGRIDcode ......... 135 5-6Comparing F t resultsfromthemulti-blockandSGRIDcodes ........... 136 5-7Energyruxcomputedwiththemulti-blockcode .................. 136 5-8EnergyruxcomputedwiththeSGRIDcode .................... 137 5-9Dependenceofthe(equilibrium)energyruxaccuracyone xtractionradius ... 138 5-10Evolutionof E usingvariousextractionradii .................... 138 5-11Energyruxextrapolationtoinniteextractionradius ............... 140 9

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5-12 F r computedatdierentangularresolutionsonthemulti-bloc kcode ...... 141 5-13 F r computedatdierentangularresolutionsontheSGRIDcode ......... 142 5-14 F r computedat2dierentradialresolutionsonthemulti-bloc kcode ...... 142 5-15 F r computedat2dierentradialresolutionsontheSGRIDcode ......... 143 5-16Comparing F r resultsfromthemulti-blockandSGRIDcode ........... 143 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THEDYNAMICSOFPOINTPARTICLESAROUNDBLACKHOLES By MichaelFrancisIanG.VegaII August2009 Chair:StevenDetweilerMajor:Physics Apointparticlemovinginacurvedspacetimegivesrisetoe ldsthatinturnaect itsmotion.Oneconvenientlythinksofthisinterplayasthe responseoftheparticleto its self-force .Todate,modelsofpointparticlemotioninthevicinityofb lackholeshave ignoredpartsofthisself-forcebecauseitissuchachallen getocalculate.Thisworkispart ofalargereorttodevelopsystematictoolsfortheecient calculationofsuchself-forces. Thisdevelopmentismadewiththeaimofaccuratelysimulati ngtheinspiralingmotion ofcompactobjectsontosupermassiveblackholes(alsoknow nasextreme-mass-ratio binaryinspirals,orEMRIs),andofobtaininggoodpredicti onsofthegravitationalwaves theyemit.EMRIsarethemaintargetsfortheproposedspacebasedgravitationalwave detector,theLaserInterferometerSpaceAntenna(LISA).F orthemissiontosucceed, accuratetemplatesofthegravitationalwavesitwillpicku parenecessary.Thisworkisan attempttoaddressthisneed. Themaincontributionofthisdissertationisthedesignand testingofanovelmethod forsimultaneouslycalculatingself-forcesandradiation ruxesduepointparticlesources using(3+1)codes.Concretecalculationsofself-forcesfo rparticlesinstrong-eldgravity haveonlypreviouslybeendonethroughmodesumapproaches, which,whilehavingbeen criticaltothedevelopmentofthesubject,appearsinconve nientfortheeventualgoalof usingacalculatedself-forcetoupdateparticletrajector ies.Thenewmethodavoidsa modedecompositionentirely,andinsteadproperlyreplace sthedistributionalsourceof 11

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thecurvedspacetimewaveequationbyaneectiveregularso urce.Theresultingregular solutionofthewaveequation,underappropriateboundaryc onditions,resultsinthe physicalretardedeldwhenevaluatedinthewavezone,whil eitsgradientatthelocation oftheparticlegivesthefullself-force. Thisprescriptionisfoundedonthepossibilityofproperly smearingoutorregularizing deltafunctionsourcesusinganelegantdecompositionofpo intsourceretardedelds, introducedbyDetweilerandWhiting.Concreteimplementat ionsofthemethodare presentedhere,focusingexclusivelyontheidealtestcase ofascalarpointchargein acircularorbitaroundaSchwarzschildblackhole.Foraqui ckproof-of-principle,the methodisrstimplementedintime-domain,usinga4th-orde r(1+1)algorithmfor evolvingthewaveequation.Thiswasusedtocalculatethese lf-forceandtheretarded eldinthewavezone.Toassessthequalityofthenumericalr esults,theywerecompared withtheresultsofhighlyaccuratefrequency-domaincalcu lationsfoundintheliterature. Encouragingagreementtowithin 1%isachieved. Thisworkalsopresentstherstsuccessfulself-forcecalc ulationsperformed with(3+1)codes.Forthistask,twoindependent(3+1)codes (nitedierenceand pseudospectral)developedoriginallyforfull-redgednum ericalrelativityapplicationswere adaptedtoimplementournewtechnique.Again,goodagreeme ntof 1%errorinthe self-forceandruxesisachieved. Theseresultsopenthedoortowardsemployingthewell-deve lopedmachineryof numericalrelativityintacklingtheextreme-mass-ratior egimeofblackholebinaries,and consequentlypavethewaytowardslongsoughtself-forcewa veformsforEMRIs. 12

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CHAPTER1 INTRODUCTION 1.1Self-ForceandGravitationalWaveAstronomy Thesubjectofself-forceandradiationreactioninclassic aleldtheoryhasbeen aroundforalongtime.Theproblemisoftenrstencountered inthecontextof electromagnetism,whereonesaskswhathappenstoanaccele ratingchargeasit radiates.Howisthedynamicsofthepointchargeaectedbyt heemittedradiation? Andsubsequently,howdoestheradiationchangeasaresulto fthemodiedparticle dynamics?Thegoalofaconsistentframeworkinwhichtotrea tbothparticleandeld dynamicshasbeenalong-soughtgoalinclassicaleldtheor y[ 1 ]. Whileformostapplications,theeectsofself-forceandra diationreactionarelargely irrelevant,therehasbeenaresurgenceofinterestinthepr oblembroughtaboutbythe prospectofdetectinglow-frequencygravitationalwavesw ithspace-basedinterferometers [ 2 ].Oneofthemostpromisingsourcesoflow-frequencyGWsist heinspiralofacompact object(likeastellar-massblackhole,neutronstar,orwhi tedwarf)ontoamuchheavier supermassiveblackhole.Theseeventsarereferredtoasext reme-mass-ratioinspiralsor EMRIs,andarebelievedtobequiteubiquitousintheUnivers e,giventheconsensusthat mostnormalgalaxiesharborsupermassiveblackholesinthe ircentralcores.Adetection andsubsequentanalysisofGWsignalsfromEMRIsholdstreme ndousscienticpotential {fromanimprovedcensusofBHmassesingalaxiestotheconr mationoftheblack holeno-hairtheoreminclassicalgeneralrelativity[ 3 ].Forthistobecomeareality,not onlywillGWsneedtobeunambiguouslydetected,physicalpa rametersforthesources thatemittedtheGWswillneedtobeestimatedwithsomeaccur acy.Theextractionof astrophysicalinformationfromanEMRIsignalwillrequire precisetheoreticalmodelsof EMRIsandtheGWstheyemit.Thisiswhatunderliesmuchofthe theory/mo:qdeling componentofgravitationalwavescience{thetheoreticala nalysisoflikelyGWsources 13

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thatwillbeofassistancetothegravitationalwaveastrono merinhersearchforGWsand hersubsequenttaskofunderstandingwhatitisthatshehasf ound. Fortheorists,EMRIsareexquisitebecauseoftheirsimplic ity.Whilethemodeling ofmostobjectsintheUniverserequiresaretremendousamou ntofphysics,guess-work, andne-tuning,EMRIsarecomparativelyclean.Itisbeliev edthatbecauseofthe extremediscrepancyinmassscales,itisreasonabletoigno rethesmallcompactobject's internaldegrees-of-freedom,andthereforetreatitaspoi ntmassmovinginthe(perturbed) spacetimeofthesupermassiveblackhole.Atzeroth-order, theresultingmotionisjust geodesicmotion,butwiththeaccuracyrequirementsofGWda taanalysis,specically withrespecttoparameterestimation,correctionstogeode sicmotionturnoutto beimportant.Thesecorrectionsarebroughtaboutprecisel ybytheself-force.The ultimatemotivationbehindrevisitingthephenomenonofse lf-forceisthereforetobuild accuratemodelsofthemotionofcompactobjectsinastrophy sically-interestingblackhole spacetimes.Thesearetoserveasthebasisforthemostaccur atetheoreticalwaveforms duetoEMRIsources,theavailabilityofwhichwillsurelyex tendtheutilityofplanned space-bournegravitationalwavedetectors. Thisintroductorychapterprovidesacursoryoverviewofth eself-forceproblem anditsrelevancetothescienceofgravitationalwaves.Ito penswithareviewof whatisunderstoodfromelectromagnetism,demonstratingp articularlywhyinmost practicalcircumstances,self-forceeectsarenegligibl e.Itisalsoshownwhyitbecomes importantforextrememassratioinspiralingblackholes.T hen,itproceedstosettingthe astrophysicalcontextoftheproblem,describingEMRIsour cesfortheplannedspace-based LISAmission,inwhichitisarguedthatself-forcebecomesn ecessaryifonehopestodo preciseastronomywithlow-frequencyGWs.Aftersuchisaro ughsummaryofprevious EMRImodelsandEMRIGWcalculationsthathavebeendoneviat heformalismofblack holeperturbationtheory.Finally,thisisfollowedbyadis cussionofexistingwaveforms 14

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currentlybeingusedbytheLISAdataanalysiscommunity,an dwhywestilleagerlyawait waveformsthatconsistentlyincorporatetheeectsofthes elf-force. Thisintroductionsetsthecontextforwhatwasachievedint hiswork.Enumerated aretheconsequencesoftheseachievements.Finally,asumm aryisprovidedfortherestof thisdissertation.1.1.1RadiationReactioninElectromagnetism Twotraditionalproblemsinelectromagnetismareasfollow s:(1)Givensome prescribedchargesandcurrents,( ; ~ j ),calculatetheelectromagneticeld( F )they produce,and(2)Givenapreassignedelectromagneticeld( F ),determinethemotionof chargesandcurrents,( ; ~ j ),interactingwithit. Comingupwithasimultaneous(consistent)solutiontoboth problemsforeventhe simplestcaseofastructurelesselectronhasbeenthefocus ofcountlessinvestigations inthepast.(SeethebookbyRohrlich,[ 1 ]).Butfortunately,thelackofafullsolution israrelyanimpedimenttoonewhoseinterestinelectromagn etictheoryismotivated byapplications.Inmanyphysicalsituations,whenonekeep stheproblemsseparate, whatisachievedistypicallyahighlyaccurateapproximati on.Asanexample[ 4 ],wecan investigatetheimportanceofradiationreactionforachar gemovingquasi-periodically withatypicalamplitude d ,andacharacteristicfrequency o .Radiationreactionbecomes importantwhentheradiatedenergy(bytheLarmorformula), E rad ,becomescomparable tothecharacteristicenergyofthesystem, E o .Inthisexample, E o m! 2 o d 2 ,whilethe characteristicaccelerationis a 2 o d ,andthecharacteristictimescaleis T 1 =! o .For radiationreactiontobeunimportant,wethenhavethecrite rionthat E rad 2 e 2 a 2 T 3 c 3 bemuchsmallerthan E o m! 2 o d 2 ; 15

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where e and m isthechargeandmassoftheparticle,respectively.Thisim pliesthenthat thesystemonlyneedstosatisfy o 1 ; where 2 e 2 = 3 mc 3 isthecharacteristictimeforradiationreaction. turnsouttobe oftheorder 10 24 s,whichisgenerallymuchsmallerthanmostapplicationsof interest. Thisindicateswhywecansafelyignoreradiationreactioni ncommonpractice. 1.1.2TheExtreme-Mass-RatioTwo-BodyProbleminGeneralR elativity Ingeneralrelativity,thespacetimemetric g ab becomesthemainplayer,andthe dynamicalequationbecomesthemuchmorecomplicatedEinst einequation: R ab 1 2 g ab R =8 T ab ; (1{1) acoupledsystemofsecond-order,nonlinear,hyperboliceq uationsforthecomponents of g ab ,whichundoubtedlymakesthegamemuchmorecomplicatedtha nMaxwell's. Nonetheless,dynamicsinGRturnsouttobeconceptuallyver ysimilartothatof electrodynamics. Thegeneraltwo-bodyproblemingeneralrelativityisacomp licatedsubjectdeserving reviewsofitsown,soweshallrefrainfromdiscussingitinf ullgeneralityhere.Werestrict ourselvesrsttovacuum, T ab =0,freeingourselvesfromthecomplicationofhaving matterintheproblem;andwefurtherfocusonaspecialcaseo fthetwo-bodyproblem whereinwehavethe\masses"ofthesegeometricobjectsdie ringbyordersofmagnitude, =M 1.Forsuchacasewesaythatwehavean extrememassratio ,andfromnowon weshallcalltheseobjectsblackholes(BH). Asinelectrodynamics,anybeginningstudentofGRrstlear nsofthedynamics ofpointmassesinitssimplestform:pointmassesrespondto anexistingeldthat doesnotitselfbudgebecauseoftheirpresence.Inthiscont ext,onespeaksofthe reasonablectionofapointmassmovinginageodesicofabac kgroundspacetime. Fortheextreme-mass-ratiotwo-bodycase,thegeodesiciso fthespacetimeofthebigger 16

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BH.ThegravitationaleldofthesmallBHitselfisignored, equivalenttoassumingthata chargeresponds(accordingtotheLorentzlaw)tosomeprede terminedexternaleld,but isotherwisecompletelyunaectedbyitsownelectromagnet iceld.Hence,ourdilemmain Maxwell'stheorypersistsinEinstein'sgravitationalphy sics. Likeinelectromagnetismalso,gravitationalself-forcee ectsareirrelevantinmany importantastrophysicalcircumstances.Infact,inthers tcalculationsoftheemissionof gravitationalradiationbypointmasses[ 5 { 8 ]assumedgeodesicmotionfortheparticle's dynamics,inexactlythesamewaythatoneprescribesthetra jectoriesofchargesinEM tocalculateemittedradiation.Sometimesthough,adomain isencounteredforwhichthis approximationcannotbeexpectedtobevalid.Thefollowing exampleexploresjustwhat thisdomainis. Consideramass inaquasicircularorbitoffrequency o anddistance d around anothermass M ,suchthat M> .JustasinMaxwell'stheory,weshallconsider self-forceeectstobeimportantwhentheresultingenergy lossinasystemover n characteristictimeperiodsiscomparabletoitscharacter isticenergy.Atlowestorder,GW luminosityisapproximatedtobe L ( ... Q ) 2 ,where Q isthesystem'squadrupolemoment. Theradiatedenergythroughaperiod2 1 o n isthenroughly: E rad n ( ... Q ) 2 1 o n 2 d 4 5 o : Throughthevirialtheorem,thekineticenergyofthemass willberoughlyequaltothe gravitationalbindingenergy.Thecharacteristicenergy, E o ,ofthissystemis E o 2 o d 2 Self-forceeectswill not beimportantif E rad E o .With 2 o = M=d 3 ,thisisequivalently n M M d 5 = 2 = n M M 5 = 2 1 : 48km d 5 = 2 1 ; (1{2) 17

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where isthemassratio.FortheSun-Mercurysystem, 2 : 7 10 7 ,andthedistance ofseparation d isroughly5 : 8 10 7 km.Thesenumbersimplythatforself-forceeectsto beappreciable,Mercuryhastoundergoroughly 10 24 orbits(i.e. 10 23 years),much longerthantheexpectedlifetimeoftheSun( 10billionyears). Fora10 6 M 10 M BHbinarysystemthough,( M=d )cangettoaslargeas1/6 (foranon-spinningcentralBH),sothatwithabout10 3 10 5 orbitsself-forceeectscan becomequitesevere.Itisforasystemlikethisthatgravita tionalself-interactioncannot beignored. Goingbeyondthetestmassapproximationbyincludingthee ectsofthesmall BH'sinteractionwithitsowngravitationaleldisthegoal ofmodern self-forceanalyses Suchstudiesseektoextendourunderstandingofthedynamic softheextreme-mass-ratio 2-bodysystems,andareultimatelyexpectedtoresultinmor eaccuratetheoreticalmodels. Tosome,thetheoreticalsignicanceofself-interactioni nextreme-mass-ratio two-bodydynamicswarrantsperhapsnofurtherjusticatio n.However,therealimpetus behindrecenteortsonthisproblemgoesbeyondapurist'sm otivations,andthishas beentheprospectofusheringanew,fundamentallydierent approachtoastronomy.To date,astronomershavedependeduponinformationaboutthe cosmosthatisbroughtto themviaelectromagneticwavesfromavarietyoffrequencyr egimes.Generalrelativity predictsthatlargeinteractingmassesintheuniverseemit gravitationalwaves;andthese GWsencodeinformationaboutthecosmos,someofwhichcanno tbebroughttousby electromagneticwaves[ 9 10 ].ThesuccessfuldetectionanduseofGWspromisestoreveal manyaspectsoftheuniversetowhichwearecurrentlyblind. Thispromisehastransformedgravitationalphysicsintoad ynamiceldhalf-owned byexperimentalists.Ground-basedinterferometerssucha sLIGO,VIRGO,andGEO arenowonline,whileaspace-basedLISAispoisedtobelaunc hedwithinthenext decade.Impressivestridesarebeingmadeinloweringtheno iseroor(andthusimproving thesensitivity)oftheseGWdetectors.However,despiteth esefantasticadvancesin 18

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technology,thesheerweaknessofexpectedGWs(oftheorder 10 21 inrelativestrain, L=L )currentlyconstrainsdetectionand,morecrucially,para meterestimation(bywhich weextractusefulinformationfromthem)tobesignicantly reliantonaparadigmatic dataanalysistechniquecalled matchedltering .(GWburstsandstochasticGWsources, however,falloutsidethisparadigm.) Inthistechnique,atheoretically-derivedtemplatewavef orm,isintegratedagainstreal datafromtheinterferometersinordertondaGWsignalthat wouldotherwisebeburied innoise.Thepresenceofanactualsignalinthedatamatchin gaparticulartemplate closelyenough(andlongenough)wouldberevealedbyahighr esultingvalueforthe integral,relativetosomecarefullypredeterminedthresh old;anditwouldsubsequentlybe interpretedasadetection.Thisintegrationprocedureisd oneoverallavailabletemplates insometemplatebank,untilthedataiseitherconrmedtoma tchoneormoreofthe templatesorjustdeterminedtobepurelynoise.Ifadetecti onismade,thenthenext stepistoextracttheencodedinformationoutofit.Thisisd onebycarefullyidentifying whichofthetemplatesmostcloselymatchesit,sinceeachte mplateactuallycorresponds toatheoreticalmodelofanastrophysicalsourcewithauniq uesetofparameters.Alikely scenario,however,isthatmorethanonetemplatematchesit ,sofurtheranalysismust donetoreallyascertainwhattherealparameterscorrespon dingtothedetectedsignalare. Inaccuratetemplates,however,willseverelyhinderthisp artoftheprocess,andultimately limittheutilityofanydetection[ 11 ].Thepushformoreaccuratemodelsofgravitational wavesourcesisthusintimatelytiedtoparameterestimatio n. 1.2AstrophysicalContext:LISAandGWsfromEMRIs Anextreme-mass-ratioinspiralwouldbecomprisedofacomp actobject(white dwarf,neutronstar,orblackhole)ofmass & 1 M andasupermassiveBHofmass 10 4 M M 10 7 M .Theinspiralingbodyhastobeacompactobjectbecauseits internalgravityhastobestrongenoughtowithstandtidald isruptionbythesupermassive blackhole,whilealsoproducingGWsofsucientamplitudef orittobedetectable. 19

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Theothermemberofthebinaryalsohastobeasupermassivebl ackholebecausethe characteristicfrequencyofanemittedGWscalesas M 1 ,whileplannedspace-borneGW detectorswillbemostsensitiveinthelow-frequencyregim e,i.e.0.1mHz-100mHz[ 12 ]. Thereexistvariousscenarioswhichleadtotheformationof ERMIs.[Acomprehensive reviewofEMRIastrophysicscanbefoundin[ 13 ].]However,thestandardpicture involvestwo-bodyrelaxationbringingacompactobjectont oanorbitwithasmall enoughpericenterdistancethatdissipationbytheemissio nofGWsbecomesdominant. Otherwise,compactobjectssimplyorbitthesupermassiveb lackholeswiththeirorbital parameterschangingslowlyduetwo-bodyrelaxation. TheastrophysicsofEMRIsiscurrentlyarichsubjectofinte nseactivity,mostof whichthoughisbeyondthescopeofthiswork.Oneaspectwort hpointingout,however, isthatestimatesofthedistributionofeccentricitiesfor one-bodyinspiralsshowthatit isskewedtowardshighe values,withapeakofthedistributionaround e 0 : 7[ 14 ]. StrategiesaimedatcalculatingEMRIwaveformsmusttheref orebegearedtowardsthe high-eccentricityregimeofparameterspace. TheplannedLISAmissionwillbeaconstellationofthreesat ellitesmovingaround theSun,formingtheverticesofanequilateraltriangleofa rm-lengthroughly5 10 6 kilometersfromeachother,whosenormalisinclined60 o withrespecttotheecliptic.The verylongarmlengthspracticallyprohibitclassicalinter ferometry,whereinlaserlightis bouncedofmirrors.Aftertraveling5millionkilometers,a tiny10 6 -radbeamdivergence becomesahugespotroughly20kmindiameter[ 13 ],resultinginsignicantlossofpower ifthecapturedlightismerelyrerectedback.Tocircumvent thisdiculty,LISAwill employactivemirrorswhosejobitistocapturephaseinform ationfromanincomingbeam andthenorderthere-emissionofanampliedbeamwiththesa mephaseinformationthat wasjustcollected. Todetectdistancevariationsofpico-meterprecisionbetw eenanytwospacecraft(in themHzband),LISAwillhavetofollowanoptimallystableor bitinwhichthethree 20

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spacecraftgoarounditsbarycenterwithaperiodofoneyear ,whileitsbarycenterlags theEarthby20 o asitgoesaroundtheSun.Andwhilethearm-lengthswillcert ainlyvary (byasmuchas 120 ; 000km),thesevariationsoccurontimescalesofmonths,whe reas GW-inducedvariationswillhappenontimescalesofhours.T hisiswhatpermitsthe possibilityofpico-meterinterferometryinspace.(Thein terestedreaderisinvitedtolook up[ 13 ]formoredetails.) Sun 60 o Figure1-1.PlannedorbitforLISA. EstimatesforEMRIdetectionsbyLISAlieintherangeofafew toafewthousand [ 3 ].Althougharrivingattheseestimatesisquiteaninvolved process(andonenotwithout muchuncertainty),often,aback-of-the-envelopecalcula tionwillsuceinassessingthe relevanceofaparticularGWsourcetoeitherLIGOorLISA.Tw ooftheseusefulnumbers arethestrengthoftheGWsignal h andthecharacteristicfrequency. Thestrengthofagravitationalwavesignalcomingfromanis olatedsourcefarawayis roughlyestimatedwiththeso-calledquadrupoleformula[ 15 ]: h ( TT ) ij ( t; ~ x )= 2 r G c 4 I ( TT ) ij ( t r c ) ; (1{3) 21

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where h ( TT ) ij isthegravitationalwaveintransverse-tracelessgauge,( t; ~ x )arethe coordinatesoftheobserverwiththesourceatthespatialor igin, r = j ~ x j ,and I ( TT ) ij ( t r c )is justthetransverse-tracelessquadrupolemomentofthesou rceevaluatedatretardedtime. G and c are,respectively,justNewton'sconstantandthespeedofl ightinvacuum. Inorderofmagnitude,thequadrupolemomentofasourcescal esas Mv 2 ,with M and v beingitsmassandcharacteristicspeed.Thequadrupolefor mulaisthenoften quotedas h r Schw r v 2 c 2 ; (1{4) where r Schw =2 GM=c 2 ( 3( M=M )km)isthecorrespondingSchwarzschildradiusofthe source.Fora10 6 M 10 M blackholebinary1Gpcaway,weexpect h 1 : 5 10 20 ThefrequencyoftheGWfromaparticularsourcewillberough lyrelatedtoits naturalfrequency[ 10 ]: f o = p = 4 ; (1{5) where isthemeandensityofthemass-energyofthesource.Forthes amebinaryabove, wecanthenexpectafrequencyofroughly 4mHz.ThesenumbersputtheGWfrom thissourcerightwithintheLISAband. 1.3EMRIModelingandWaveforms ThemodelingofEMRIscanbesaidtohavebeguninthe70switht hepioneering calculationsbyDaviset.al.,[ 5 6 ]andDetweiler[ 7 8 ]ofthegravitationalradiation emittedbysmallpointmassesorbitingblackholes.Thiswas doneforradialinfalland circularorbitsaroundbothSchwarzschild(non-rotating) andKerr(rotating)BHs. Acriticalsimplicationusedinallthesecalculationswas theassumptionofgeodesic trajectoriesfortheorbitingpointparticle.Hence,nothi ngabouttheeectsofthepoint's interactionwithitsowngravitationaleldcanbefoundint heircalculatedwaveforms. Whiletheeectofthisinteractionissmall,itaccumulates appreciablyoverthecourseof 10 4 10 5 orbits. 22

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Inthe90s,withtheapprovaloftheconstructionofLIGOandV IRGO,similar calculationsofgravitationalwaveemissionweredone[ 16 { 19 ]toaddress,amongother issues,theanticipatedwaveformaccuracyneedsforthesef acilities.Butagain,noself-force eectswereincorporatedinthesourcetrajectoriestheyus ed.Post-Newtoniancalculations wouldalsobecarriedoutthatincludesomeradiationreacti oneects;butbecausethis approximationassumesweakinterbodygravitationalelds andsloworbitalspeeds(e.g. [ 20 ]),itspredictionsholdonlyintheslowinspirallimit(i.e .,whentheblackholesareso farapart).Probingthestrong-eld,high-speedregimeofa nEMRIbinaryrequiresthe richformalismofblackholeperturbationtheory[ 21 ].Thenextfewsubsectionssketchhow thisisdoneforthecaseofaSchwarzschildblackhole.1.3.1PerturbationTheoryinGeneralRelativity PerturbationtheoryinGR[ 21 ]seeksapproximatesolutionstoEinstein'sequationby askinghowthemetricbehavesclosetosomeknownsolution.T histypicallyproceedsfrom adecompositionoftheunknownsolution{thefullmetric{in toaknownexact(usually vacuum)solution,whichweshallcallthe\background"metr ic,andaresidualperturbing metric, h ab ,whichweshalljustcallthe\perturbation". g ab = g ( B ) ab + h ab (1{6) Withthisdened,onecanthenexpandouttheEinsteintensor ( G ab R ab (1 = 2) g ab R ) inpowersof h ab ,andthenignorehigherpowersdependingonhowaccurateaso lution oneneeds.Thiscanbedonewiththeassumptionthatthepertu rbationisinsomesense \small".Theresultwillbeanequationmoretractablethan( 1{1 ). The\ N "in N -thorderperturbationtheorygenerallyreferstohowmanyp owers of h oneadmitsintheequationsonedecidestosolve.Presumably ,solvinghigherorder equationsgivesyoumoreaccurateapproximations;butthis isdiculttoshow,as convergencetothetruesolutionisoftenjustassumedinmos tcalculations.Ingeneral,the 23

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rst-orderperturbedEinsteinequationwouldbe G (1)ab = 1 2 [ r b r a h + r 2 h ab 2 r c r ( a h b ) c 2 R d ( a h b ) d + g ab ( r 2 h r c r d h cd )]=8 T ab ; (1{7) wherethecurvaturetermsaredenedonthebackground.Inth ecaseofavacuum backgroundspacetime,thissimpliesto G (1)ab = 1 2 [ r b r a h + r 2 h ab 2 r c r ( a h b ) c + g ab ( r 2 h r c r d h cd )]=8 T ab : (1{8) Second-orderperturbationtheorywouldentailworkingwit h: G (1)ab = G (2)ab +8 T ab (1{9) where G (2)ab isthepartoftheEinsteintensorquadraticintheperturbat ion.Thisequation isquitelong,andratherunenlightening,sotheyareshowni nsteadinAppendixC. Intheseequations, h = g ab ( B ) h ab ,\ r "isthecovariantderivativecompatiblewiththe background,andindicesareraisedandloweredwiththeback groundmetric. 1.3.2First-OrderPerturbationTheoryonaSchwarzschildB ackground PerturbationtheorywasrstappliedtoblackholesbyRegge andWheelerintheir pioneeringworkonthelinearstabilityoftheSchwarzschil dgeometry[ 22 ].(Thisstudy, however,wouldonlybecompletedbyZerilli[ 23 ]nearly15yearslater).Inthisanalysis, thebackgroundspacetimeisgivenby g ( B ) ab dx a dx b = 1 2 M r dt 2 + 1 2 M r 1 dr 2 + r 2 d 2 + r 2 sin 2 d 2 ; (1{10) whichispluggedintothelinearized,vacuumEinsteinequat ion: r 2 h ab + r b r a h 2 r c r ( a h b ) c + g ab ( r 2 h r c r d h cd )=0(1{11) 24

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togivetheequationfortherst-orderperturbation h ab .Notethatthisequationis invariantunderthetransformation: h ab h ab + L g ab = h ab + r a b + r b a ; (1{12) where a isanarbitraryvectorand L istheLie-derivative.Thisisknownasa gauge transformation ,and a ,a gaugevector BytakingadvantageofthesphericalsymmetryoftheSchwarz schildbackground,one canconvenientlydecomposetheperturbation h ab asasumover multipoles (seeAppendix B): h ab = X lm h lm; even ab + X lm h lm; odd ab ; (1{13) whereeach h lm; even ab and h lm; odd ab hasauniqueangulardependencespeciedbythe( l;m ) sphericalharmonics(scalar,vector,andtensor),andaref urtherdistinguishedfrom eachotheraccordingtoparity.Byxingtheangulardepende nceateachmultipole,the decompositionallowsonetoseehowthe( t;r )dependenceoftheperturbationcompletely decouplesfromitsangulardependenceateachmode.Eacheve nandoddmodewill dependonsevenandthreefunctionsof( t;r ),respectively,asfollows: h lm; even ab = 0BBBBBBBBBBB@ H 0 H 1 [ j 0 ] H 1 H 2 [ j 1 ] 266664 j 0 377775 266664 j 1 377775 266664 K;G 377775 1CCCCCCCCCCCA (1{14) 25

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h lm; odd ab = 0BBBBBBBBBBB@ 00 [ h 0 ] 00 [ h 1 ] 266664 h 0 377775 266664 h 1 377775 266664 h 2 377775 1CCCCCCCCCCCA : (1{15) Whatthisachievesthenisadescriptionoftheperturbation notintermsoften components f h tt ;h rr ;h ;h ;h tr ;h t ;h t ;h r ;h r ;h g thatarefunctionsof( t;r;; ),butinsteadintermsoftenfunctionsof( t;r )only: odd-paritysector: f h 0 ;h 1 ;h 2 g ; even-paritysector: f H 0 ;H 1 ;H 2 ;j 0 ;j 1 ;K;G g per( l;m )mode. Furthermore,thesethreeodd-parityfunctionscanbecombi nedtoformagauge-invariant, odd-parity, masterfunction Q odd ,whichfromthenon-trivialpartsof( 1{11 )canbeshown tosatisfyagauge-invariantwaveequation.Apossiblechoi cefor Q odd wouldbe: Q odd = r ( l 1)( l +2) @ t h 1 r 2 @ r h 0 r 2 ; (1{16) whichistheso-called Moncrieffunction .(Notethatthisisdierentfrom,thoughrelated to,theRegge-Wheelerfunctionusedin[ 22 ].)Thesamecanalsobedonewiththeseven even-parityfunctions,thoughthecorrespondingevenmast erfunction, Q even ,looksmore complicatedsoitisnotshownexplicitlyhere.(Seehowever [ 23 { 25 ].) ThecoupledsystemofPDEswhichis( 1{11 )thenendsupbeingjusttwowave equationsateach( l;m )(oneforeachparitysector).Fortheodd-paritysector,we have: @ 2 @t 2 + @ 2 @r 2 V RW ( r ) Q odd =0 ; (1{17) 26

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where V RW ( r )= 1 2 M r l ( l +1) r 2 6 M r 3 ; (1{18) and r = r +2 M ln( r 2 M 1);whilefortheeven-paritysector: @ 2 @t 2 + @ 2 @r 2 V Z ( r ) Q even =0 ; (1{19) where V Z ( r )=2 1 2 M r 2 r 2 [( +1) r +3 M ]+9 M 2 ( r + M ) r 3 ( r +3 M ) 2 ; = 1 2 ( l 1)( l +2) : (1{20) Q even and Q odd arefunctionsof( t;r )fromwhichthetenfunctions f h 0 ;h 1 ;h 2 g and f H 0 ;H 1 ;H 2 ;j 0 ;j 1 ;K;G g canbeextracted,respectively.Therst-orderperturbati on problemforSchwarzschildthenboilsdowntosolvingforthe potentials Q even and Q odd with theappropriateboundaryconditions.Radiationatnullinnity Fromthemasterfunctions Q even and Q odd onecancalculatethe\plus"and\cross" polarizationamplitudes h + h ofagravitationalwaveas r !1 ; inthetransverseand tracelessgauge[ 24 25 ].Thisisgivenby: h + ih = 1 r X ( l;m ) s ( l +2)! ( l 2)! ( Q evenlm + iQ oddlm ) 2 Y lm ( ; )+ O 1 r 2 ; (1{21) where, 2 Y lm ( ; )aretheso-called( s = 2) spin-weightedsphericalharmonics denedby 2 Y lm ( ; ) s ( l 2)! ( l +2)! W lm iX lm (1{22) W lm = @ 2 @ 2 cot @ @ 1 sin 2 @ 2 @ 2 Y lm (1{23) X lm = 2 sin @ 2 @@ cot @ @ Y lm : (1{24) (ForEMRIwaveformsofinKerr,gravitationalwaveformsare computerthrough theso-calledTeukolsky-Nakamura-Sasakiformalism,whic hisbasedoncurvature 27

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perturbationsratherthanmetricperturbations.Itmakesu seoftheasymptoticvalue of 4 ,thegauge-invariantperturbationinoneofthetetradcomp onentsoftheWeyl curvaturetensor.WelimitthediscussiontoSchwarzschild ,however,justtoprovidea senseofhowthingsaredone). WiththisRegge-Wheeler-Zerrillimachineryinplace,onec ancalculatewaveforms ofEMRIsmodeledastestmassesontheSchwarzschildspaceti me.Thiswouldentail choosingaspecictrajectoryforthetestmassanddecompos ingthestress-energyforthe pointmass T ab = Z 1 1 u a u b p g (4) ( x a X a ( s ))d s (1{25) intoitssphericalharmoniccomponents.Here, X a ( s )istheprescribedworld-lineforthe testmass.EMRIwaveforms DataanalysisforEMRIsearcheswillprovequiteintensived uetothelargedimensionality ( D =14)ofitsparameterspace[ 26 ].Infact,evenasemicoherentsearch(whereby matchedlteringisappliedtoshorterstretchesofdataand thesignal-to-noiseratio issummedalongpathsinparameterspacethatareexpectedto correspondtoEMRI evolution)stillrequiresahugetemplatebank(10 12 )[ 27 ].Lessaccuratewaveforms(with respecttofullself-forcewaveforms)thatareeasiertocom putewillhavetobeusedin ordertodetectEMRIwavesinitially,andtoconstrainthesp acefromwhichtoextract sourceparametersviamatched-ltering.Itthetimeofwrit ing,existing,lessaccurate EMRIwaveformshavebeenoneofthefollowing:adiabatic/Te ukolskywaveformsand (analyticalornumerical)kludgewaveforms. Adiabatic/Teukolskywaveforms[ 28 29 ]areproducedbysolvingtheTeukolsky equationforradiationatinnityduetoatest-masssourcem ovingonaboundgeodesicof Kerr.Theradiationatinnityprovidestheorbit-averaged evolutionoftheparameters, f E;L;K g (energy,angularmomentum,andtheCarterconstant),thatd enethegeodesic 28

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orbit.WhiletheredoesnotexistaruxfortheCarterconstan t,ithasbeenrecentlyshown thatitsorbit-averagedevolutioncanalsobederivedfromt hesameTeukolskycoecients thatproducetheenergyandangularmomentumruxes[ 30 ].Theseevolutionequationsare thenusedtoproduceasequenceofgeodesicsforwhichemitte dradiationbyatestmass iscomputed.Byevolvingfromonegeodesictothenext,while computingtheradiationat innity,awaveformisthusproduced. Producingtheseadiabaticwaveformsisalsoratherexpensi ve.Andmoreover,by relyingonruxestodeterminetheevolutionofthepointmass ,itignoresallconservative eects(i.e.eectsnotassociatedwiththeemissionofradi ationatinnity).Theclaim isthatthismissingconservativepieceaveragesouttozero ,anddoesnotcontribute secularly.Butthisassumptionhasbeenrefutedatleastint hecontextofanelectric chargeweaklycurvedspacetime[ 31 ].Theimportanceoftheseconservativeeectsin strong-eldgravityiscurrentlybeingdebated,andisoneo fthereasonsjustifyingtheneed forself-force-correctedorbitsandwaveforms. Kludgewaveformsarethefastesttoproduce.Analytickludg ewaveforms[ 26 ]are Peters-MatthewswaveformsforaparticleonaKeplerianorb itwithPost-Newtonian corrections.Buttheyarequiteinaccurateparticularlyin thelatterstagesoftheinspiral, whichisexpectedsinceaKeplerianorbitisabadapproximat iontoatrueKerrorbit. Numericalkludgewaveforms[ 32 33 ]trytoremedythisdeciency,followingaroute similartotheadiabaticwaveforms.Prescriptionstotheev olutionof f E;L;K g are developedthroughPost-NewtonianexpansionsoftheTeukol skyfunctionandtsto solutionsoftheTeukolskyequation.Thisprescriptiondet erminestheinspiral.This amountstointegratingthegeodesicequationinKerrwithti me-dependent\constants" f E;L;K g .Withtheinspiraldetermined,thenextstepinvolvesusing aweak-eldGW emissionformula(suchasthequadrupoleradiationformula ,quadrupole-octupoleformula, etc.)tocomputethewaveform.Thisisperformedbyrstiden tifyingtheBoyer-Lindquist coordinatesinwhichthetrajectoryisdescribedwithratsp acesphericalcoordinates. 29

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Numericalkludgewaveformsarequiteeectiveatmatchingw aveformscomputed withsolutionsoftheTeukolskyequation(i.e.adiabaticwa veforms).Theyhavetaken centerstageintheeortsofthedataanalysiscommunitytot esttheirsearchstrategies. Thequestionstillremainsthoughastohowcorrectkludgewa veformsreallyare,thatis, whethertheycaptureenoughofthefeaturesofatrueself-fo rcewaveform.Thisquestion appearsanswerableonlywithanactualself-forcewaveform athand. Insummary,itisclearthattheseexistingEMRIwaveformsha vetheirdeciencies withrespecttothegoldstandard,whichare(future)self-f orcewaveforms.Nevertheless, theyarelikelytoplayanimportantroleinLISAdataanalysi s(e.g.detection)because theyaremucheasiertoproduce.However,weshalltakethepe rspectivethatwhile self-forcewaveformsmayeventuallyendupservingaminorr oleinLISA'sactualdata analysispipeline,anykindofalternativeEMRIwaveformwi llhavetobeassessedwith respecttohowwellitrepresentsself-forcewaveforms.Ass uch,therepersistsaclearneed forthetoolsthatwilleventuallyproducethem. 1.4AchievementsofthisWork Theaimofthisdissertationistospelloutanovelmethodfor thecalculationofthe self-forceonparticlesmovingaroundastrophysically-re levantblackholespacetimes,one thatisparticularlysuitedfor(3+1)codesthatarecurrent lyofuseinnumericalrelativity forsimulationsofcomparable-massblackholebinaries.Ov erthecourseofthreedecades, thesecodeshavebeenconsistentlyrenedandalmostperfec tedbynumericalrelativists. Theiradditiontothetoolboxofself-forceresearcherswil lprovetobeatremendous advantage.Moreover,itisina(3+1)contextthatthetaskof updatingparticleorbits usingcalculatedself-forcesappearsmostconvenient.The developmentofaframework throughwhichthetoolsofnumericalrelativitycanbereadi lyappliedtosolveself-force problemsisseenasanimportantstepintherightdirection. AtthecoreofourmethodistheDetweiler-Whitingdecomposi tionoftheGreen's functionforthecurvedspacetimewaveequationintoasmoot hpartthatcompletely 30

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determinestheself-forceandasingularpartthatmakesnoc ontributiontoit.Theeld thatresultsfromthesingularpiececanbeanalyticallyapp roximatedclosetothelocation ofthecharge,andthereforeprovidesanaturalregularizat ionprocedurewhichleaves behindasucientlyregularpieceofthe(otherwisediverge nt)physicalretardedeld, whichwouldthenbecompletelyresponsiblefortheself-for ce. Thisworktakestheseideasanddevelopsamethodforthesimu ltaneouscalculation oftheself-forceonparticlesandtheiremittedwaveform.W hereaspreviouscalculationsof theself-forceweredonewiththeintentofthenincorporati ngtheireectsintowaveform calculationsasapost-processingstep,themethodwehaved evelopedallowsoneto directlyusethecomputedself-forceinmodifyingtheparti cleorbit.Themethodwas designedspecicallytoavoidslowlyconvergingmodesumst hataretypicalinself-force calculations.Instead,aself-forceiscomputedbyjusttak ingderivativesoftheregulareld atthelocationofthecharge. Mainlytodemonstratethatthemethodworks,andtoundersta ndwhatissuesmay ariseinanactualimplementation,werstdevelopeda4th-o rder(1+1)characteristic codetoevolvethereducedscalarwaveequationincurvedspa cetime.Thiswasusedin achievingoneofthersttime-domaincalculationsofaself -force.Then,havinggained condencewithour(1+1)work,twoexisting(3+1)codesfrom numericalrelativity werealsoemployedtodemonstratethatthemethodworksinth iscontextaswell.With this,wehavecarriedouttheveryrstself-forcecalculati onswitha(3+1)code.All implementationswerecarriedoutonthesimpletestcaseofa scalarchargemovingina circulargeodesicofaSchwarzschildblackhole.Thesetest sprovideampleevidencethat therenowexistsacorrectframeworkthroughwhichthe(3+1) infrastructureofnumerical relativitycanbeutilizedtohandleself-forceproblems. 1.5OutlineofthisManuscript Thenextchaptershallbedevotedtotheideasthathavebeenm ostcriticaltothe developmentofourmethodofcalculation.Inparticular,we rstpresentasummary 31

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ofDirac'sderivationfortheequationofmotionofanelectr icchargeinratspacetime, whichhasbeenasourceofinspirationforalotoftheworksth atwouldfollow.Dewitt andBrehme'sextensionofDirac'sequationofmotiontothec aseofcurvedspacetime issummarizedinthefollowingsection.Thisisthenfollowe dbyareviewofvarious derivationsofthegravitationalself-force(ortheMiSaTa QuWaequation)forapoint massmovingincurvedspacetime.AdiscussionofGreen'sfun ctionsforthecurved spacetimewaveequationisthenpresented,highlightingan importantdeciencyinpast interpretationsoftheself-force,whichhavekeptthemfro mbeingtheappropriatecurved spacetimegeneralizationofDirac'sresults. TheseideassetupthebackgroundforDetweilerandWhiting' sapproachtoself-force. Byfocusingontheanalogousscalarcase,itisindicatedhow onecanviewtheresultsof Dewitt-Brehme,andtheMiSaTaQuWaequationasaresultofth einteractionofapoint charge/masswithalocalsolutionofthehomogeneouswaveeq uation,whichisonethe maininsightsarisingfromDirac'soriginalanalysisforth eratspacetimecase. Finallywediscussthemodesumapproachtocalculatingself -forces,whichis inarguablythemostusefulcalculationalprocedureinthes elf-forceliteraturetodate. Wesummarizethestepsthatgointosuchacalculation,andth enstresshowitmaybe understoodthroughtheDetweiler-Whitingdecomposition. Weemphasizethatwhilethe modesumschemeisexcellentforcalculatingself-force,it possessesfeaturesthatmakeit ill-suitedforimplementingbackreactiononparticleorbi ts. Followingthemotivatingideasofthepreviouschapter,Cha pter3expoundsonour newmethodforself-forcecalculation.Weprovideadetaile ddiscussionforthecaseofa movingscalarcharge.Latersectionsofthischapterthengo throughthetheoreticaldetails uponwhichourprescriptionisfounded. InChapter4,weprovidetherstconcretedemonstrationoft heviabilityofour calculationalmethod.Here,wedescribeanumericalexperi mentinwhichwecomputethe self-forceonthescalarchargeanditscorrespondingretar dedeldinthewavezonefora 32

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scalarchargeinacircularorbitaroundanon-rotatingblac khole.Allthiswasdonein time-domain.Detailsonthedevelopmentofa4th-order(1+1 )waveevolutioncodeare presented,alongwithresultsthatmatchwellwithonesprev iouslycalculatedwithhighly accuratefrequencydomaincodes. WithcondencegainedfromtheresultsofChapter4,present edintheChapter 5areresultsoftheveryrstsuccessfulattemptatcalculat ingaself-forcewith(3+1) codes.Thisagainisperformedonthetoycaseofascalarchar geincircularorbitofa Schwarzschildblackhole,whichisidealfortestingmethod s.Afewmoredetailsofthe simplephysicsforthisphysicalsystemarepresented,alon gwithabriefdescriptionofthe codesthatwereused.Adiscussionthenfollowswhichstress esthemeritsofourapproach, andthemainhurdlesthatneedtobeovercomeforittonallyp roduceself-consistent modelsofparticlemotionaroundblackholesandtheiremitt edradiation. Finally,wereviewthemaincontributionsofthiswork,ande xplorethefutureresearch itmakespossible. 33

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CHAPTER2 IDEASINRADIATIONREACTIONANDSELF-FORCE Thetheoryofradiationreactionandself-forcehashadalon ghistory.Inthischapter, wesurveyafewofthedevelopmentsthatweremostinruential toourcurrentstateof understanding,andarecrucialtounderstandingthecalcul ationprescriptionforselfforce calculationthatisreshedoutinlaterchapters.Inparticu lar,wehighlightcontributions thatleadtothederivationoftheequationsofmotionforasc alarcharge,electriccharge, andpointmassinacurvedspacetime,andthemotivationsbeh indtheDetweiler-Whiting decompositionoftheretardedGreen'sfunctionforthewave equation. Asaconcretecalculationalprescriptionforself-forces, thereisnonethatsurpasses thecontributionofthemodesumapproach.Wedevotethelast sectionofthischapterto thismethod,andpointouthowitmaybeunderstoodinlightof theDetweiler-Whiting decomposition.Wealsopayattentiontothepropertiesthat makethemode-sumscheme ill-suitedforthetaskofsimulatingself-consistentdyna micsofparticlesandelds(i.e. dynamicsthatincludestheeectsoftheself-force),which willmotivatethenovelmethod describedinthenextchapter. 2.1MotionofaChargedParticleinFlatSpacetime TheAbraham-Lorentz-Dirac(ALD)equationdescribeshowac hargedparticlemoves undertheinruenceofanexternalforceanditsownelectroma gneticeld.Originallydue toAbraham[ 34 ],ittakesinspirationfromtheclassicalLorentzmodeloft heelectron asasmallspherepossessingamassdeterminedbytheenergyo ftheelectriceld surroundingit.Dirac[ 35 ]reproducedtheequationbyimposinglocalenergyconserva tion onatubesurroundingtheparticle'sworldline,andhidingd ivergentcontributionsby \renormalizing"theparticlemass.Wereviewthisderivati onindetailbelow,sinceall modernstudiesoftheself-forceareguidedbyDirac'sorigi nalanalysis. Thesimpleideathatleadstotheequationisthepostulateth atthetotalfour-momentum P a (consistingoftheelectromagneticmomentum P a em andasuitablydenedmechanical 34

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momentum P a mech )beconservedontheworldlineofthecharge.TheALDequatio nisas follows: ma a = F a ext + 2 3 q 2 ( a b + u a u b )_ a b (2{1) where q isthechargeoftheparticle, u a itsfour-velocity,and F a ext .Notethatwrittenthis wayitisathird-orderequationin x a ,whichhasbeentherootofmuchofthepathological solutionsassociatedwithit. Inwhatfollows,wesummarizeDirac'sderivationin[ 35 ].Asimilarderivation(witha moregeometricalravortoit)canbefoundin[ 36 ]. Considerachargeonaworldlinespeciedby z a ( ),where isthepropertimealong theworldline.Surroundthisbyathinworldtube,whichisa 3-cylinderofarbitrary shape. Nowweconsiderthestress-energytensorofMaxwell'stheor y T ab = 1 4 F ac F b c + 1 4 g ab F cd F cd (2{2) andwiththisseektocomputethefour-momentumruxacross, whichwedenoteby P a : P a = Z T ab d b (2{3) whered b istheoutward-directedsurfaceelementon. Wecanverifythat P a isindependentoftheshapeof.Consideranotherworld tube 0 thatsharesthesame\caps"as.Let P 0 a bethemomentumruxacross 0 Thenconsideringthe4-volume V boundedbyand 0 wehave: P a P 0 a = Z @ V T ab d b = Z V @ b T ab d V =0 ; (2{4) 35

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wherewehaveusedGausslaw,andthefactthatin V theelectromagneticeldis sourceless.Thisprovestheassertionthatthechangeinthe momentum P a depends onlyonthestartandend\caps",andisindependentofthe3-c ylindersjoiningthem. Thisfactpermitstwothings:(1)Wecanchoosethesimplests hapefortheworldtube (e.g.theconstant r surface)incalculating P a ,and(2)ifwecanexpresstheruxacross intheform Z T ab d b = Z G a d; (2{5) withtherestrictiontoaverythinworld-tube,thenwecanco ncludethatthat G a isatotal dierential: d d B a = G a : (2{6) Withthis,allthatremainstobedoneisthecalculationofth emomentumrux.Before proceedingthough,itisinstructivetodiscusstheeldsin volvedthatmakeupthe stress-energy T ab Theelectromagneticeldsurroundingthechargeisdetermi nedbyMaxwell's equations: F ab ;b =4 j a ; (2{7) wherethecurrentdensityisgivenby j a = q Z d u a (4) ( x z ) ; (2{8) u a =d z a = d istheparticle'sfour-velocity,andthedeltafunctionisf our-dimensional. Theintegrationinthesourceisovertheentireworldline.T heequationscanbeexpressed intermsofavectorpotential( F ab = @ a A b @ b A a )forwhichtheMaxwell'sequationsin Coulombgaugebecomethesimultaneousequations: 2 A a = 4 j a (2{9) @ a A a =0(2{10) 36

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Diraccallstheeldwithwhichtoevaluate T ab theactualeld, F ab act .Forthecaseofa pointcharge,thiswillbecomprisedoftheretardedeld F ab ret (theknowneldgeneratedby amovingpointcharge),andahomogeneouseldrepresenting incidentradiation,denoted by F ab in .Thus, F ab act = F ab ret + F ab in (2{11) Theactualeldwouldcouldalsoberepresentedbyadvancedp otentials: F ab act = F ab adv + F ab out ; (2{12) wheretheadvancedeldissimplythecounterpartofthereta rdedeldgeneratedbythe advancedGreen'sfunctionofthewaveequation,and F ab out isdenedasthedierencein theactualandtheadvancedelds.Asthecounterpartofthe F ab in ,itmaybeinterpretedas theeldofoutgoingradiation. Thedierenceinoutgoingandincomingeldsgivestheeldo fradiationproducedby thechargeitself: F ab rad = F ab out F ab in = F ab ret F ab adv : (2{13) Whilethismaylooklikejustasillycyclingofvariables,it ssignicanceliesinthefact thatwenowhaveanexpressionfortheradiativeeldoftheel ectrongivenintermsof componentsthatarefullydeterminedbytheworldline.Note alsothatthisiswell-dened locally,thusoeringanidenticationoftheradiativeel dclosetothechargewhereinthe usualtreatmentsitisinextricablywovenwiththeCoulombn ear-eld. WenoteforfuturereferencethatDiracdenesasingularity -freeeld f ab as: f ab F ab act 1 2 ( F ab ret + F ab adv ) : (2{14) = F ab in + F ab ret 1 2 ( F ab ret + F ab adv ) (2{15) Thiseldisasolutiontothesource-freeequationsandisde terminedbyconditions at t = 1 and t = 1 .Itwillturnoutthatthemotionofthechargeiscompletely 37

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determinedby f ab ,whichiscomprisedoftheincidentelddeterminedbycondi tionsinthe distantpast,and(inthesquarebrackets)anappropriately regularizedretardedeld. Thegoalnowistocomputetheactualeld F ab act anditsstress-energytensornearthe worldline.Webeginwiththeretardedeld. Tocompute F ab ret welooktotheretardedLienard-Wiechertpotential: A reta = qu a u b ( x b z b ) (2{16) where u a and z a areevaluatedattheretardedproper-time ,orthevalueof satisfying ( x a z a ( ))( x a a ( ))=0(2{17) with( x 0 z 0 ) > 0. Thiscanbewritteninthemoreconvenientform: A reta =2 q Z int 1 u a (( x b z b )( x b z b ))d (2{18) where int issomepropertimebetweentheretardedandadvancedtimeso f x a .After dierentiatingandafewmanipulationsthisbecomes: A reta;b =2 q Z d d u a ( x b z b ) u c ( x c z c ) (( x d z d )( x d z d ))d (2{19) sothat: F ret ab = 2 q Z d d u a ( x b z b ) u b ( x a z a ) u c ( x c z c ) (( x d z d )( x d z d ))d = q u c ( x c z c ) d d u a ( x b z b ) u b ( x a z a ) u d ( x d z d ) (2{20) wheretheworldlinequantities u a and z a aretobeevaluatedattheretardedpropertime. Nowthetaskathandistondanapproximationof( 2{20 )alongtheworldtubein termsofquantitiesdenedontheworldline.Diracdoesthis byexpressingaeldpointon 38

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, x a ,intermsofasmallvector r a : x a = z a ( o )+ r a : (2{21) Here, o isapropertimealongtheworldlinechosensothat u a r a = u a ( o )( x a z a ( o ))=0 : (2{22) Thenlettheretardedpropertimebe o ,where isanothersmallquantityofthesame orderas r a .Forbookkeepingpurposesweshallexpressourexpansionsi ntermsofasmall quantity ,sothat r a r a and .Withthese,weTaylor-expand( 2{20 )upto thirdorderin (whichisallthatisrequired),whilesimplifyingtermsasm uchaspossible with( 2{22 )andthefollowingtrivialidentities: u a u a =1, u a u a =0,and u a u a +_ u a u a =0. NotethatDiracemploysasignature(+---). Asexamples,wehave: x a z a ( o )= ( r a + u a )+ 2 1 2 u a 2 + 3 1 6 3 u a (2{23) u a ( o )= u a u a + 2 1 2 u a + 3 1 6 3 ... u a (2{24) Notethatalltheworldlinequantitiesinthecoecientsoft heseexpansionsarenowto beevaluatedat o .Dierentiatingexpansionssuchasthesewithrespectto wouldbe tantamounttodierentiatingwith ,since asyethasnotbeenxed.Whatresults thenisanexpansionof( 2{20 ): F ret ab = 2 q (1 r a u a ) 2 2 u [ a r b ] 3 + 1 1 2 u [ a u b ] + 1 2 ( r c u c ) u [ a r b ] 2 + 1 2 u [ a r b ] + 2 3 u [ a u b ] (2{25) Inthisoneneedstoplug-intheappropriate ,whichisxedusing( 2{17 ).Using ( 2{23 ),andsetting r a r a = 2 ,thisconditionbecomes: 2 + 2 (1 r a u a )+ 1 3 3 ( r a u a ) 1 12 4 u a u a =0 : (2{26) 39

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Solvingfor ,weget: = (1 r a u a ) 1 = 2 1 1 6 r b u b + 1 24 2 u b u b : (2{27) Pluggingthisin,andkeepingonlytermsthatdon'tvanishas 0,givesthenal expressionfortheretardedeld: F ret ab =2 q (1 r a u a ) 1 = 2 3 u [ a r b ] + 1 1 2 u [ a u b ] (1 r c u c )+ 1 8 u c u c u [ a r b ] + 1 2 u [ a r b ] + 2 3 u [ a u b ] (2{28) Gettingasimilarexpansionfortheadvancedeldwillgothr oughthesameprocess. Theonlychangewillbethatinsteadofevaluatingworldline quantitiesatretardedtime o ,theyareevaluatedatadvancedtime o + .Toarriveattheadvancedeldthen, allonehastodoisperformtheswitch ,whichwouldimplyswitching in( 2{28 )andputtinganover-allminussigninfrontoftheresulting expression.[Recall thatinthecalculationfortheretardedeld,thederivativ ewithrespectto in( 2{20 )was performedasaderivativewithrespectto .Hencetheover-allsignchange.] Following( 2{14 ),theactualeld F act ab willsimplybe: F act ab = 1 2 ( F ret ab + F adv ab )+ f ab : (2{29) Becauseoftherelationshipbetweentheapproximationsfor theretardedand advancedeld,whatsurvivesintheirmeanaretermsofoddpo wersof (i.e.allexcept thelasttermin( 2{28 )).Theactualeldisthenfoundtobe: F act ab =2 q (1 r a u a ) 1 = 2 3 u [ a r b ] + 1 1 2 u [ a u b ] (1 r c u c )+ 1 8 u c u c u [ a r b ] + 1 2 u [ a r b ] + f ab (2{30) 40

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Fromthisonebuildstheassociatedstress-energytensor( 5{35 ),focusingprimarilyon thecomponentinthedirectionof r a 1 T ab r b ,sinceweareinterestedintheruxacross theworldtubewhoseunitnormalis 1 r a Alongcalculationeventuallyyields: 4 T ab r b = q 2 [1 r c u c ] 1 1 2 4 + 1 2 2 u c u c r a 1 2 2 1+ 3 2 r c u c u a + q 1 u b f ab (2{31) Tocalculatetheruxacross,werstneedtodeterminethesu rfaceelementd a isdenedbythesetofequations: ( x a z a ( ))( x a z a ( ))= 2 (2{32) ( x a z a ( )) u a ( )=0 : (2{33) Thisissetofvevariables f x a ; g constrainedbytwoequations,whichthusclearlydenes ahypersurface. Consideranarbitraryshiftalongthesurfacebyd x a .Supposethatthepoint x a +d x a correspondstothepropertime +d (i.e.varying x a correspondstoapropertime variationd ).From( 2{32 )and( 2{33 )weget: ( x a z a ( ))(d x a u a ( )d )=0(2{34) (d x a u a ( )d s ) u a ( )+( x a z a ( ))_ u a d s =0(2{35) Thesebecome: r a d x a =0 (2{36) u a d x a =(1 r c u c )d : (2{37) Equation( 2{36 )simplyimpliesthat r a isnormalto.Sinceisitofmagnitude ,the unitnormaltoisjust 1 r a ,asclaimed.Equation( 2{37 )givesthelineelementfor 41

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displacementsalongthesurfaceparallelto u a : j d x ajj j =(1 r c u c )d : (2{38) Nowforaxed ,onecangotoaframemomentarilycomovingwith u a ( ) : =(1 ; 0 ; 0 ; 0) (where : =meansequalityonlyinthisframe).Thismakes( 2{33 )become( t z 0 ( ))=0, whichthenreduces( 2{32 )to j ~x ~z ( ) j 2 = 2 ,theequationfora2-sphere.Therefore,for displacementsorthogonalto u a ,forwhich isxed,theareaelementisnothingbutthat ofa2-sphereofradius Puttingeverythingtogether,theruxacrossis: Z T ab d b = ZI 1 T ab r b ( 2 dn)(1 r c u c )d = ZI q 2 1 2 3 + 1 2 1 u c u c r a 1 2 1 1+ 3 2 r c u c u a + qu b f ab dnd : (2{39) Tosimplify,rstnotethattermslinearin r a justvanishuponaveragingoverthe 2-sphere.Thenweignorealltermsthatvanishinthelimit 0.Thisleavesuswith: Z T ab d b = Z 1 2 q 2 1 u a qu b f ab d : (2{40) Wehavethuswrittentheruxthroughintheformexpressedin ( 2{5 ).Sincethe ruxdependsonlyontheendpoints,theintegrandmustbeatot aldierential.Thistotal dierentialwouldbethetotalmechanicalmomentumofthech arge.Thatis,thereexistsa vector B a suchthat d B a d = 1 2 q 2 1 u a qu b f ab (2{41) Whatwehavecomputedisthentheruxperunitpropertime.Dir acimposesthatthis mustalsomustalsobehowthetotalmechanicalmomentumofth echargechangesper unitpropertime.Thatis, d P a d = 1 2 q 2 1 u a qu b f ab (2{42) 42

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Toturnthisintoanequationofmotionforthecharge,onemus tthenassumeaformfor themechanicalmomentumofthecharge.Thechoiceisarbitra ryexceptfortheconstraint that u a d P a d = 1 2 q 2 1 u a u a =0 : (2{43) Diracchoosesthesimplestformsatisfyingthisconstraint : P a = ku a ,arguingthat\one wouldhardlyexpect[moredicultforms]toapplytoasimple thinglikeanelectron". Withthischoice,theequationofmotionisthensimply m u a = qu b f a b (2{44) where m =(1 = 2) q 2 1 k .Theresultappearstobethestandardequationofmotionfor a chargeinanexternalelectromagneticeld.Werecallthoug hthat f ab isdenedas f ab = F act ab 1 2 ( F ret ab + F adv ab ) : (2{45) Itistheactualeldbeingregularizedbythemeanofthereta rdedandadvancedelds, andasmoothsolutiontothehomogeneouseldequations.Thi sshallbearecurringtheme inlaterchapters. Tomaketheequationofmotionmoretransparent,weexpress f ab inanotherform: f ab = F (in) ab + 1 2 F rad ab (2{46) wheretheradiationeldisjustthedierencebetweenretar dedandadvancedelds.Using theretardedeld( 2{28 )andknowinghowtheadvancedeldisrelatedtoit,weseetha t onlyevenpowersof surviveintheirdierence{whichisjustthelasttermof( 2{28 ). Thistermiswell-denedattheworldline,soevaluatingitt here(therebyignoringtheany r -terms),wehave: F rad ab = 4 q 3 ( u a u b u b u a ) : (2{47) 43

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Theequationofmotionthenbecomes: m u a = qu b f a b = qu b F a b (in) + 2 q 2 3 (_ a a a b u a u b )(2{48) or ma b = qF bc (in) u c + 2 q 2 3 ( bc u b u c )_ a c : (2{49) ThisistheAbraham-Lorentz-Diracequationforthemotiono fachargeunderthe inruenceofanexternaleldanditsselfforce. F bc in isunderstoodtobeanincidentexternal elddeterminedbyconditionsinthefar-awaypast.Theseco ndtermiswhatisassociated withradiationreaction. ThemostcrucialaspectofDirac'scalculationisthathewas abletoidentifythepiece oftheactualelectromagneticeldthatcontributestothes elfforce,whichis 1 2 F rad ab ,often referredtoasthe\half-retardedminushalf-advanced"el d,orhalfDirac'sradiativeeld. ThisradiativeeldisasolutionofthehomogeneousMaxwell 'sequations,andcanalsobe expressedconvenientlyas 1 2 F rad ab = F ret ab 1 2 F ret ab + F adv ab ; (2{50) wherethelasttermisatime-reversalinvariantsolutionof thesameinhomogeneous Maxwell'sequations,butsatisfyingstandingwaveboundar yconditionsatinnity.Its time-reversalinvarianceimpliesthatithasnothingtodow iththeradiationdamping(or self-forceeects)ofthecharge.Thus,Diracisalsoableto isolatethepieceofthephysical retardedsolutionthathasnothingtodoself-force.Weshal lreturntothispointinlater sections. 2.2MotionofaChargedParticleinCurvedSpacetime AnattemptatgeneralizingDirac'sanalysistocurvedspace timewasrstprovidedby DeWittandBrehme(henceforthDB)[ 37 ].Wereviewtheircalculationhere,withoutgoing intomuchdetail,focusinginsteadontheconceptualaspect srelevanttolatersections. 44

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DeWittandBrehme'sstrategywastoessentiallyfollowtheD iraccalculationbutwith therestrictionofkeepinggeneralcovariancethroughout( justasDiracinsistedonLorentz invariance).Inkeepingwiththisconstraint,theyrelyonm athematicalobjectscalled bi-tensors,whicharegeneralizationsofordinarytensors ,thatareessentialforanysort ofgenerally-covariantcalculationinvolvingnon-localq uestions.Forinstance,aGreen's functionisabi-tensorsinceitconnectstheresponseatae ldpoint x toadelta-function sourceat x 0 .Thegeneraltheoryofbi-tensorsisthoroughlydiscussedi nPoisson'sreview [ 38 ],andwefollowmuchofthestyleofhisreviewintheensuingd iscussions. Bi-tensorsaretensorialfunctionsoftwopointsinspaceti me.Letthesetwopointsbe x and x 0 ,andwelabelthecomponentsofthebi-tensorbyunprimedind ices ;; etc.,and primedindices 0 ; 0 ; etc.dependingonhowtheytransformwithrespecttooperati ons carriedouton x and x 0 ,respectively.Asimpleexampleistheproductofavectorat x and adualvectorat x 0 : C a b 0 ( x;x 0 )= A a ( x ) B b 0 ( x 0 ) : (2{51) Suchisabi-tensorthatbehavesasavectorat x andadualvectorat x 0 .Atransformation from x to x simplygives: C a b 0 ( x;x 0 )= @ x a @x a C a b 0 ( x;x 0 ) ; (2{52) whileatransformationfrom x 0 to x 0 wouldcorrespondinglybe: C a b 0 ( x; x 0 )= @x b 0 @ x b 0 C a b 0 ( x;x 0 ) : (2{53) Onemaywishtotakecovariantderivativeswithrespecttoei therspacetimepoint, rememberingthatwhenitiswithrespectto x ,say,allprimedindicesmustbeignored. Thus, C a b 0 ; c = C a b 0 ;c + aec C e b 0 (2{54) C a b 0 ; c 0 = C a b 0 ;c 0 e 0 b 0 c 0 C a e 0 : (2{55) 45

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Fromthis,wecandrawthefactthattheorderingofindicesin bitensorsisonlyimportant forlikeindices.Thatis,primedandunprimedindicescommu te. Aparticularlyusualquantityisthebi-scalarknownasSyng e'sworldfunction, ( x;x 0 ).Thisisdenedas ( x;x 0 )= 1 2 ( 1 0 ) Z 1 0 g ab ( z ) t a t b d (2{56) where z ( )isageodesicconnecting x and x 0 ,suchthattheaneparameter rangesfrom 0 to 1 ,and z ( 0 ) x 0 and z ( 1 ) x t a = d z a d isjustthetangentvectortothegeodesic, anditobeysthegeodesicequation Dt a =d =0.Thegeodesiclinkingthetwospacetime pointswillbeuniqueinasucientlysmallneighborhood.Th isregionshallbereferredto asthenormalneighborhoodoftheworldline. Duetothegeodesicequation,thequantity g ab t a t b isconstantonthegeodesic. Synge'sworldfunctionisthennumericallyequalto 1 2 ( 1 0 ) 2 .Itiseasytoverifythat isinvariantwithrespecttoananetransformationof .Thus,dependingonthetype ofgeodesicconnectingthetwopoints x and x 0 wecanswitch toeitherthepropertime, ,orproperdistance, s ,fortimelikeandspacelikegeodesics,respectively.Ofco urse,when thegeodesicisnull, =0.Thus, =0denesthelocusofpoints x thatmakeupthe lightconeof x 0 Itcanbeshownthat a r a isjustthetangentvectorat x ofthegeodesic connecting x and x 0 ,rescaledbytheaneparameterlength( 1 0 ).Specically, a ( x;x 0 )=( 1 0 ) t a (2{57) a 0 ( x;x 0 )= ( 1 0 ) t a 0 : (2{58) Notethatinratspacetime,where(inLorentziancoordinate s)geodesicsarejust x a = x a0 + t a ,thesejustbecome: a = ab ( x x 0 ) b and a 0 = a 0 b 0 ( x x 0 ) b 0 .Thus, a and a 0 ,justliketheiranaloguesinratspacetime,aretobeuseful inperformingcovariant near-coincidenceexpansionsofquantitiesincurvedspace time. 46

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Inlightofperformingcovariantexpansions,anotherimpor tantbi-tensoristheparallel propagator, g a a 0 ( x;x 0 )orwhatDBcallthebi-vectorofgeodeticparalleldisplace ment. (Thismustnotbemistookforthemetrictensor,sinceitstwo indicesrefertodierent points).Thisbi-tensorisdenedbytheequations: g a a 0 ; b b = g a a 0 ; b 0 b 0 =0 ;g a 0 a ; b b = g a 0 a ; b 0 b 0 =0 ; (2{59) andtheinitialcondition: lim x x 0 g a b 0 = a 0 b 0 : (2{60) Asolutiontotheseequationscanbeconvenientlyrepresent edbyerectinganorthonormal basis, f e a( ) ( x ) g ,thatisparallel-transportedalongthegeodesicjoining x and x 0 .Theframe indices( ) ; ( ),runfrom0...3,andsatisfy: g ab e a( ) e b( ) = ( )( ) ; De a( ) d =0 : (2{61) Intermsofthisbasis,theparallelpropagatorisjust: g a a 0 ( x;x 0 )= e a( ) ( x ) e ( ) a 0 ( x 0 ) ; (2{62) where e ( ) a = ( )( ) g ab e b( nu ) .Sincethetetradisparallel-transportedalongtheconnec ting geodesic,theequationsin( 2{59 )areallsatised. Consideravectorat x A a ( x ).Thiscanbedecomposedas A a = A ( ) e a( ) .If A a ( x ) istobeparallel-transportedto x 0 ,thetetradcomponents A ( ) A a e ( ) a willberemain constant.Thatis, A ( ) = A a 0 e ( ) a 0 = A a e ( ) a ,sothatwehave A a ( x )= A ( ) e a( ) = A a 0 ( x 0 ) e ( ) a 0 ( x 0 ) e a( ) ( x ) = e a( ) ( x ) e ( ) a 0 ( x 0 ) A a 0 ( x 0 ) = g a a 0 ( x;x 0 ) A a 0 ( x 0 ) : (2{63) 47

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Therefore,itisclearthattheparallelpropagatortakesav ectorat x 0 andparallel-transports itto x alongtheuniquegeodesicthatlinksthesetwopoints.There existsalsotheinverse operation A a 0 ( x 0 )= g a 0 a ( x 0 ;x ) A a ( x ) ;g a 0 a ( x 0 ;x ) e a 0 ( ) ( x 0 ) e ( ) a ( x )(2{64) whichtakesavectorat x andparalleltransportsitbackto x 0 Theparallelpropagatorisusefulfortheexpansionofbi-te nsorsnearcoincidence,i.e. lim x x 0 .Forinstance,ifonewantstoexpressabi-tensorn a 0 b 0 ( x;x 0 )nearcoincidence,this maybedonewith a 0 : n a 0 b 0 = A a 0 b 0 + A a 0 b 0 c 0 c 0 + 1 2 A a 0 b 0 c 0 d 0 c 0 d 0 + O ( 3 ) ; (2{65) wheretheexpansioncoecientsareallordinarytensorsat x 0 .Todeterminethem, onesimplydierentiatesandtakescoincidencelimitsofEq .( 2{65 ).Knowingthe coincidencelimitsof a 0 anditscovariantderivatives,allowsonetocomputeallthe expansioncoecients.Afulldiscussionofthiscanbefound in[ 38 ]. Inthecaseabove,forinstance,weget: A a 0 b 0 =lim x x 0 n a 0 b 0 (2{66) A a 0 b 0 c 0 =lim x x 0 n a 0 b 0 ; c 0 A a 0 b 0 ; c 0 (2{67) A a 0 b 0 c 0 d 0 =lim x x 0 n a 0 b 0 ; c 0 d 0 A a 0 b 0 ; c 0 d 0 A a 0 b 0 c 0 ; d 0 A a 0 b 0 d 0 ; c 0 (2{68) Onelastquantitythatfrequentlyappearsistheso-calledV anVleckdeterminant, whichisdenedas: ( x;x 0 ) det h a 0 b 0 i ; a 0 b 0 g a 0 a ( x 0 ;x ) a b 0 ( x;x 0 )(2{69) whichcanalsobewrittenas ( x;x 0 )= det[ ab 0 ( x;x 0 )] p g p g 0 ; (2{70) 48

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where g and g 0 arethemetricdeterminantsat x and x 0 ,respectively.ThevanVleck biscalarindicateswhethergeodesicsemanatingfrom x 0 are(intheneighborhoodof x ) eitherfocusingordefocusing.If( x;x 0 ) > 1,thenthegeodesicsarefocusing,andif ( x;x 0 )theyaredefocusingat x Havingintroducedthesemathematicalpreliminaries,were turntotheDeWitt-Brehme derivation.Weconsiderachargewhoseworldlinethroughcu rvedspacetimeisdescribed by z a ( ),where isthepropertimealongtheworldline. Recallrsttheformalderivationoftheequationsfortheco upleddynamicsofthe eld F ab andthecharge q throughanactionprinciple.Thisactionisjustthesumof theactionsforafreeelectromagneticeldandafreepartic le,andaninteractionterm: S = S eld + S particle + S interaction .Explicitly,theseare S eld = 1 16 Z F ab F ab p g d 4 x (2{71) S particle = m Z r d = m Z r r g ab ( z ) d z a d d z b d d (2{72) S interaction = q Z r A a ( z ) d z a d d = q Z r A a ( x ) g a b 0 ( x;z )_ z b 0 4 ( x;z ) p g d 4 x d (2{73) wheretheprimedindicesinEq.( 2{73 )pertaintopointsontheworldline, z b 0 ,and 4 ( x;z )istheinvariantDiracdistribution 4 ( x;z )= 4 ( x z ) g ( x ) 1 = 4 g ( z ) 1 = 4 : (2{74) Notethat 4 ( x z )istheordinary(coordinate)Diracdistribution. Requiringthattheactionbestationaryunderavariation A a ( x )and z a ( ), respectivelygivesthecoupledsetofequations: F ab ; b =4 j a (2{75) m Du a d = qF a b ( z ) u b ; (2{76) 49

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where j a = q Z r g a b 0 ( x;z )_ z b 0 4 ( x;z )d : (2{77) Theseequationshoweverareill-dened,sincetheelectrom agneticeldwillbesingularat thelocationofthecharge.Inasense,thetaskofDBwastond thecorrectregularization of F a b atthelocationofthecharge,sothatEq.( 2{76 )makessense. ThestepsintheDBcalculationareasfollows: 1. Recallstress-energyforthecoupledsystem, T ab = T ab P + T ab em ,where T ab em = 1 4 F a c F bc 1 4 g ab F cd F cd (2{78) isthestress-energyoftheelectromagneticeld,and T ab P = m Z g a a 0 g b b 0 u a 0 u b 0 4 ( x;z )d (2{79) isthestress-energyoftheparticle. 2. SolveforretardedandadvancedGreen'sfunctionsforMaxwe ll'sequationinLorenz gauge: 2 G a b 0 ( x;x 0 ) R a b ( x ) G b b 0 ( x;x 0 )= 4 g a b 0 ( x;x 0 ) 4 ( x;x 0 ) : (2{80) (AdiscussionofGreen'sfunctionsforwaveequationsincur vedspacetimeisprovided inalatersection).Theirresult: G a+ a 0 ( x;x 0 )= ( x; ( x 0 )) G a b 0 ( x;x 0 )(2{81) G a a 0 ( x;x 0 )= (( x 0 ) ;x ) G a b 0 ( x;x 0 )(2{82) wherethe+and signsrefertoretardedandadvancedGreen'sfunctions, respectively. ( x; ( x 0 ))isonewhere x 2 I + ( x 0 )andzeroelsewhere, (( x 0 ) ;x )= 1 ( x; ( x 0 )),and G a b 0 ( x;x 0 )= g a b 0 ( x;x 0 ) 1 = 2 ( x;x 0 ) ( )+ V a b 0 ( x;x 0 ) ( )(2{83) ( x;x 0 )istheVanVleckdeterminantintroducedabove,and istheHeaviside function. 3. Computetheretardedandadvancedpotentialsduetothechar ge.Maxwell's equationinLorenzgaugeisjust: 2 A a R a b A b = 4 j a ; (2{84) 50

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where 2 = g ab r a r b isthewaveoperatorand R a b istheRiccitensor.Intermsofthe Green'sfunctions,thesolutionisjust A a ( x )= Z G a b 0 ( x;x 0 ) j b 0 ( x 0 ) p g d 4 x 0 ; (2{85) where j b 0 ( x 0 )isjust( 2{77 ).TheresultofsucharethecovariantLienard-Wiechert potentials: A aadv/ret ( x )= q [ g a b 0 u b 0 1 = 2 ( u a a ) 1 ] j adv = ret Z 1 adv = ret V a b 0 u b 0 d : (2{86) 4. Computetheretardedandadvancedeldsfromthepotentials : F ab = r a A b r b A a 5. Characterizetheworldtube.Inthiscase,itisacylinderof distance awayfromthe worldline. 6. Approximate F ret/adv ab ontheworldtubeasanexpansionin ,keepingonlytermsup to O ( ). 7. Use r b T ab =0.Butrst,tobeabletouseGauss'slawon r b T ab =0,parallel transportthistotheworld-line g a a 0 r b T ab =0.Nowallthequantitiesbelongingto thissumpertaintovectorson z ,anditisnowascalarwithrespectto x .Therefore Gauss'slawcannowbeusedtoyield: 0= Z V g a a 0 r b T ab d 4 x (2{87) 0= Z + Z 1 + Z 2 g a a 0 T ab d b Z V ( r b g a a 0 ) T ab d 4 x (2{88) where ; 1 ; and 2 arethewallsandcapsoftheworldtube,respectively,while V istheinteriorofthisworldtube.Forthestress-energyins tep(1),theintegralover thewallspickuponlytheeld.Therestofintegrals,DBthen claimtopickuponly theparticlestressdensity.Thisisnotcorrectthough,and infacttheseintegrals aredivergentduetothecontributionoftheeldstress-ene rgy.Theintegrationof thecaps,however,canbehandledthroughamassregularizat ionsimilartowhatis shownbelow.Theinteriorintegralisoutrightlydiscarded withoutjustication.In theend,however,theyachievethecorrectequationsofmoti on(whichhassincebeen derivedthroughothermeans[ 38 ]). Inanycase,inthelimitthat 0andthecapsbeinginnitesimallyclose(in ), thisbecomes: 0= m u a 0 d +lim 0 d I n g a a 0 T ab d b ; (2{89) wherenreferstothe2-spherecenteredat z 51

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8. Performthe(lengthy)expansionofthesecondtermintermso f ,andtakethelimit 0.Itturnsouttobeconvenienttoexpressthetotalelectrom agneticeldasa sum: F ab = f ab + F ab ; (2{90) where F ab = 1 2 ( F ab ret + F ab adv ) ; (2{91) and f ab isthereforeasingularity-freeeld. Whatresultsisthefollowing: lim 0 I n g a a 0 T ab d b = lim 0 q 2 2 u a 0 qf a 0 b 0 u b 0 q 2 u b 0 Z ret 1 r [ b 0 G a 0 ] (ret) c 00 u c 00 ( 00 )d 00 + Z + 1 adv r [ b 0 G a 0 ] (adv) c 00 u c 00 ( 00 )d 00 (2{92) 9. Returnto( 2{89 ),andrenormalizethemassbythedivergent 1 -piece: m = m 0 +lim 0 q 2 2 : (2{93) yieldingtheequationofmotion:ma a 0 = qf a 0 b 0 u b 0 + q 2 u b 0 Z ret 1 r [ b 0 G a 0 ] (ret) c 00 u c 00 ( 00 )d 00 + Z + 1 adv r [ b 0 G a 0 ] (adv) c 00 u c 00 ( 00 )d 00 (2{94) AswiththeDiraccalculation,itisusefultoexpresstheequ ationofmotioninterms oftheincidenteld F ab in ,whichrepresentsanincomingwavedeterminedbycondition sin thedistantpast.Recallthat f ab canbeexpressedas f ab = F ab (in) + 1 2 F ab (rad) : (2{95) Intermsoftheincidenteld,theequationofmotionbecomes : ma a = qF (in) a b u b + q 2 ( a b + u a u b ) 2 3 a a + 1 3 R a b u b + q 2 u b Z ret 1 r [ b G a ] (ret) c 0 ( z ( ) ;z ( 0 )) u c 0 ( 0 )d 0 (2{96) ThisisthegeneralizationoftheAbraham-Lorentz-Diraceq uationincurved spacetime.Noticethatitsrsttwotermsareexactlywhatwo uldbefoundintherat spacetimecase(seeEq.( 2{49 )).Thereisalsoanewcurvatureterm,butthisvanishes 52

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forvacuumspacetimeslikeSchwarzschildandKerr.Themost importantdierenceisthe presenceoftheintegralovertheentirepasthistoryofthec harge.Thisiswhatisknown asthe\tail"term,whichisaninherentlynonlocaleect.Th istermpersistseveninthe absenceofanexternaleld,whichimpliesthatevenintheca seoffree-fall,radiation dampingoccurs. 2.3MiSaTaQuWaEquation Clearly,theresultsofDirac,DeWittandBrehmehavebeenkn ownforalongtime. Recentinterestintheirtechniqueshasbeenstimulatedbyt heneedtoestablishsimilar resultstothecaseofgravitationalelds.Asmentioned,EM RIsforLISAcanbeviewedas pointmassesmovinginaKerrbackground.Therefore,precis emodelingofEMRIsrequires knowingtheequationofmotionforapointmassinacurvedspa cetime. Derivingtheanalogousequationofmotionforthegravitati onalcasewasaccomplished bytwogroupssimultaneouslyin1997.Workingseparately,t hegroupsofQuinnandWald (QW)[ 39 ],andthatofMino,Sasaki,andTanaka(MST)[ 40 ],providedthreeindependent derivationsofthesameequationofmotion.Thisequationha scometobeknownasthe MiSaTaQuWaequation.Adescriptionoftheireortsisprovi dedhere.Goingthrough theirdetailedcalculationswillnotbenecessarytounders tandtherestofthisdissertation. Theinterestedreaderisinvitedtorefertotheoriginalpap ersandPoisson'sreview[ 38 ]. Eachofthesederivations,however,havebeenpointedoutto makesomeunjustied assumptions.Foremostofwhichisallowingaviolationofth elinearizedEinsteinequation inordertoaccommodatenon-geodesicsources.Whendealing withthelinearizedEinstein equation: G [1]ab [ h ]=8 a u b (3) ( x i z i ( t )) p g ; (2{97) thelinearizedBianchiidentityimpliesconservationofth epoint-particlestress-energy, whichinturnforcestheworldlineoftheparticletobeageod esicofthebackground spacetime.Thus,therearenosolutionstoEq.( 2{97 )forparticlesonaccelerated world-lines,whichmakesituselessasastartingpointford erivingcorrectionstogeodesic 53

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motion.Tocircumventthisdiculty,eachofthederivation schoosestoworkinLorenz gauge,therebysplittingEq.( 2{97 )into 2 h ab 2 R c ab d h cd = 16 Mu a u b (3) ( x i z i ( t )) p g (2{98) r b h ab =0 ; (2{99) where h ab h ab 1 2 g ab h isthetrace-reversedmetricperturbation.Withthissplit ,itis thenpossibletondsolutionstoEq.( 2{98 )forpoint-particlesourcesonanyworldline. ItisonlywhenonesimultaneouslyimposesEq.( 2{99 )thatgeodesicmotionbecomes inescapable. Assuch,thestrategyofmostderivationshasbeentoallowfo rviolationsofEq. ( 2{99 ),knownasLorenzgaugerelaxation,whilearguingthatsinc edeviationstogeodesic motionaresmall,theviolationoftheLorenzgaugeshouldli kewisebesmallsothatto thedesiredaccuracy,whatresultsisstillasolutiontothe originalsystemofequations.A recentderivationaddressingalltheseconcernscanbefoun din[ 41 ]. Inanycase,allderivationsultimatelyyieldthesameresul t(inLorenzgauge): u b r b u a = ( g ab + u a u b ) r d h tailbc 1 2 r b h tailcd u c u d (2{100) h tailab ( )= M Z 1 G +aba 0 b 0 1 2 g ab G +c c a 0 b 0 ( z ( ) ;z ( 0 )) u a 0 u b 0 d 0 (2{101) Theaccelerationduetotheself-forcerepresentsadeviati onawayfromgeodesic motioninthebackgroundspacetime,butassuggestivelywri ttenhere,canbeviewedas geodesicmotionintheperturbedspacetime.Wenotethatunl iketheelectromagneticcase, theself-forceisentirelyduetothetailpartofthelineari zedmetricperturbation. 2.3.1AxiomaticApproach ThederivationbyQWismotivatedbythefactthatthediverge ntterms(inthe r 0limit)ofthefulleldsurroundingapointcharge(ormass) dependsonlyonthe four-velocityandfour-accelerationoftheparticle.Ther efore,solongasoneappropriately 54

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identies u a and a a ofparticlesmovingalongdierentworld-lines,thediere nceintheir eldsmustbefreeofthesedivergences.QWpromotethisobse rvationtothelevelofan axiom. TherstoftheQWaxiomsisthatatanygivenpointinspacetim e,thedierence betweengravitationalforces f a and ~ f a onpointmassesmovingalongdierentworld-lines (onpossiblydierentspacetimes),isgivenbythefollowin g: f a ~ f a =lim r 0 1 2 r a h bc r b h a c 1 2 r a ~ h bc r b ~ h ac u b u c ; (2{102) where h ab and ~ h ab arethemetricperturbations(onthedierentbackgrounds) thatarise duetothepointmasses.Itisessentialthoughforthe f u a ;a a g and f ~ u a ; ~ a a g tomatch. ThesecondaxiomisthatinMinkowskispacetime,iftheworld lineisuniformly acceleratingandif h ab isthehalf-retarded,half-advancedsolution,then f a =0. Tobeabletogettheself-forceatapointalonganytrajector yinanarbitrary spacetime,allonehastodoistake u a and a a atthatpoint,andthenconsideranother pointalongauniformlyacceleratingworldlineinMinkowsk ispacetimewheretheparticle hasthesamefour-velocityandfouracceleration.Toproper lycomparetheeldsaround thetwopointsoneusesRiemannnormalcoordinatesaroundth ecurvedspacetime (therebymakingitlocallyMinkowski)andexpressestheHad amardexpansion(tobe discussedbelow)of h ab intermsofthem. Bythesecondaxiom ~ f a =0,sothatthegravitationalself-forceispreciselyjust therighthandsideofEq.( 2{102 ),where h ab isjustwhatyougetfromtheHadamard expansionwritteninRiemannnormalcoordinates,and ~ h ab isjustthehalf-retarded, half-advancedsolutionofthelinearizedEinsteinequatio nwithapointmasssourcein Minkowskispacetime. Thisderivationofthegravitationalself-forceistheleas tcumbersome.Andit essentiallyentailsusingtheMinkowskihalf-retarded,ha lf-advancedeldasaregularizing eldincurvedspacetime.Byapproachingthisaxiomaticall y(andnotfromphysical 55

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principles),whattheQWderivationachievesisarigorousd erivationthatunfortunately cannotbethenalwordonthegravitationalselfforce.Itma inlyshiftstheburden awayfromndinganexpressionforthegravitationalselffo rcetoguringoutarigorous justicationoftheaxiomstheyemployed.2.3.2ConservationofEnergy-Momentum TherstoftwoderivationspresentedbyMSTessentiallyrep eatsthestepsof DeWittandBrehme.Theyareabletodothisbytreatingthepro blemasthatofapoint massinteractingwithalineareldtheory(wheretheeldis themetricperturbation) propagatingonacurvedspacetime.Undertheassumptiontha tthemassofthemoving pointissmallcomparedtothecentralblackhole,thespacet imecanbethoughttobe g (B) ab + h ab ,where h ab isthedistortioninducedonthebackground(blackhole)spa cetime g ab .Withthis,onecanthenlinearizetheEinsteinequationin h ab .Moreover,wheninstead dealingwiththetracereversedmetricperturbation r a b ,andthenimposingtheLorenz gauge r ab ; b =0,theequationgoverning r ab isjust: 2 r ab +2 R c a d b r cd = 16 T ab : (2{103) where T ab isthestress-energyoftheparticlegivenby T ab = m Z r g a a 0 ( x;z ) g b b 0 u a 0 u b 0 4 ( x;z )d : (2{104) Exceptforextraindices,thisisequivalenttotheelectrom agneticcase.Sothe procedureforgettingtheequationofmotionisthesame.The ydeterminethecorresponding Green'sfunctiontothelinearizedEinsteinoperator,andc omputetheadvancedand retardedmetricperturbationsduetothepointmasssource. Thenexttaskthenwouldbe tondaconservationlaw,whichwhenintegratedoverthewor ldtubethatisthenmade innitesimallythin,yieldstheequationofmotion. 56

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2.3.3MatchedAsymptoticExpansions Inthisderivation,rstperformedby[ 40 ],andrepeatedin[ 38 ],thegoalistonda perturbativesolutiontotheEinsteineldequations.Thep roblemofasmallblackhole movinginanexternaluniverseischaracterizedbytwolengt hscales:themassofthesmall blackhole m andtheradiusofcurvatureoftheexternaluniverse R .Theproblem,of course,callsfor m< R .Theselengthscalesnaturallysplitthedomainintotwomai n regions,theinnerzone( r R )andouterzone( r m ),relyingondierentexpansion parameters( r= R and m=r )withwhichtodetermineanapproximatesolution.Intheinn er zone( r R ),thepictureisthatofablackholebeingtidally-distorte dbytheexternal universe.Aninnermetricisfoundbysolvingtheequationso fblackholeperturbation theorytorstorderin r= R .Thenintheouterzone,wherethepresenceofablackhole cannotbedistinguishedfromtheperturbationproducedbya pointmass,onesolvesthe linearizedeldequationfortheperturbationsduetothepo intmasstorstorderin m=r Theassumptionisthenthatthereexistsabuerregion( m r R )whereinboth solutionsarevalid.Andsotheremustexistacoordinatetra nsformationthattakesone fromtheinnersolutiontotheoutersolution.Thiscoordina tetransformationwilldepend onthepropertiesoftheparticle'sworldline.Itthenturns outthat,whileinsistingon regularityofthesolution,thiscoordinatetransformatio nisnotalwayspossible.Thatis, auniformlyvalidsolutionoftheperturbedEinsteinequati onwillnotexistforarbitrary motionsofthepointmass.Itwillonlyexistwheneverthewor ldlinehasanacceleration preciselygivenbytheMiSaTaQuWaequation. Thisderivationisbyfarthemostrigorousofthethreeandth emostelegant.Itis well-knownthatingeneralrelativitytheintegrabilityof theeldequationsimposesthe specicequationsofmotionofitssources.Thisisthespiri tofthematchedasymptotic expansionstechnique.Theequationofmotionforasmallbla ckholeincurvedbackground spacetimearisesfromanattempttondauniformlyvalidsol utionoftheeldequations overtheentirephysicaldomain. 57

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2.4Green'sFunctionsfortheWaveEquationinCurvedSpacet ime Theprevioussectionsmakeclearthat,eversinceDirac'spi oneeringstudyofradiation reactiononachargemovingthroughratspacetime,self-for ceanalyseshavefoundit convenienttousetheGreen'sfunctionsofthewaveoperator .Inthissection,borrowing muchfrom[ 38 ],wecollecttheGreen'sfunctionsthatarecommonlyencoun teredinthe self-forceliterature.Someofthemhavebeendiscussedint heprevioussections,buthere weprovideanextendeddiscussiontobettermotivatetherem ainderofthisdissertation. Fornotationalsimplicity,allexpressionsareforthescal arwaveequation: 2 ( x )= 4 ( x ) : (2{105) Retarded( G + ( x;x 0 ))andadvanced( G ( x;x 0 ))Green'sfunctionsaresolutionsto 2 G ( x;x 0 )= 4 4 ( x;x 0 )(2{106) thatcorrespondtooutgoingandingoingwavesatinnity,re spectively.Theyshare thesamesingularitystructureatcoincidencesincetheird ierenceisasolutiontothe homogeneouswaveequationandisthusguaranteedtobesmoot hatcoincidence.They alsosatisfythereciprocityrelation: G ( x 0 ;x )= G + ( x;x 0 ) ; (2{107) aproofofwhichcanbefoundin[ 38 ]. Foranysource ( x 0 ),theretardedGreen'sfunctiongivesthephysicallyrelev ant solutiontoeldequations: ret ( x )= Z G ( x;x 0 ) ( x 0 ) p g ( x 0 )d 4 x 0 : (2{108) Inratspacetime,theretarded/advancedGreen'sfunctions assumeaverysimple form: G ( x;x 0 )= ( ( t t 0 )) ( )(2{109) 58

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whichhavesupportonlyonthenullconeof x ,thelocusofpoints x 0 ,suchthat =0. Forascalarpointcharge ( x )= q Z 4 ( x;z ( ))d ; (2{110) theseyieldthesimpleretarded(+)andadvanced(-)elds: = q Z t z 0 ( ) ( ) d (2{111) = q ( x z ( )) a u a ( ) = ret/adv (2{112) whichmaybecomparedwiththeLienard-Wiechertpotentials duetoanelectricchargein eq.( 2{16 ).Theyareclearlysingularontheworld-line. Forcurvedspacetime,theGreen'sfunctionsbecomeabitmor ecomplicated, extendingitssupporttothecausalpastandfutureofthe x .Scalarwavesgetbackscattered bythecurvatureofspacetimeandinteractwiththeemitting chargeatalatertime.For self-forceconsiderations,theseGreen'sfunctionsareon lyneedednearcoincidence, wellwithinthenormalneighborhoodofthecharge.Itissuc ientthentoconsidera constructionoftheseGreen'sfunctionswell-denedonlyi nanormalneighborhoodof x 0 suchistheHadamardconstructionexplainedbelow. Following[ 38 ],westartwiththeansatz G ( x;x 0 )= U ( x;x 0 ) ( )+ V ( x;x 0 ) ( ) ; (2{113) where U ( x;x 0 )and V ( x;x 0 )areassumedtobesmoothbiscalars. ( )and ( )are denedtobe: ( )= ( x; ( x 0 )) ( )(2{114) ( )= ( x; ( x 0 )) ( ) ; (2{115) where + ( x; ( x 0 ))isdenedtobeonewhen x isinthefutureofthespacelikehypersurface ( x 0 )that x 0 belongsto,while ( x; ( x 0 ))=1 + ( x; ( x 0 )). 59

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Theadditionofthesecondterm(whichisabsentinratspacet ime),whosesupportis withinthelightcone,ismotivatedbythephysicalconsider ationsmentionedabove. Thetasknowistodetermineoratleastcharacterizethebisc alars U and V .Plugging theansatzintoEq.( 2{106 ),weget: 2 G = 4 4 ( x;x 0 ) U + 0 ( ) f 2 U ;a a +( a a 4) U g + ( ) f 2 V ;a a +(2 a a ) V + 2 U g + ( ) f 2 V g (2{116) = 4 4 ( x;x 0 ) : (2{117) Fromthisitisclearthat lim x x 0 U ( x;x 0 )=1 : (2{118) Also,tomakethe 0 termvanish,wemustimpose 2 U ;a a +( a a 4) U =0 : (2{119) Itisshownby[ 38 ]thatthesetwoequationscompletelydetermine U ( x;x 0 ),forwhichthe uniquesolutionis U ( x;x 0 )= 1 = 2 ( x;x 0 ) : (2{120) NowwemustofthetermsinEq.( 2{116 )tozero,whichwillcharacterizeour remainingundeterminedcoecient V .Lookingatthe term,whichhassupportonlyon thelightcone,wedemandthat V ;a a + 1 2 ( a a 2) V = 1 2 2 U =0 (2{121) butonlyonthelightcone ( x;x 0 )=0;imposingthiselsewherewillunnecessarilyrestrict V .Therstterminthisequationisjust d V= d ,soitcanbeintegratedalonganynull geodesicemanatingfrom x 0 .Aninitialconditionneedstobeprovided,however,forthe resulttobeunique.From[ 38 ],weseethat U ( x;x 0 )nearcoincidencecanbeexpandedas U =1+ 1 12 R a 0 b 0 a 0 b 0 + O ( 3 )(2{122) 60

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where R a 0 b 0 istheRiccitensorat x 0 and istheaneparameterdistanceto x .Dierentiating this,using a 0 c 0 = a 0 c 0 + O ( )and a 0 c = g b 0 c a 0 b 0 :weget U ;a = 1 6 g a 0 a R a 0 b 0 b 0 + O ( 2 )(2{123) U ;a 0 = 1 6 R a 0 b 0 b 0 + O ( 2 ) : (2{124) Fromwhichwend 2 U = 1 6 R ( x 0 )+ O ( ) : (2{125) Therefore,takingthecoincidencelimitofEq.( 2{121 ),andnotingfrom[ 38 ]that lim x x 0 a a ( x;x 0 )=1,wehave lim x x 0 V ( x;x 0 )= 1 12 R ( x 0 ) : (2{126) Thisistheinitialconditionrequiredfortheintegrationo fEq.( 2{121 )alonggeneratorsof thelightcone.Wehavethusdemonstratedthat V isuniquelyspeciedonthelightcone. Finally,thelasttermofEq.( 2{116 )thatneedstovanishisthe ( )term,which issupportedontheinteriorofthelightcone.Thus,weneedt oimposethat 2 V ( x;x 0 )=0(2{127) ontheinteriorofthelightcone.Thatis, V ( x;x 0 )isahomogeneoussolutionofwave equationinthecausalpastandfutureof x 0 .Since V isuniqueonthelightcone,thesecan serveascharacteristicdataforthewaveequation. Beyondthis,wecangonofurther.Ouransatz,Eq.( 2{113 ),isavalidlocal constructionoftheretardedandadvancedGreen'sfunction sprovidedthat U ( x;x 0 )= 1 = 2 ( x;x 0 ),andthat V ( x;x 0 )isahomogeneoussolutionofthescalarwaveequationin curvedspacetime.Notethatthisisanexpressionvalidonly inthenormalneighborhoodof x 0 ,whichisclearfromthefactthatituses ( x;x 0 )thatitselfrequiresauniqueconnecting geodesicbetween x and x 0 .Also,fromthereciprocityofretardedandadvancedGreen' s 61

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functions,itisclearthatboth U and V aresymmetricwithrespecttointerchangingtheir arguments. Acompletely symmetricGreen'sfunction canbeeasilyfoundfromtheconstruction justprovided.Simplyremovingthedistinctionsbetweenre tardedandadvancedinthe ansatz,wegethalfthesumoftheretardedandadvancedGreen 'sfunctions: G sym = 1 2 [ G ( x;x 0 )+ G + ( x;x 0 )]= 1 2 U ( x;x 0 ) ( )+ 1 2 V ( x;x 0 )( )(2{128) BybeingaGreen'sfunction,itsharesthesamesingularitys tructureas G + and G .Its support,however,wouldincludealleventscausallyconnec tedto x .SeeFigure( 2-1 ). ThesymmetricGreen'sfunctionisimportantbecauseinrats pacetimeitisthepiece Diracidentiedtobeirrelevanttothemotionofthecharge. Sinceitsharesthesingularity structureoftheretardedeld,itcouldbethusbeusedtoreg ularizetheretardedeld, leavingthehomogeneouseldinvolvedinradiationdamping (recallEq. 2{50 ). Inratspacetime,theGreen'sfunctionfullydeterminingth ishomogeneouseldthat eectsradiationdampingis: G rad = G ret G sym ; (2{129) whichisknownastheradiativeGreen'sfunction.Theeldth isgivesrisetoislikewise knownastheradiativeeld.(NotethatinDirac'spaper,the radiativeeldisactuallyhalf whatwepresenthere.) Unfortunately,tosimplyadoptthesameprescriptionforcu rvedspacetimewould bemisguided.Aswehaveseen,thesymmetricGreen'sfunctio nextendsitssupportto theinteriorofthelightcone.A(naive)radiativeGreen'sf unctiondenedas G rad = G ret G sym wouldinheritthesupportofthesymmetricGreen'sfunction (Fig.( 2-1 )). Whilesuchachoicesucientlyregularizestheretardedel d,theradiativeeldat x will havethisphysicallyunpleasantfeatureofdependingnoton lyoneventsinitscausalpast, butonalleventsinitscausalfutureaswell.Intheself-for cecontext,thiswouldimply thatthescalarchargeinteractswithaelddependingoneve ntsyettohappen! 62

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Clearly,ndingthecorrectgeneralizationofDirac'sregu larizationisnon-trivial. Infact,thediscoveryofthecorrectprescriptionwasmadem uchlater[ 42 ]andisthe foundationformuchofthiswork. 2.5Self-ForcefromaGreen'sFunctionDecomposition ThecorrectsymmetricGreen'sfunction, G S ( x;x 0 ),isonethatwillbesimilarto (1 = 2)( G + + G ),buthavetheappropriatecausalcharacter.Specically, itmust vanishwhen x 2 I ( x 0 ),thechronologicalfutureandpastof x 0 ,toensurethatupon regularizationoftheretardedGreen'sfunction,thesuppo rtofthe(proper)radiative Green'sfunctionwillnotincludethechronologicalfuture of x Thiscanbeachievedbysimplysubtractingasuitablehomoge neoustwo-point function H ( x;x 0 )fromthenaiveradiativeGreen'sfunction. H ( x;x 0 )mustalsobe symmetricinitsarguments.Wenotethatthenaiveradiative Green'sfunctionisjusthalf theadvancedGreen'sfunctionwhen x 2 I ( x 0 ),anditisjusthalftheretardedGreen's functionwhen x 2 I + ( x 0 ).Therefore, H ( x;x 0 )mustagreewiththeretardedGreen's functionwhen x 2 I + ( x 0 ),andwiththeadvancedGreen'sfunctionwhen x 2 I ( x 0 ).These propertieswouldensurethat G S ( x;x 0 )= 1 2 ( G + ( x;x 0 )+ G ( x;x 0 )) 1 2 H ( x;x 0 )(2{130) isanothersymmetricGreen'sfunction,whosesupportisres trictedtowhere x= 2 I ( x 0 ).To avoidconfusingthisnewversionwiththeoriginalsymmetri cGreen'sfunction G sym ,thisis referredtoasthe singular Green'sfunction. Usingthistoregularizetheretardedeldyieldsanewradia tiveGreen'sfunction G R = G ret G S (2{131) thatvanisheswhen x 2 I ( x 0 ).(SeeFigure( 2-1 )).Again,toavoidconfusingthiswith theoriginal(naive)radiativeGreen'sfunction,thisisre ferredtoasthe regular Green's function. 63

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A. G + ( x;x 0 ),curved. B. G ( x;x 0 ),curved. C. G rad ( x;x 0 ),rat. D.(Naive) G rad ( x;x 0 ),curved. E. G S ( x;x 0 ),curved. F. G R ( x;x 0 ),curved. Figure2-1.ThesupportofthevariousGreen'sfunctionsare theshadedareas.Thered curveistheworldlineofthepointsource,thusshowingwhic hpartsofit contributetotheeldpoint x atthecenteroflightcone. 64

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Thequestionthenbecomeswhethersuchatwo-pointfunction H ( x;x 0 )existsatall. Aplausibilityargumentforglobalexistenceisprovidedin [ 38 ],butsinceourinteresthere istouse G S mainlyasaregularizingGreen'sfunction(andthereforeim portantonlyina smallneighborhoodofthecharge),weshallbecontentwith nding H ( x;x 0 )inaconvex normalneighborhoodof x 0 ReturningtotheHadamardconstructionoftheretardedanda dvancedGreen's functions,Eq.( 2{113 ),weseethat G + ( x;x 0 )= V ( x;x 0 ) ; if x 2 I + ( x 0 )(2{132) G ( x;x 0 )= V ( x;x 0 ) ; if x 2 I ( x 0 ) : (2{133) Since V ( x;x 0 )isahomogeneoussolutionofthewaveequation(seeEq.( 2{127 )),andis symmetricinitsarguments,itisclearthatitsatisesallt heconditionsfor H ( x;x 0 ). Therefore,solongaswerestrictourselvestoanormalneigh borhoodof x 0 ,weare guaranteedthelocalexistenceof H ( x;x 0 ),whereitissimply V ( x;x 0 ).Inthisnormal neighborhood,wendthatthesingularGreen'sfunctiontak estheform G S ( x;x 0 )= 1 2 U ( x;x 0 ) ( ) 1 2 V ( x;x 0 ) ( ) : (2{134) TheregularGreen'sfunctionsisthenlikewise G R ( x;x 0 )= 1 2 U ( x;x 0 )[ + ( ) ( )]+ V ( x;x 0 ) + ( )+ 1 2 ( ) : (2{135) Wehavethusachievedalocalregularizationoftheretarded Green'sfunctionforthe waveequationincurvedspacetime,analogoustowhatwasdis coveredinDirac'sanalysis ofthemotionofanelectricchargeinratspacetime.Whatrem ainstobeshownisthat theeldarisingfromthisnewregularGreen'sfunctionsisa llthatisresponsibleforthe self-forceonthemovingcharge,muchlikeratspacetimecas ewherein(intheabsenceof otherexternalelds)amovingchargeisseentointeractsol elywiththeradiativeeld. 65

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Todemonstratethatthisisso,werevisittheequationsofmo tionthatbeenderived previously(Eqs. 2{96 and 2{100 ).Forcompleteness,wealsoincludetheequationsof motionforascalarchargerstderivedbyQuinn[ 43 ].Supposingthatthebackgroundisa vacuumspacetimetheseare: a a = 1 m f a ext + q 2 m ( a b + u a u b ) 1 3 m f a ext + 1 6 R a b u b + Z 1 r b G + ( z ( ) ;z ( 0 ))d 0 (2{136) a a = 1 m f a ext + q 2 m ( a b + u a u b ) 2 3 m f a ext + 1 3 R a b u b + 2 q 2 m u b Z 1 r [ a G + b ] c 0 ( z ( ) ;z ( 0 )) u c 0 d 0 (2{137) a a = ( g ab + u a u b ) r d h tailbc 1 2 r b h tailcd u c u d where h tailab ( )= M Z 1 G +aba 0 b 0 1 2 g ab G +c c a 0 b 0 ( z ( ) ;z ( 0 )) u a 0 u b 0 d 0 (2{138) forascalarpointcharge,electricpointcharge,andpointm ass,respectively,wherethe termsinvolvinginvolving_ a a havebeenreplaced f a ext =m ,accordingtothereduction-of-order techniquementionedabove. Giventheformofeachoftheseequations,itistemptingtoin terprettheself-forceas theresultofthechargeinteractingwitharegularself-el d self consistingofa\direct" part(fromeldcontributionsonthelightconeitself,whic hgivethelocalterms)anda \tail"part(fromeldcontributionsintheinteriorofthel ightcone,givingthenon-local eects).Thus,forthecaseofthescalarcharge,theself-fo rceonit[whichisallbutthe rsttermofEq.( 2{136 )]isdescribedsimplyaslim x z r a self ,where self = dir + tail (2{139) dir qU ( x;z ( )) 2_ adv ret (2{140) tail q Z ret 1 V [ x;z ( )]d =lim 0 + q Z ret 1 G + [ x;z ( )]d ; (2{141) where ret and adv are,respectively,theintersectionsofthepastandfuture lightconeof x withtheworld-lineofthecharge. 66

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Thisinterpretationisrathershaky,sincethegradientoft hetailpiecedoesnot haveawell-denedcoincidencelimit.Whilethedirectpiec eturnsouttobeniteand dierentiableatcoincidence,whosegradientgivestheana logueoftheAbraham-Lorentz-Dirac forceonthecharge,wemaycalculatethegradientofthetail pieceandget: r a tail = q Z ret 1 r a V d + qV ( x;z ( ret )) r a ret : (2{142) With V beingasmoothfunction,thecoincidencelimitoftherstte rmiswell-dened. Thesecondtermhoweverbehavesas qV ( x; ret ) r a ret = q [ V 1 r a ] j ret = qR ( x ) 12 r ( x a z a )+ O ( r ) ;x a z a ; (2{143) whichisadirection-dependentlimit.Togetaroundthispro blem,onerstaverages r a self aroundasmall,spatialtwo-spheresurroundingtheparticl ebeforetakingthe coincidencelimit.Thisaveragingremovestheproblematic contributionofthetailpart. Andso,theinterpretationofthechargeinteractingwithth e self dependscriticallyonthis averagingstep.Moreover,itisclearthat self isnotahomogeneoussolutionofthewave equation,sothatanobservercomovingwiththeparticlewou ldexpectascalarcharge distributioninitsneighborhoodneededtoactasthesource for self Consider,however,theeldsarisingfrom G S and G R .FromEqs.( 2{134 )and ( 2{135 ),weget S ( x )= qU ( x;z ( )) 2_ ret + qU ( x;z ( )) 2_ adv q 2 Z adv ret V ( x;z ( ))d (2{144) and R ( x )= ret S = qU ( x;z ( )) 2_ adv ret + q Z ret 1 + 1 2 Z adv ret V ( x;z ( ))d : (2{145) 67

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Byconstruction, R isahomogeneoussolutionofthewaveequation.Moreover,it isclear that self and R arerelatedaccordingto R = self + q 2 Z adv ret V ( x;z ( ))d : (2{146) Fromthisitisapparentthatatcoincidence R and self match,whichprovesthat R is solelyresponsiblefortheself-forceonthechargejustlik e self .Theintegraltermcloseto coincidenceisapproximatedtobe qrV ( x;x )+ O ( r 2 )= 1 12 qrR ( x )+ O ( r 2 ) : (2{147) Itsderivativegivesaninward-pointing,spatialunitvect orneartheworldline.This ispreciselywhatisneededtocancelthenon-dierentiable pieceof self ,shownin Eq.( 2{143 ).Therefore,itismorenaturaltointerprettheself-force asaresultofthe interactionbetweenthechargeand R ,asitreliesonnoaveraging.Moreover,sincethe regulareldisasmoothhomogeneoussolutionjustlikeDira c'sradiativeeldinrat spacetime. Thisnewperspectiveontheself-force,asonearisingfromt heinteractionofthe chargewithahomogeneouseld R wasdiscoveredandrstenunciatedbyDetweilerand Whiting[ 42 ].Itisthecorrectcurvedspace-timegeneralizationofDir ac'spicture. Mode-sumcalculationoftheself-force Inpracticalterms,foronetocalculateaself-force,theDW perspectivetakestheview thatonemustaimtoretrievethesmooth,homogeneous,regul areld R fromwhichthe self-forceisjustitsgradient.Aswewillseeinthenextcha pter,thisperspectivelends itselftoaprescriptionforself-forcecalculationwithti me-domainandpowerful,existing (3+1)codes.PriortoDW'sdiscovery,thefocushadbeendie rent.Sophisticatedmethods weredevisedtocomputethetailpartoftheself-force,whic hisjustthegradientofthetail 68

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eldinEq.( 2{141 ). F tail a =lim 0 + Z ret 1 r a G + ( x;z ( ))d (2{148) andofthesemethodsthemostsuccessfulistheso-calledmod e-sumscheme,pioneered byBarackandOri[ 44 { 46 ].Mostconcretestrong-eldcalculationsofself-forceha ve,in fact,beenperformedusingthisapproach,anditisbutappro priatethatweprovideabrief discussionhere. Themode-sumschemeexploitsthefactthatwhiletheretarde delddivergesatthe locationofthecharge,its l -modes(inaspherical-harmonicdecomposition)donot.It startswithsplittingthetailcontributioninEq.( 2{148 )intowhattheyrefertoasthe \full"and\direct"parts. F tail a = F full a F dir a (2{149) F full a =lim 0 + Z ret + 1 r a G + ( x;z ( ))d (2{150) F dir a =lim 0 + Z ret + ret r a G + ( x;z ( ))d : (2{151) [Notethatthe\direct"partoftheforcehereisnotthesamea sthat\direct"partof Eq.( 2{140 ).Whiletheyarebothduetowavespropagatingalongthenull coneofthe world-lineof x ,ourprevious\direct"piecewasthedierencebetweenadva ncedand retardedcontributions.Thisdierenceisregularanddie rentiableatcoincidence,even whiletheadvancedandretardedpartsareindividuallydive rgent.] Onenoticesimmediatelythatbothelds F full a ( x )and F dir a ( x )aredivergentat coincidence.Theymustsharethesamesingularitystructur eatcoincidence,fortheir dierencetoyieldthewell-denedtailcontributionofthe self-force. Thebenetofthissplittingintotwodivergentpiecesliesi nthefactthat,at coincidence,the\full"force(oftencalledthebareforce) isjustthegradientofthe divergentretardedeldattheparticle'slocation.Toavoi dthedivergence,oneformally 69

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employsamultipoledecompositionof F full a : F full a ( x )= 1 X l =0 F (full) l a ( x ) ; (2{152) where,atcoincidence, F (full) l a =lim x z l X m = l r a ret lm ( x ) ; (2{153) and ret lm ( x )arejustthespherical-harmoniccomponentsoftheretarde deld.Wesay formal,becauseofcoursethesuminEq.( 2{152 )isdivergent.However,eachofitsmodes inEq.( 2{153 )turnouttobenite(butdiscontinuous)at x z .Moreover,thesemodes areaccessiblebynumericallysolvingthereduced(1+1)wav eequationresultingfroma spherical-harmonicdecompositionappliedtothequantiti esenteringthefullscalarwave equation. Withthis,thechallengethenistondaregularizationfunc tion h la thatwouldmake P l [lim x z F (full) l a ( x ) h la ](or P l [lim x z F (ret) l a ( x ) h la ])convergent.Inwhichcase,onehas F self a = X l (lim x z F (full) l a ( x ) h la ) X l (lim x z F (dir) l a ( x ) h la )(2{154) = X l (lim x z F (ret) l a ( x ) h la ) X l (lim x z F (dir) l a ( x ) h la )(2{155) andonehastwoconvergentinnitesums,therstofwhichisn owfullycalculableonce themodecontributionsoftheretardedeldareavailable. Inprinciple,onecandetermine h la bylookingthelargel behaviorofthemodesof F (ret) l a .Butsince F (dir) a sharesthesamesingularitystructureas F (ret) a (byconstruction), itisdeducedbyanalyzingthedirectpieceinstead,sinceit isalocalobject(unlikethe retardedpiece),whosebehavioristhusamenabletoanalyti caltreatment. Inallcasesconsidered,theregularizationfunctionisfou ndtohavethegeneralform h la = A a l + 1 2 + B a + C a l + 1 2 ; (2{156) 70

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wherethecoecients f A a ;B a ;C a g are l -independentquantities,calledregularization parameters,whoseexpressionsdependonthedetailsofthet rajectoryofthecharge. Calculatingtheself-forcethenamountstoevaluating F self a = 1 X l =0 lim x z F (ret) l a ( x ) A a l + 1 2 B a C a l + 1 2 # D a ; (2{157) where D a 1 X l =0 lim x z F (ret) l a ( x ) A a l + 1 2 B a C a l + 1 2 # : (2{158) Itcanbeshownthatas l !1 ,the D a -termcontributionbecomesatelescopingseries thatvanishes.Therefore,inpractice,onejustneedstogoa shighan l aspossibletogeta goodestimatefortheself-force. Foreaseofexposition,welistthestepsinvolvedinamode-s umcalculationofthe self-force: 1. Foraspecicorbitonaspecicbackground,determineanaly ticallytheregularization parameters. 2. Expandtheretardedeld ret ( x )intermsofsphericalharmonics,withcoecients ret lm ( t;r ) ret = X ret lm ( t;r ) Y lm ( ; ) ; (2{159) aswellasthedeltafunctionpoint-source. 3. Solvethereducedwaveequationforthe lm coecientsoftheretardedeld. 4. Compute F ret l a P lm = l r a ret lm Y lm .Thesewillbeniteatthelocationofthe charge,butthesumover l diverges. 5. Regularizethe l -modeexpansionofthe r a ret P l F ret lm bysubtracting l + 1 2 A a + B a + C a l + 1 2 (2{160) foreach l 6. Sumtheregularized l contributionstoashighan l asdesired.Byconstruction, thissumwillconverge,butitdoessoas1 =l n ,where n dependsonhowmany regularizationparametersoneuses. 71

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OnehasachoiceinhowtoperformStep2ofthemode-sumrecipe {tosolvethe (1+1)reducedscalarwaveequationeitherintime-domainor frequency-domain. Infrequency-domain,onegoesonestepfurtherinFourierde composingthespherical harmoniccomponents,andsolvesanordinarydierentialeq uationforeach lm! -component oftheretardedeld.Forrelativelysimpleorbits(e.g.cir cularorbitsandslightlyeccentric orbits),frequency-domainissuperior,becausethesimple temporalbehaviorofthesource restrictsitsFouriercomponentstoonlyafewfrequencies .Thenumericalmethods neededtosolveODEsarealsoconsiderablymuchsimplerandl esscomputationally demandingthantheevolutioncodesforhyperbolicpartiald ierentialequations.Forthe samesetofresourcesthen,aFDapproachwouldyieldexcepti onallymoreaccurateresults Butfrequency-domaincalculationssoonbecomeimpractica lfororbitsthataremore complicated(e.g.highlyeccentricorbits),becausetheso urcegetsspreadouttomany moretermsinaFourierseries.Inthisinstance,itbecomesr easonabletoavoidtheFourier decompositionaltogether,andjustsolveforeach lm -componentbyevolvinga(1+1)wave equation.Thislatterapproachiswhatwouldbecalledatime -domaincalculation. Manyself-forcecalculationshavebeendonethroughtheFDa pproach,whileonly recentlyhaveresearchersfocusedontime-domaincalculat ions(atthetimeofwriting,we knowofonlythree[ 47 { 49 ]),motivatedbytheexpectationthatEMRIsarelikelytobe highlyeccentric. Thelaststepofthemode-sumrecipeisalsoworthnoting:amo de-sumcalculation ofaself-forcerequireshavingtosumthecontributionsofe achregularizedmode,which convergesslowly.StudiespriortotheDWdecompositionrep ortthissumconvergeas1 =l whichconsequentlyrequiredsolvingthe(1+1)waveequatio nformanymodesinorderto reachadesiredaccuracy. ThroughtheDWdecomposition,amode-sumschememayalsobep ursued.However, sincethegoalistoconstruct R atthelocationofthecharge,themultipoledecomposition ofthesingulareldthenservesasanaturalregularization functionforthedivergent 72

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retardedeld.Thatis,ifone,say,breaksupthesingulare ldinsphericalharmonics: S ( x )= 1 X l =0 m = l X m = l S lm ( t;r ) Y lm ( ; )(2{161) thenonemayview h l as: h la = l X m = l lim x z r a ( S lm Y lm ) ; (2{162) where z isthelocationofthecharge. Fromthisperspective,thersttaskistondananalyticfor mfor S ( x )uponwhich amultipoledecompositioncanbeappliedinordertodetermi neregularizationparameters. Ingeneral,onecanonlyapproximatethesingulareld,soth atoneislimitedinthe numberofregularizationparametersthatcanbedetermined aheadoftime.Suchan implementationwasperformedby[ 50 ]. Onemightnaturallyaskwhatadvantageliesinadoptingthis point-of-view.One thingitachievesisfurtherclarifytheroleofregularizat ionparameters.Sincethe exactregulareldisahomogeneoussolutionandhencesmoot hatthelocationofthe charge,thesingulareldisseentocontainallthepartsoft heretardedeldthatarenot onlydivergentbutalsoofnitedierentiability.Therefo re,sincetheregularization parametersarisefrommultipolecomponentsofthegradient ofthesingulareld, theyarearepresentationofthesedivergentandnon-smooth pieces.Usingmoreof theregularizationparametersyieldsaremainderofhigher dierentiability,andthusa mode-sumoffasterconvergence. Themode-sumofa C 1 -eld(liketheregulareld R )convergesexponentiallyin l .Butaccesstotheexact R ispossibleonlywhenoneknowstheexact S ,whichisin generalnotpossible.ForSchwarzschildandKerrspacetime s,inparticular,onecanonly locallyapproximatethesingulareldtoacertainorderin = R (where isameasureof distanceawayfromthechargelocationand R istheradiusofcurvatureofthebackground spacetime).Therefore,forcasesofastrophysicalinteres t,onecanonlycomputealimited numberofregularizationparameters.Thisinturnmeanstha ttherewillalwaysbea 73

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non-smoothpieceintheremainderwhichkeepstheconvergen ceofthemode-sumfrom beingexponential. Thisaspectofthemode-sumschemeisoneofitsmaindiculti es.Thebestonecan doiscalculatehigherorderregularizationparameters,in ordertoimproveconvergence. Butdoingthisisnotatrivialtask.Inaddition,themode-su mschemeappearsill-suitedto thetaskofeventuallyusingthecomputedself-forceinupda tingtheorbitoftheparticle. Toputthisbackreactionintoeect,itisenvisionedthatac atalogueofself-forcevalues wouldbeproducedforavarietyoforbitalparameters.Thisc atalogueshallthenbeused toupdateorbitsinlatersimulations.Thus,inthisscheme, backreactionis(ineciently) renderedasapost-processingstep. Adesirableprescriptionwouldbeonethatavoidsbothprobl ems;onethatdoesnot havetogothroughamodedecompositionoftheeldsattheout set,andoneallowing quickaccesstotheself-forceateveryinstantsothatitmay bedirectlyusedtoupdatethe trajectoryofthecharge. However,withthedivergenceoftheretardedeldattheloca tionofthecharge,it appearsthatgoingthroughamodedecompositionisnecessar yforonetodealexclusively withnitequantities.Weshallshowinthenextchapterthat thisneedisonlyapparent. ByemployingtheDWdecomposition,thereisawaytoavoidthe divergentretarded eldaltogethersothatonedealsentirelywithregulareld sandregularsourcesforpoint particles.Suchisthekeycontributionofthisdissertatio n.Itisthiscapabilitythatopens thedoortocalculationsofself-forceandsimulationsofth eself-consistentdynamicsof pointparticlesincurvedspacetimewithexisting(3+1)com putationalinfrastructure. 74

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CHAPTER3 REGULARIZEDPOINTSOURCESFORSELF-FORCECALCULATIONS [Portionsofthischaptercanbefoundinthefollowingartic le:Regularizationofelds forself-forceproblemsincurvedspacetime:Foundationsa ndatime-domainapplication. I.VegaandS.Detweiler,Phys.Rev.D 77 ,084008(2008).Copyright(2008)bythe AmericanPhysicalSociety.] Inthepreviouschapter,weintroducedthemode-sumapproac htocalculating self-forcesonpointparticlesmovingincurvedspacetime. Thisprescriptionisthemost reliablemethodforself-forcecalculationandhasbeenapp liedtoawidevarietyofcases (e.g.scalar/electricchargesingenericgeodesics;point massesinradialinfallandcircular orbits),andhasledtomanyinsightsconcerningself-force phenomena.However,the methodappearsill-suitedforthegoalofsimulatingtheful lyself-consistentdynamicsof pointchargesincurvedspacetime,because(a)itrequiresa slowly-convergentmodesum toretrievetheself-forceonachargeatacertaininstant,a nd(b)isratherinconvenientfor self-consistentlymodifyingtheparticletrajectoryusin gthecomputedself-force. Inthischapter,wepresentanalternativemethoddesigneds pecicallytoaddress theseconcerns.Theprescriptionwehavedevelopedisasimp lesmearingoutofthe delta-functionsourcethatismadepossiblebytheappropri atesplittingoftheretarded eldintoregularandsingularparts.Thetechniqueisbesti mplementedwitha(3+1) evolverofthewaveequation,withwhichit(a)allowseasyac cesstotheself-forceona particle(withnomode-sumrequired)andtotheradiatione ldinthewavezone,and(b) avoidsdelta-functionsandsingulareldsaltogether.Wes eebothpropertiesasessential towardsachievingsimulationsoftheself-consistentmoti onofpointparticles,inclusiveof self-forceeects. 3.1GeneralPrescription Foreaseofexpositionweshallillustratetheprescription usingascalarchargemoving incurvedspacetime.Analogoustotheelectromagneticandg ravitationalcases,fora 75

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scalarcharge,thegeneralstrategyforcomputingtheselfforcerstinvolvessolvingthe minimally-coupledscalarwaveequationwithapointcharge q source, r a r a ret = 4 q Z r (4) ( x z ( )) p g d; (3{1) fortheretardedeld ret .Here r a isthederivativeoperatorassociatedwiththemetric g ab ofthebackgroundspacetimeand r istheworldlineofthechargedenedby z a ( )and parameterizedbythepropertime .Thephysicalsolutionoftheresultingwaveequation willbearetardedeldthatissingularatthelocationofthe pointcharge.Asmentioned previously,aself-forceformallyexpressedas F a ( )= q r a ret ( z ( ))(3{2) willneedaregularizationprescriptiontomakesense.Earl yregularizationprescriptions [ 37 39 40 ]werebaseduponaHadamardexpansionoftheGreenfunction, andshowed thatforaparticlemovingalongageodesictheselfforcecou ldbedescribedintermsofthe particleinteractingonlywiththe\tail"partof ret ,whichisniteattheparticleitself. Later[ 42 ]itwasrealizedthatasingularpartoftheeld S whichexertsnoforceonthe particleitselfcouldbeidentiedasanactualsolutiontoE q.( 3{1 )inaneighborhoodof theparticle.Aformaldescriptionof S intermsofpartsoftheretardedGreen'sfunction [ 42 ]ispossible(seeEq. 2{134 ),butgenerallythereisnoexactfunctionaldescriptionfo r S inaneighborhoodoftheparticle.Fortunately,aswillbesh owninthenextsection,an intuitivelysatisfyingdescriptionfor S resultsfromacarefulexpansionaboutthelocation oftheparticle: S = q= + O ( 3 = R 4 )as 0 ; (3{3) where R isaconstantlengthscaleofthebackgroundgeometryand isascalareld whichsimplysatises 2 = x 2 + y 2 + z 2 inaveryspecialMinkowski-likelocallyinertial coordinatesystemcenteredontheparticle,rstdescribed byThorne,HartleandZhang [ 51 52 ]andappliedtoself-forceproblemsinRefs.[ 50 53 54 ].Thesecoordinatesare 76

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explainedinthenextsectionandinAppendixA.Notsurprisi nglythesingularpartofthe eld,whichexertsnoforceontheparticleitself,appearsa sapproximatelytheCoulomb potentialtoalocalobservermovingwiththeparticle. OurproposalforsolvingEq.( 3{1 ),anddeterminingtheself-forceactingbackonthe particlenowappearselementary.Firstwedene ~ S q= (3{4) asaspecicapproximationto S .Byconstruction,weknowthat ~ S issingularatthe particleandis C 1 elsewhere.Also,withinaneighborhoodoftheworldlineoft heparticle r a r a ~ S = 4 q Z r (4) ( x z ( )) p g d + O ( = R 4 ) ; as 0 : (3{5) ItmustbepointedoutthatforEqs.( 3{3 )and( 3{5 )tobevalid,theThorne-Hartle-Zhang (THZ)coordinatesmustbeknowncorrectlyto O ( 4 = R 3 ).IftheTHZwereknowntobe correctonlyto O ( 3 = R 2 ),thiswouldspoiltheremainderinEq.( 3{5 ),whichwould thenhaveanon-vanishingdirectiondependentlimitas 0.Thispointsthen totheimportanceofdeterminingthelocalcoordinateframe preciselyenoughforthe Coulomb-likepotentialtobeagoodrepresentationofthelo calsingulareld. Next,weintroduceawindowfunction W whichisa C 1 scalareldwith W =1+ O ( 4 = R 4 )as 0 ; (3{6) and W 0sucientlyfarfromtheparticle,inparticularinthewave zone.The requirementthat W approaches1thisway,i.e. O ( 4 ),isexplainedbelow. Finallywedenearegularremaindereld R ret W ~ S (3{7) 77

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whichisasolutionof r a r a R = r a r a ( W ~ S ) 4 q Z r (4) ( x z ( )) p g d (3{8) fromEq.( 3{1 ). R wouldbeanapproximationofDetweiler-Whiting'ssmoothre gular eldthatisresponsiblefortheself-force.Becauseofouru seof ~ S asanapproximationof thefullsingulareld, R wouldbecontaminatedwith O ( 3 = R 4 )-piecesthatareonly C 2 at thelocationofthecharge. Theeectivesourceofthisequation S e r a r a ( W ~ S ) 4 q Z r (4) ( x z ( )) p g d (3{9) isstraightforwardtoevaluateanalytically,andthetwote rmsontherighthandsidehave delta-functionpiecesthatpreciselycancelatthelocatio nofthecharge,leavingasource whichbehavesas S e = O ( = R 4 )as 0 : (3{10) Thustheeectivesource S e iscontinuousbutnotnecessarilydierentiable, C 0 ,at theparticlewhilebeing C 1 elsewhere 1 .Fig. 3-1 showsthesourcefunctionwhichis actuallyusedinthenumericalanalysisdescribedinChapte r4.Thisisforascalarcharge movinginacircularorbit( R =10 M )aroundaSchwarzschildblackhole.Themodest non-dierentiabilityof S e attheparticleisrevealedinFig. 3-2 Asolution R of r a r a R = S e (3{11) 1 With 2 x 2 + y 2 + z 2 ,afunctionwhichis O ( n )as 0,isatleast C n 1 where = 0. 78

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isnecessarily C 2 attheparticle.Itsderivativewouldbe r a R = r a ( ret W ~ S ) ~ S r a W = r a ( ret S )+ O ( 2 = R 4 ) 0(3{12) wouldbetheapproximateself-forceactingontheparticlew henevaluatedatthelocation ofthecharge.Fromthisitisclearwhythebehaviorof W ischosenasinEq.( 3{6 ).A windowfunctionwiththisbehaviorwouldnotaddtothe O ( 2 = R 4 )-erroralreadyincurred byusingthe q= approximationforthesingulareld.Also,inthewavezone W eectively vanishesand R isthenidentically ret andprovidesboththewaveformaswellasany desiredruxmeasuredatalargedistance. Generalcovariancedictatesthatthebehaviorof S e inEq.( 3{9 )maybeanalyzedin anycoordinatesystem.But,onlyinthespeciccoordinates ofRefs.[ 51 ]and[ 52 ],orthe THZcoordinates,isitsoeasilyshown[ 50 ]thatthesimpleexpressionfor S inEq.( 3{3 ) leadstothe O ( = R 4 )behaviorinEq.( 3{10 )andthentothe C 2 natureofthesolution R ofEq.( 3{11 ). TheprescriptionthenissimplytosolveEq.Eq.( 3{11 ),oncetheeectivesource isappropriatelydetermined.Wedescribethismethodas eldregularization ,sinceit eectivelyregularizestheeldsratherthanthegradiento ftheeld(whichthemode-sum schemedoes).Notehoweverthatregularizationisimplicit inourmethod.Anumerical implementationwillhavetodonoregularizationatall,but willhavetoevolveawave equationwithanalreadyregularizedsource.Thederivativ esof R determinethe self-force,providinginstantaccess;while R isidenticalto ret inthewavezone,allowing directaccesstoruxesandwaveforms. Withthismethodthereisnoapparentreasontodeterminethe actualretardedeld. However,ifonewantstocompareresultsfromeldregulariz ationwithresultsfroma traditionaldeterminationoftheretardedeldthensimply adding W ~ S totheremainder 79

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R resultsintheretardedeld ret .Suchacomparisonforourtrialofeldregularization inChapter4appearsinFigs.( 4-7 )and( 4-8 ). 6 7 8 9 10 11 12 13 14 r/M -0.9 -0.6 -0.3 0 0.3 0.6 0.9 f/p -0.5 0 0.5 1 1.5 2 S eff Figure3-1.Theeectivesource S e ontheequatorialplane.Theparticleisat r=M =10, = =0,where S e appearstohavenostructureonthisscale.Thesmooth \doublebump"shapefarfromthechargeisacharacteristico fanyfunction similarto r 2 ( W= j ~r ~r 0 j )inratspace,withawindowfunction W asgivenin Eq.( 4{7 ). 3.2ApproximatingtheLocalSingularField Inthissectionprovidedetailsleadingtotheapproximatee xpressionofthesingular eldinTHZcoordinates.Weshallseethatthedierencebetw eentheactualdelta functionsourceandanapproximationtothesingulareldis asourcewithsomedegreeof non-dierentiabilityatthelocationofthepointcharge.H ere,weshalltakeguidancefrom thediscussioninSectionIVof[ 50 ]. Letbeageodesicthroughabackgroundspacetime g ab ,andlet R bearepresentative lengthscaleforthisbackground{thesmallestoftheradius ofcurvature,thescaleof inhomogeneities,andthetimescaleforchangesincurvatur ealong.Itisalwayspossible toerecta normal coordinatesystemalong,forwhichthemetricanditsrstd erivative 80

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9.9 9.95 10 10.05 r/M -0.003 -0.002 -0.001 0 0.001 0.002 0.003 f/p -1.5e-06 -1e-06 -5e-07 0 5e-07 1e-06 1.5e-06 S eff Figure3-2.Theeectivesource S e intheequatorialplaneinthevicinityofthepoint sourceat r=M =10, = =0.Notethesignicantdierenceofscaleswith Fig. 3-1 matchthatoftheMinkowskimetric,andthecoordinate t measurethepropertimealong thegeodesic.Normalcoordinatesforageodesicarenotuniq ue,andtheTHZcoordinates areaspecialkindofnormalcoordinatesconvenientindescr ibingtheexternalmultipole momentsofavacuumsolutionoftheEinstein'sequations.Am oredetaileddescriptionof thesecoordinatesarepresentedinAppendixA. InTHZcoordinates,metrictakestheform: g ab = ab + H ab = ab + 2 H ab + 3 H ab + O ( 4 = R 4 ) ;= R! 0(3{13) 81

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with 2 H ab d x a d x b = E ij x i x j (d t 2 + kl d x k d x l ) + 4 3 kpq B q i x p x i d t d x k 20 21 E ij x i x j x k 2 5 2 E ik x i d t d x k + 5 21 x i jpq B q k x p x k 1 5 2 pqi B j q x p d x i d x j (3{14) and 3 H ab d x a d x b = 1 3 E ijk x i x j x k (d t 2 + kl d x k d x l ) + 2 3 kpq B q ij x p x i x j d t d x k + O ( 4 = R 4 ) ij d x i d x j ; (3{15) wherethe\dot"isaderivativewithrespectto t ab istheratMinkowskimetricinthe THXcoordinates( t;x;y;z ), ijk istheratspaceLevi-Civitatensor, 2 = x 2 + y 2 + z 2 andtheindices i;j;k;l;p and q areallspatial,andraisedandloweredwiththethree dimensionalratspacemetric ij .CoordinatesthatmatchEq.( 3{14 )aretobecalled third-orderTHZcoordinates,sinceknowledgeofthesecoor dinatesupto O ( 3 = R 3 )is required.Theyarewell-deneduptotheadditionofarbitra ryfunctionsof O ( 4 = R 4 ). WhenEq.( 3{15 )isalsomatched,thenthesewouldbefourth-orderTHZcoord inates. Theexternalmultipolemomentsarespatial,symmetric,tra cefreetensorsrelatedtothe Riemanntensorofthebackgroundevaluatedon: E ij = R titj (3{16) B ij = i pq R pqjt = 2(3{17) E ijk =[ r k R titj ] STF (3{18) B ijk = 3 8 [ i pq r k R pqjt ] STF (3{19) whereSTFmeanstotakethesymmetric,tracefreepartwithre specttothespatialindices i;j; and k 82

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Thegoalistoexpresstoexpressthesingulareldintermsof THZcoordinates. Recallthatthisisjust S ( x )= Z G S ( x;z ( ))d (3{20) where G S isgiveninEq.( 2{134 )tobe: G S ( x;x 0 )= 1 2 U ( x;x 0 ) ( ) 1 2 V ( x;x 0 ) ( ) : (3{21) anditwaspreviouslydeterminedthat U ( x;x 0 )= p ( x;x 0 ),and V ( x;x 0 )isahomogeneous solutionofthewaveequation. For g ab = ab + H ab ,itisshowninEqs.(39)and(40)of[ 55 ]thatonecanwritethe vanVleckdeterminantas ( x;x 0 ) 1 1 2 x a x b Z 0 C H ab;c c (1 )d =1+ O ( 4 = R 4 )(3{22) where 0 C isthecoordinate\straightline"connecting x a and x a 0 givenby a ( )= x a 0 + ( x a x a 0 ) ; (3{23) whichisan O (1)-approximationtotheactualgeodesicconnectingthetw opoints.InTHZ coordinates,thislinetakestheform a ( )= ( x a t 0 a t ) ; (3{24) since x a 0 willbeon,wherethespatialcoordinatesarezero.Thescal ingoftheerror termin( 3{22 )arisesfromthefactthateachderivativeintroducesafact orof O (1 = R )and H ab = O ( 2 = R 2 ).Theerrorincurredby inusingastraightlinepathinsteadoftheexact geodesicbetween x a and x a 0 is O ( H 2 )= O ( 4 = R 4 ),whichisconsistentwiththeerrorterm inEq.( 3{22 ). 83

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Clearly,Eq.( 3{22 )impliesthat U ( x;z )=1+ O ( 4 = R 4 ) : (3{25) WecanalsoapproximateSynge'sworldfunction( 2{56 )byrstrescalingtheane parametersothat 0 =0and 1 =1.Theresultwouldthenbe: ( x;z )= Z C 1 2 g ab d a d d b d d = 1 2 a b ab + Z 0 C H ab d + O ( 6 = R 4 ) ; (3{26) where a =( x a t 0 a t ).Thelinearizedmetric ab + H ab asgivenin( 3{13 )diers fromthetruemetricbytherstnon-linearpiece,whichscal esas O ( H 2 )= O ( 4 = R 4 ). Thisisconsistentwiththeerrorintroducedbyusingthestr aight-linepathforthe integration.Theuseofthelinearizedmetricandthestraig ht-linepathbothcontributeto the O ( 6 = R 4 )errorintheworldfunction. Dening H ab = Z 0 C H ab d ; (3{27) wecanwriteSynge'sworldfunctionas: ( x;z )= 1 2 a b ab + 1 2 ( t t 0 ) 2 H tt +( t t 0 ) x i H it + 1 2 x i x j H ij + O ( 6 = R 4 )(3{28) = 1 2 (1 H tt ) ( t 0 t + x i H it ) 2 x i x j ( ij + H ij ) 1 H tt + O ( 6 = R 4 ) : (3{29) Therearrangementdoneinthesecondequalityisjustiedby notingthat H ab = O ( 2 = R 2 ), andthatclosethenullcone(i.e. 0) j t t 0 j = O ( ). Wecanconvenientlyrearrangethisfurtherbyrstwritingt hefollowingexpressions: H tt = Z 0 C E ij i j + 1 3 E ijk i j k d + O ( 4 = R 4 ) = 1 3 E ij x i x j 1 12 E ijk x i x j x k + O ( 4 = R 4 )(3{30) 84

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and H ij = Z 0 C E kl k l ij + 5 21 i jpq B q k p k 1 21 kl k l pqi B j q p 1 4 E klm k l m ij d + O ( 4 = R 4 ) H ij = 1 4 E kl x k x l + 1 12 E klm x k x l x m ij + 5 84 x i jpq B q k x p x k 1 84 kl x k x l pqi B j q x p + O ( 4 = R 4 ) : (3{31) Usingthese,wecanwrite x i x j ( ij + H ij )= 2 + x i x j H ij (3{32) and x i x j H ij = 1 4 E kl x k x l + 1 12 E klm x k x l x m 2 + O ( 4 = R 4 ) = 2 H tt + O ( 6 = R 4 ) : (3{33) wherethe B termsvanishduetotheantisymmetryoftheLevi-Civitatens or.Thus, x i x j ( ij + H ij )= 2 (1+ H tt )+ O ( 6 = R 4 ) : (3{34) SubstitutingEq.( 3{34 )intoEq.( 3{29 ),onecanfactor into ( x;z )= 1 2 (1 H tt ) ( t 0 t + x i H it ) 2 2 (1+ H tt ) 1 H tt + O ( 6 = R 4 ) = 1 2 (1 H tt ) ( t 0 t + x i H it ) 2 2 (1+ H tt ) 2 + O ( 6 = R 4 ) = 1 2 (1 H tt ) t 0 t + x i H it (1+ H tt ) t 0 t + x i H it + (1+ H tt ) + O ( 6 = R 4 ) : (3{35) Atretardedtime,forwhich z isonthepastnullconeof x and =0,( t 0 t ) Therefore, t 0 t + x i H it (1+ H tt )= O ( ) : (3{36) 85

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Fromthiswecanimmediatelyconcludethat t 0 t + x i H it + (1+ H tt )= O ( 5 = R 4 ) ; (3{37) inordertomatchtheerrorterm O ( 6 = R 4 ),sothatthe canvanishexactlyonthenull coneof x .Thuswehave, d ( x;z ) d t 0 ret = 1 2 (1 H tt )[ t 0 t + x i H it (1+ H tt )]+ O ( 6 = R 5 ) = 1 2 (1 H tt )[ 2 (1+ H tt )+ O ( 5 = R 4 )] = [1+ O ( 4 = R 4 )] ; (3{38) wherethesecondequalityfollowsfromevaluatingatretard edtime,andthethirdfollows fromthefactthat H tt O ( 2 = R 2 ),Eq.( 3{30 ).Asimilarcalculationgivescanbedone withanevaluationattheadvancedtime d ( x;z ) d t 0 adv = 1 2 (1 H tt )[ t 0 t + x i H it (1+ H tt )]+ O ( 6 = R 5 ) = 1 2 (1 H tt )[2 (1+ H tt )+ O ( 5 = R 4 )] = [1+ O ( 4 = R 4 )] : (3{39) DewittandBrehmeshowthat V ( x;z )= 1 12 R ( z )+ O ( = R 3 ) ;x z: (3{40) Butinvacuumspacetimes, R =0, V ( x;z )= O ( 2 = R 4 ) : (3{41) Whenintegratedoverthepropertime,thedominantcontribu tionfromthistermis O ( 3 = R 4 )as x z .Withthis,wenowhaveeverythingnecessarytoarriveatan approximationfor S inTHZcoordinates. 86

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ReturningthentoEq.( 3{21 ),wewrite S = 1 2 Z U ( x;z ( ))( + ( )+ ( ))d + 1 2 Z V ( x;z ( )) ( )d : (3{42) Notingthat_ j ret (d = d t 0 ) j ret > 0and_ j adv < 0,wecanuseEqs.( 3{25 ),( 3{38 )and ( 3{39 )toapproximatethetermsoftherstintegral: 1 2 U ( x;z ) ret ( )= 1+ O ( 4 = R 4 ) 2_ ret ( t t ret )= ( t t ret ) 2 + O ( 3 = R 4 )(3{43) 1 2 U ( x;z ) adv ( )= 1+ O ( 4 = R 4 ) 2_ adv ( t t adv )= ( t t adv ) 2 + O ( 3 = R 4 ) : (3{44) FromEq.( 3{41 ),thesecondintegralisalready O ( 3 = R 4 ),thesameorderastheerrorof ourapproximationoftherstintegral.Weshallthereforei gnoreit. Ourapproximationthencomesentirelyfromthedirectpiece ofthesingularGreen's function,whosecorrespondingsingulareld S wecannowevaluatetobe S = q + O 3 R 4 ; 0(3{45) ThisisexactlywhatwasclaimedinEq.( 3{3 ).TheCoulombpieceispreciselythelocal approximationforthesingularpieceneededfortheeectiv esourceinEq.( 3{9 ). 87

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CHAPTER4 NUMERICALEXPERIMENTINTIME-DOMAIN [Thecontentsofthischaptercanbefoundinthefollowingar ticle:Regularization ofeldsforself-forceproblemsincurvedspacetime:Found ationsandatime-domain application.I.VegaandS.Detweiler,Phys.Rev.D 77 ,084008(2008).Copyright(2008) bytheAmericanPhysicalSociety.] Previously,wedevelopedamethodforself-forcecalculati onswhichwasatitscore ausefulsmearingout(orregularization)ofthedeltafunct ionrepresentationforapoint charge.Asaresult,wehaveintroducedawayofrepresenting pointchargeswithregular functions(whichisapropertyusefulfor3+1evolutioncode s).Becausetheregularization wasbasedonthelocalsplittingoftheretardedeldintoits regularandsingularparts, ortheDetweiler-Whitingdecomposition,ourmethodhasthe additionalbenetthat computingself-forcesamountstosimplytakingderivative softhesolutiontooureective waveequation.Thischapterprovidestherstconcretedemo nstrationoftheviability ofourmethod.Inparticular,weshowthatweareabletocompu tecorrectlyboththe self-forceonthechargeandtheretardedeldinthewavezon efromwhichruxesmaybe computed. Thisdemonstrationisachievedforthewell-studiedtoypro blemofascalarcharge movinginacircularorbitaboutaSchwarzschildblackhole. Wechoose q=m =1forthe chargetomassratiooftheparticle,andacirculargeodesic atSchwarzschildradii, R = 10 M and R =12 M ,where M isthemassoftheblackhole.WeworkinSchwarzschild coordinates,inwhichthemetricisexpressedas g ab =diag( (1 2 M=r ) ; 1 = (1 2 M=r ) ;r 2 ;r 2 sin 2 ). Forthistask,wehavedevelopedcodethat(a)solvestheregu larizedwaveequation ( 3{11 )and(b)computesthescalarself-force.Forsimplicity,we havechosentosolvethe regularizedwaveequationusinga(1+1)-approach.Weexplo itthesphericalsymmetry ofthebackground,decomposephysicalquantitiesintosphe ricalharmonics,andthen 88

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solvetheresultingsetof(1+1)D-waveequations(one`time '+one`space')forthe spherical-harmoniccomponents. Itmustbestressedatthispointthatthenumericalimplemen tationpresentedin thischapterdoesnothighlighttheadvantagesofourprescr iption.Thesimplicityofthe orbitweconsiderandthesphericalsymmetryofourbackgrou ndgeometrynaturally lendthemselvestoasignicantlymoreecientfrequency-d omaintreatment.Butthe pointhereistoprovideaquick,rstcheckofourideas.Onei scautionednottoletthe natureofourtoyproblemobscurethegeneralityofourpropo sedmethod,anditspotential forcaseswithgenericorbitsandspacetimeslackingsymmet ry,andforself-consistent evolutionswhicharelikelytorequireself-forcecalculat ionsinrealtime(asopposedto beingapost-processingstep).Testingtherobustnessofou rmethodagainstthesemore dicultproblemswillbeaddressedinfuturework.Thecurre ntgoalsaremainlyto establishplausibilityandgleansomeoftheinsightandexp erienceneededfortackling problemsofincreasedsophistication. 4.1ScalarFieldsinaSchwarzschildGeometry Waveequationsinsphericallysymmetricbackgroundssimpl ifyconsiderablywitha sphericalharmonicdecompositionoftheeld.Inthecaseof aSchwarzschildgeometry expressedinSchwarzschildcoordinates,thisdecompositi onistypicallyperformedas follows: = X lm 1 r f lm ( r )( t;r ) Y lm ( ; ) : (4{1) With r = r +2 M ln( r= 2 M 1),thisyieldsequationsfor f lm ( t;r ): @ 2 f lm @t 2 + @ 2 f lm @r 2 V ( r ) f lm = S lm ( t;r )(4{2) where V ( r )isimplicitlygivenintermsof r as: V ( r )= 1 2 M r l ( l +1) r 2 + 2 M r 3 ; (4{3) 89

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whilethesource S lm ( t;r )is S lm ( t;r )=( r 2 M ) Z ( x 0 ) Y lm ( 0 ; 0 ) d n 0 (4{4) Inafrequency-domainapproach,onefurtherchoosestoFour ier-decompose f lm ( t;r )= R F lm! ( r )exp( i!t ) d! ,andtherebysolvetheresultingsetofordinarydierentia l equationsfor F lm! ( r ),foreachmode .Thismethodtendstobenumericallyexpensive, however,forsourceswithacontinuous -spectrum.Instead,wechoosetosolveEq.( 4{2 ) asaninitialboundaryvalueproblem,inatime-domainfashi on,foreach( l;m ).Thisis donewith S lm computedbeforehandasthesphericalharmoniccomponentso ftheeective sourcefoundinEq.( 3{9 ). Anovelfeatureofourapproachistheuseofaneectivesourc ethatpermitsthe easycalculationofbothself-forcesandruxes.Asdiscusse dintheprecedingchapter,this eectivesourceisformally S e = r 2 ( W ~ S ) 4 q Z r (4) ( x z ( )) p g d: (4{5) Tolowestorder,thesingulareldtakesontheform S ~ S = q : (4{6) Wetakeadvantageoftheresultsin[ 50 ],where isexpressedexplicitlyas = p ij x i x j inThorne-Hartle-Zhangcoordinatesforaparticlemovingi nacircular orbit.UsingthecoordinatetransformationfoundinAppend ixBof[ 50 ],weareable toexpressthesingulareldinSchwarzschildcoordinates. (Adiscussionofthiscoordinate transformationisprovidedinAppendixDofthisdissertati on).Tocompleteoureective source,weselectawindowfunctionwhoseroleistokillosm oothlythesingulareld inregionswhereitisnotneeded.Consequently,theeectiv esupportofthewindowed singulareld W ~ S isconnedtoacompactregionsurroundingtheparticle'swo rldline. 90

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Ourchosenwindowfunctionisspherically-symmetricwithr especttothecenterofthe blackhole.Thischoicewasnotnecessarybutguaranteestha t W wouldnotunnecessarily modifythe( l;m )-spectrumofthesource,andtherebyallowsustomakemorec ontrolled comparisonswithexistingfrequency-domainresultsonthe sameproblem.Oursimple choiceof W is W ( r )=exp h ( r R ) N N i : (4{7) Inthiswindowfunction,theconstant setsthewidth,andtheexponent N controlshow quickly W and r a W reachtherequiredvaluesof1and0,respectively,asoneapp roaches theparticle.Weuse =2 M and N =8inalltheresultspresentedinthischapter.It isnecessarythat N isaneveninteger,andtakingfulladvantageoftheaccuracy ofour approximationfor S requiresthat W =1+ O ( 4 = R 4 )as 0.Thuswerequirethat N 4.Infactweused N =8inanticipationofimprovingtheapproximationfor S in thefuture. Ourchoiceforthewindowfunctionleadstotheeectivesour ce S e displayedin Figs.( 3-1 )and( 3-2 ).Alargerchoicefor wouldspreadthebumpsoutfurther,anda smallerchoicefor N wouldsmooththebumps.Butif N werelessthan4,then W ~ S wouldnotadequatelymatchthebehaviorof S as 0. Withtheeectivesourceconstructedasabove,itsspherica l-harmoniccomponents werethencomputed.Circularorbitsprovedadvantageoushe rebecauseofwhichthetime dependenceofthecomponentscouldthensimplybeinferred. Thesphericalharmonic componentswereevaluatedwitha4th-orderRunge-Kuttaint egratorwithself-adjusting stepsize,whichwasderivedfromaroutinein[ 56 ]. 4.2EvolutionAlgorithm TheintegrationschemeweuseinevolvingEq.( 4{2 )followsatechniquerst introducedbyLoustoandPrice[ 57 ],andlaterimprovedtofourth-orderaccuracyby Lousto[ 58 ]andHaas[ 47 ].Unliketheirschemes,however,wedonotdealwithsourced and 91

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vacuumregionsofournumericaldomainseparately.Theirus eofasingulardelta-function sourcemeantthattheresultingeldwasnon-dierentiable atthelocationofthecharge, whilesmootheverywhereelse.Forus,theeectivesourceis C 0 ,implyingthattheeldis atleast C 2 .Whilethisisstillofnitedierentiability,wendthatt heeectivesourceis dierentiableenoughnottowarrantatreatmentdierentfr omthevacuumcase. Inthe( t;r )-plane,weintroduceastaggeredgridwithstepsizes t = 1 2 r = h Inthisgrid,aunitcellisdenedtobethediamondregionwit hcorners f ( t + h;r ) ; ( t h;r ) ; ( t;r + h ) ; ( t;r h ) g .Onlyatthesegridpointsdoweevaluate f lm .Wehenceforth dropthespherical-harmonicindicesin f lm forconvenience. t = h r =2 h ( t;r h )( t;r + h ) ( t + h;r ) ( t h;r ) Figure4-1.Staggered(characteristic)gridwithunitcell Themainideabehindthealgorithmistointegratethewaveeq uationoveraunitcell. ThisisdoneeasiestwithEddington-Finkelsteinnullcoord inates u = t r and v = t + r astheintegrationvariables. 92

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Thedierentialoperatorofthewaveequation,whenexpress edin( u;v )coordinates, isjust 4 @ u @ v .Overaunitcellthen,thederivativeterminEq.( 4{2 )canbeintegrated exactly: ZZ C 4 @ u @ v fdudv = 4[ f ( t + h;r )+ f ( t h;r ) f ( t;r + h ) f ( t;r h )] : (4{8) Integrationsofthepotentialtermandthesourcetermdonot enjoythesame simplicityasthederivativeterm.Weneedtoapproximateth eseintegralstothe appropriateorderin h soastoachievethedesired O ( h 4 )-convergenceovertheentire numericaldomain. Supposewewishtosolvethewaveequationoveraregiondene dby T and R .In thisregion,therewillbe N = T R =h 2 cells.Achieving O ( h 4 )-convergenceforevolution meansthatweneedtointegratethewaveequationwithanover -allerrorofatmost O ( h 4 ) overtheentirecomputationaldomain.Foraunitcell,thism eansanapproximationwith anerror O ( h 4 ) =N O ( h 6 ). SuchanapproximationisachievedwiththedoubleSimpsonru le.Considera sucientlydierentiablefunction G ( t;r )tobeintegratedoveraunitcell.Thedouble Simpsonrulethenreads: ZZ C Gdudv = h 3 2 [ G corners +16 G ( t;r )+4( G ( t + h= 2 ;r h= 2)+ G ( t + h= 2 ;r + h= 2) + G ( t h= 2 ;r h= 2)+ G ( t h= 2 ;r + h= 2))]+ O ( h 6 ) ; (4{9) where G corners isjustthesumofthevaluesof G evaluatedatthecornersoftheunitcell. ThisisdirectlyappliedinintegratingthesourcetermofEq .( 4{2 ): ZZ C S e lm dudv: (4{10) Onesimplyevaluatesthesourcetermattherequiredpointsa ndthensumsthese accordinglyinordertogetan O ( h 6 )-accurateapproximationtotheintegral. 93

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However,forintegratingthepotentialterm: ZZ C Vfdudv; (4{11) werecallthatonehasonlyrestrictedaccessto f .Thedirectevaluationof f isdoneonly atthegridpoints,i.e.cornersoftheunitcell.Thusfar,on ly G corners inEq.( 4{9 )canbe explicitlyevaluated.TouseEq.( 4{9 )forthepotentialterm,weneedtodeterminehowto evaluate f atalltheotherpoints. FollowingLousto[ 58 ],weevaluate G = Vf atthecentralgridpoint(i.e. G ( t;r )) usingvaluesattheneighboringgridpointsonthesametimes lice. G ( t;r )= 1 16 [9 G ( t;r h )+9 G ( t;r + h ) G ( t;r 3 h ) G ( t;r +3 h )]+ O ( h 4 ) : (4{12) NotethatthisisdierentfromHaas[ 47 ],whousesgridpointsinthecausalpastofthe unitcell.The O ( h 4 )-errorincurredinthisapproximationistolerablebecaus eofthe h 2 -factorthatappearsinEq.( 4{9 ). Weseeksimilarapproximationsfor G intheremainingpoints.Considerrstthe pair G ( t + h= 2 ;r h= 2)and G ( t h= 2 ;r h= 2).(Theotherpair,composedof G ( t + h= 2 ;r + h= 2)and G ( t h= 2 ;r + h= 2),istreatedsimilarly).Thispairmakesup thetopandbottomcornersofasmallercell, C left ,madeupofthepoints f ( t + h= 2 ;r h= 2) ; ( t h= 2 ;r h= 2) ; ( t;r h ) ; ( t;r ) g Whatweshalldonextisndanapproximationfor G ( t + h= 2 ;r h= 2)+ G ( t h= 2 ;r h= 2)(4{13) accurateto O ( h 4 ).Again,thisissucientbecauseofthe h 2 -factorinEq.( 4{9 ). 94

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C left C right ( t;r ) t h 2 ;r + h 2 t + h 2 ;r + h 2 t + h 2 ;r h 2 t h 2 ;r h 2 Figure4-2.Unitcellofthealgorithm.Theblackdotsindica tegridpoints,whereasthe grayonesstandforthepointswhere G = Vf needstobeapproximated.The subcells C left and C right areshadedgray.Toapproximate G atsomeofthegray dots,weintegratethewaveequationineachofthesesubcell s. Considerintegratingthewaveequationoverthissmallerce ll,butthistimeonlyupto anaccuracyof O ( h 4 ).Theintegraloverthederivativetermwillagainbeexact: ZZ C left 4 @ u @ v fdudv = 4[ f ( t + h= 2 ;r h= 2)+ f ( t h= 2 ;r h= 2) f ( t;r h ) f ( t;r )] : (4{14) Theintegralsofthepotentialandsourcetermsoverthissma llercellareagainhandled asbefore,butthistimeweapproximatethemonlyto O ( h 4 ).Tothisend,thedouble trapezoidalrulewillsuce,whichreads: ZZ C left Gdudv = h 2 2 [ G ( t + h= 2 ;r h= 2)+ G ( t h= 2 ;r h= 2) + G ( t;r h )+ G ( t;r )]+ O ( h 4 ) : (4{15) 95

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Applyingthistothepotentialtermthengives: ZZ C left Vfdudv = h 2 2 [ V ( r h= 2) f ( t + h= 2 ;r h= 2) + V ( r h= 2) f ( t h= 2 ;r h= 2) + V ( r h ) f ( t;r h )+ V ( r ) f ( t;r )]+ O ( h 4 ) : (4{16) CombiningEq.( 4{14 )andEq.( 4{16 ),theresultofintegratingthewaveequationoverthis smallercellyields: f ( t + h= 2 ;r h= 2)+ f ( t h= 2 ;r h= 2) =( f ( t;r h )+ f ( t;r )) 1 1 2 h 2 2 V ( r h= 2) # 1 4 ZZ C left S e dudv + O ( h 4 ) : (4{17) Aftermultiplyingbothsidesofthislastequationby V ( r h= 2),theresulting left-hand-sidebecomestwooftheasyetmissingpiecesinth edoubleSimpsonformula: G ( t + h= 2 ;r h= 2)+ G ( t h= 2 ;r h= 2)= V ( r h= 2) f ( t + h= 2 ;r h= 2) V ( r h= 2) f ( t h= 2 ;r h= 2) : (4{18) Theresultingequationthengivesusthedesired O ( h 4 )-approximationofthemissing expression, G ( t + h= 2 ;r h= 2)+ G ( t h= 2 ;r h= 2),inEq.( 4{9 ).Followingthe samesteps,itiseasytoarriveatanequivalentapproximati onfortheothermissingpair, G ( t + h= 2 ;r + h= 2)+ G ( t h= 2 ;r + h= 2).Wesummarizethesebelow: G ( t + h= 2 ;r h= 2)+ G ( t h= 2 ;r h= 2)= V ( r h= 2)( f ( t;r h )+ f ( t;r )) 1 1 2 h 2 2 V ( r h= 2) # + V ( r h= 2) 4 ZZ C left S e dudv + O ( h 4 ) : (4{19) 96

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G ( t + h= 2 ;r + h= 2)+ G ( t h= 2 ;r + h= 2)= V ( r + h= 2)( f ( t;r + h )+ f ( t;r )) 1 1 2 h 2 2 V ( r h= 2) # + V ( r + h= 2) 4 ZZ C right S e dudv + O ( h 4 ) : (4{20) Exceptforthepresenceofintegratedsourceterms,theseeq uationsareidenticalto Lousto'sequations(32)and(33),andHaas'sequations(2.8 )and(2.9).FollowingHaas [ 47 ],wechoosetoavoidisolatedoccurrencesof f ( t;r ),whichprovetobenumerically unstableclosetotheeventhorizon.Aspointedoutin[ 47 ],thisisduetohavingrst approximated G = Vf ,whichmakesitdiculttoisolate f = G=V where V 0.This appearsunnecessaryif f weredirectlyapproximatedinsteadof G in( 4{12 ).Nevertheless, likeHaas,weavoidneedingtoisolate f byaddingupequationsEq.( 4{19 )andEq.( 4{20 ), andthenTaylor-expandingthepotentialtermsthataremult ipliedby f ( t;r ).Theresult isHaas'sequation(2.10)withextrasourceterms: X G G ( t + h= 2 ;r h= 2)+ G ( t h= 2 ;r h= 2) + G ( t + h= 2 ;r + h= 2)+ G ( t h= 2 ;r + h= 2) = 2 V ( r ) f ( t;r ) 1 1 2 h 2 2 V ( r ) # V ( r h= 2) f ( t;r h ) 1 1 2 h 2 2 V ( r h= 2) # V ( r + h= 2) f ( t;r + h ) 1 1 2 h 2 2 V ( r + h= 2) # 1 2 [ V ( r h= 2) 2 V ( r )+ V ( r + h= 2)][ f ( t;r h )+ f ( t;r + h )] + V ( r h= 2) 4 ZZ C left S e dudv + V ( r + h= 2) 4 ZZ C right S e dudv + O ( h 4 )(4{21) Thislastequationcompletesthepiecesneededfortheevolu tionalgorithm. 97

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UsingEq.( 4{8 )forthederivativetermandEq.( 4{9 )forthepotentialandsource terms,theresultofintegratingthewaveequationovertheu nitcellnallyyields: f ( t + h;r )= f ( t h;r )+ 1 1 4 ( h 3 ) 2 V ( r + h ) 1+ 1 4 ( h 3 ) 2 V ( r ) f ( t;r + h ) + 1 1 4 ( h 3 ) 2 V ( r h ) 1+ 1 4 ( h 3 ) 2 V ( r ) f ( t;r h ) 1 1+ 1 4 ( h 3 ) 2 V ( r ) h 3 2 4 G 0 + X G + 1 4 ZZ C S e dudv # + O ( h 6 ) ; (4{22) where G 0 isevaluatedaccordingtoEq.( 4{12 ),with G ( t;r )= V ( r ) f ( t;r ); P G isthe expressioninEq.( 4{21 );andthedoubleSimpsonruleEq.( 4{9 )isappliedinevaluating theremainingintegralterm RR C S e dudv .Withthisequation,onecannowdeterminethe eld f attime t + h givenitsvaluesatearliertimes t and t h ThisderivationmakesliberaluseofdoubleSimpsonanddoub letrapezoidalformulas whenapproximatingintegralsofthesourceandpotentialte rmsovertheunitcell.The formulascomefromtheirsingle-integralcounterparts: Z x 0 + h x 0 f ( x ) dx = h 2 [ f ( x 0 )+ f ( x 0 + h )] h 3 12 f (2) ( )(4{23) Z x 0 +2 h x 0 f ( x ) dx = h 3 [ f ( x 0 )+4 f ( x 0 + h )+ f ( x 0 +2 h )] h 5 90 f (4) ( ) ; (4{24) where f ( n ) denotesthe n th-derivativeof f ,and issomepointwithinthelimits ofintegration.Theserequiretheboundedness,ifnotexist enceofthesecondand fourthderivativesoftheintegrandfortheerrorestimatet obevalid.Withthelimited dierentiabilityofoursourceandpotentialterms( C 0 and C 2 ,respectively),onemight worryaboutthevalidityofourover-allconvergenceestima te.However,ourcalculations revealthat4th-orderconvergenceisachieveddespitethis deciency. 98

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4.3InitialDataandBoundaryConditions Forourevolutionwehavenoobviousmethodforchoosingapri orithecorrect initialdata,whichconsistsofthevalueof R ontwoconsecutiveconstant-timeslices. Consequentlywejustsettheinitial R tozeroeverywhereontheinitialtwoslices. Physically,thisscenariocorrespondstotheimpulsiveapp earanceofthescalarpoint chargealongwith W ~ S ,whichleadstospuriousradiationcontaminatingourcompu tational domainduringtheearlystagesoftheevolution.Fortunatel y,thisradiationpropagates outoftheregionsofinterestquickly;sotocircumventthen eedforproperinitialdata, wesimplyevolvetheequationtolongenoughtimessuchthati nitialdataeectsdonot becomepertinentinanyofourresults. Withthescalarchargemovinginacircularorbit,itisexpec tedthattheeld eventuallybecomesstationaryinaframecorotatingwithth echarge.Apracticaltestthen forthepersistenceofinitialdataeectsistosimplycheck whetherornottheeldhas alreadysettledintoaquiescentstatewhenevaluatedinthi sframe. Boundaryconditionsaretreatedsimilarly.Ratherthanhan dlingthemcarefully,we insteadmadethecomputationaldomainlargeenoughthaterr orsincurredbyunspecied boundaryconditionsdidnotaectourregionsofinterest.F orthiswork,ourchoiceof boundarieswereat r = 700 M and r =800 M 4.4Self-ForceCalculation Attheendofevolutionforeachmode,wecomputetheself-for ceatthelocationof theparticle.Sincethislocationisnotonanygridpoint,in terpolationof f lm ( T;r )andits derivativesto r = R wasrequiredusingaselectionofgridpointssurroundingit Oncethiswasdone,computingtheself-forcewasasimplemat terofperformingthe followingsums: R = 1 R L X l =0 f l 0 ( T;R ) Y l 0 2 ; n T + 2 R L X l =1 l X m =1 Re f lm ( T;R ) Y lm 2 ; n T (4{25) 99

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@ t R = 2 R L X l =0 l X m =1 ( m n)Im f lm ( T;R ) Y lm 2 ; n T (4{26) @ r R = 1 R L X l =0 @ r f l 0 ( T;R ) Y l 0 2 ; n T + 2 R L X l =1 l X m =1 Re @ r f lm ( T;R ) Y lm 2 ; n T (4{27) Here, L isthepointwherewetruncatethemultipoleexpansion.Inal lourworkwehave used L =39.Thesesumsariseprimarilybecauseourchargemovesina circularorbit. Twomethodswereemployedforinterpolation.Therstwasas impleLagrange interpolationofboth f and @ r f to r = R .However,becauseofthenitedierentiabilityof ourregulareldat r = R wealsointerpolatedusingtheform R ( r )= A 0 + A 1 x + A 2 x 2 + A 3 x 3 + ( x ) B 0 x 3 ; (4{28) where x = r R ,and ( x )isthestandardHeavisidefunction.Thisformcloselyresp ects the C 2 natureoftheregulareldat r = R byallowingforadiscontinuityinthethird derivative. Withthisform,( r r F ) j r = R = A 1 .However,thisledtoresultsnotsignicantly dierentfromtheoneachievedwithordinaryLagrangeinter polation. 4.5CodeDiagnostics 4.5.1Convergence Convergenceofatime-domaincodeiseasilydeterminedbyco mputingtheconvergence factor n asdenedbyLousto[ 58 ]: n ( r;t )=log f 4 h ( r;t ) f 2 h ( r;t ) f 2 h ( r;t ) f h ( r;t ) = log(2)+log j ( n ) ( ) j = log(2) ; (4{29) where f ( r;t )istheresultoftheevolutionforaresolutionof,and ( n ) ( )representsan errorfunction 1.An n th-orderevolutioncodeisoneforwhich = N ( h )+( ( n ) )( ) h n where N ( h )isthenumericalsolutionatresolution h 100

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Incheckingconvergence,oneevolvesthewaveequationatdi erentresolutions, h 2 h ,and4 h .Foraxed r = R ,onethenextracts f h ( R;t ) ;f 2 h ( R;t ) ; and f 4 h ( R;t )forall t Fromthese,onecancompute n ( R;t ). Theconvergencefactorwascomputedforafewrepresentativ epointsinthewavezone andintheregionclosetothepointparticle.Twooftheseare showninFig. 4-3 .These arefor r 10 M and r 100 M .Allshowthedesired4th-orderconvergenceeventually, followingatransientperiodinwhichthenumericalevoluti oniscontaminatedbythe eectsofpoorinitialdata. 0 2 4 6 8 10 100 200 300 400 500 600 700 nt/M convergence at r=10M convergence at r=100M Figure4-3.Convergenceattheparticlelocation( r =10 M )andinthewavezone ( r =100 M ).Atthestartoftheevolution,inequivalentinitialdatal eadtothe lackof4th-orderconvergence.But n graduallyapproaches4asinitial-data eectspropagateawayfromthecomputationaldomain.Notet hatthe convergencetestattheparticlelocationalreadyincludes theinterpolation step. 4.5.2Highl Fall-O In[ 44 46 50 59 ],itwasdemonstratedthattherateofconvergenceofthe l -components oftheself-forcewasdictatedprimarilybythelackofdier entiabilityoftheregular piecefromwhichtheself-forceiscomputed.Bydenition,t hedierencebetweenthe 101

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retardedeldandthesingulareldyieldsafunctionthatis C 1 .Inthisidealsituation, convergencein l oftheself-forcecomputedfromthissmoothregulareldwou ldbe exponentiallyfast.Inpractice,however,oneisalwayslim itedtoconstructingonlyan approximatesingulareld,thereforeleavingnon-dieren tiablepiecesintheresidual ret ~ S .Thedegreeofnon-dierentiabilityofthisremainderiswh atsetstherateof convergenceoftheself-forcein l Thehighl asymptoticstructureofthesingularpiece S issuchthat: lim r R ( r r S ) l = l + 1 2 A r + B r 2 p 2 D r (2 l 1)(2 l +3) + E (1) r P 3 = 2 (2 l 3)(2 l 1)(2 l +3)(2 l +5) + E (2) r P 5 = 2 (2 l 5)(2 l 3)(2 l 1)(2 l +3)(2 l +5)(2 l +7) + :::; (4{30) where P k +1 = 2 =( 1) k +1 2 k +3 = 2 [(2 k +1)!!] 2 ,and A a ;B a ;D a ;E (1) a ;E (2) a ;::: arethe l -independent regularizationparameters ,whicharecommonincontemporaryself-force studies[ 44 46 ]. Thenumberofregularizationparametersthatcanbedetermi nedinthisexpansion correspondsdirectlytotheaccuracyofthesingulareldap proximation.Convergencein l oftheself-force r r R isxedbythelowest-orderundeterminedpieceoftheapprox imate singulareld.Specically,ifthesingulareldisaccurat elydeterminedonlyuptothe B -termoftheexpansionabove,thenthe l -convergenceinthemodesoftheself-forcewould be 1 =l 2 ,correspondingtothe D -termfall-o. Theapproximationtothesingulareldhereis ~ S = q= ,andtheattendantTHZto Schwarzschildcoordinatetransformation,hasbeenshowni n[ 50 ]toincludeatleastthe D -term.Theexpectationthenwouldbeforthe l -componentsofourremainder,( r r R ) l 102

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tofalloasthe E (1) -piece: E (1) r P 3 = 2 (2 l 3)(2 l 1)(2 l +3)(2 l +5) : (4{31) Fig. 4-4 showsourresultsconrmingthisexpectation.Ourresultsa replottedwith D E (1) and E (2) fall-ocurvesfoundinEq.( 4{30 )thataremadetomatchat l =15. 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 |(F r ) l |ell result D fall-off E 1 fall-off E 2 fall-off Figure4-4.( r r R ) l versus l .Ourresultsshow l -convergenceclosesttothe E (1) fall-o. Thebluelinescorrespondtotheexpectedfall-ointhe r -componentofthe self-forcewhenoneregularizesusingasingulareldappro ximationwithoutthe D -term, E (1) -term,and E (2) -term,respectively.Ourresultismatchedtothese curvesat l =15. The t -componentoftheself-force,ontheotherhand,doesnotreq uireregularization forthecaseofachargeinacircularorbitofSchwarzschild. Anexponentialfall-oisthen expected.ThisisshowninFig. 4-5 4.5.3DependenceontheWindowFunction Theuseofawindowfunction W isapeculiarfeatureofourapproach.Itsfunction ismainlytokillo ~ S intheregionswhereitisnolongerrelevantandtherebytoha ve thecomputedregulareld R transformintotheretardedeldinthoseregions.Asitis 103

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1e-20 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 |(F t ) l |ell t-component fall-off Figure4-5.( r t R ) l versus l .Theexpectedexponentialfall-oisobserveduntilthepoi nt wherenumericalnoisebeginstodominate. amereartifactofourimplementation,itiscrucialthatthe self-forceandwaveformbe independentofthespecicchoiceofwindowfunction. Onehasconsiderablefreedominchoosing W ,theonlyrequirementsbeingthat W goesto1andthatitsgradientvanishesfastenoughinthelim itthatoneapproachesthe pointparticle.WithourspecicchoiceofWbecomingnumeri callysignicantonlyinan annularregion j r R j ,wehaveinspectedthechangesintheself-forceandruxesas onevariesthewidth Fig. 4-6 showstheeectofdoublingtheannularsupportofthewindow function.The ( l =2 ;m =2)waveequationwasevolvedforthesamelengthoftime,but witheective sourceshavingdierentwindowfunctions.Acomparisonist henmadeoftheresulting eldsovermostofthecomputationaldomain.Itisseenthatt heeldsdiersignicantly onlyinregionswherethewindowfunctionsdier.Neverthel ess,theregulareld R remainsthesame(uptofractionalchangesof 10 8 )inthemostphysically-relevant 104

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regions:thevicinityofthecharge, r =10 M (wheretheself-forceiscomputed),andthe wavezone, r 10 M (wherethewaveformistobeextracted). Asdesiredthen,thewindowfunctionappearstohavenoeect onanyofthe numericalresultsattained. 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110 120 |( DY 22 )/ Y 22 |r /M |( DY 22 )/ Y 22 | 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 9.5 9.75 10 10.25 10.5 r Figure4-6.Fractionalchangesin R 22 asaresultofusingdierentwindowfunctions.Note thatthesechangesaresignicantonlywherethewindowfunc tionsdier;they areinsignicantintheimportantregionsinthevicinityof thecharge, r =10 M ,andinthewavezone, r 10 M 4.6Results Ourspecicimplementationmadeuseofagridspacing h M= 25,sothat t 0 : 04 M and r 0 : 08 M 4.6.1RecoveringtheRetardedField Fromournumericalcalculationsweareabletoaccuratelyre covertheretardedeld. Inthewavezone,wherethesingulareldisnegligible,this retardedeldequalsourregular eld R .Since,energyruxesdependdirectlyontheretardedeldin thisregion,the accuracywithwhichwerecovertheretardedeldinthewavez onegivesusameasure ofhowwellwecancomputeruxesusingourmethod.Wedetermin ethisaccuracyby 105

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comparingourresultfor R inthewavezonewiththatobtainedfortheretardedeld usingaseparatefrequency-domaincalculation.Anexample ofsuchacomparisonisshown inFigs. 4-7 and 4-8 .Weobserverelativeerrorsthatareatworst10 6 .ShowninFig. 4-7 arethe( l =2 ;m =2)componentoftheretardedeldcomputedinthefrequency -domain andourcorrespondingtime-domainresult, R 22 +( W ~ S ) 22 ,forthecaseofachargeat r =10 M .Alsoshownarethesingulareld( W ~ S ) 22 andtheregulareld R 22 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 20 40 60 80 100 120 |f 22 |r /M TD recovered retarded field FD retarded field TD regular field singular field 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 6 8 10 12 14 r/M Figure4-7.Comparisonoftime-domainandfrequency-domai nresultsfor f 22 ( r ).The regulareldistheresultofourcode(representedbytheblu edashedline). Addingthistothe( l =2, m =2)-componentofouranalyticalsingulareld, W ~ S ,resultsintheFD-computedretardedeldtogoodagreement 4.6.2ComputedSelf-Force Weobtainthe t and r componentsoftheself-forceforanorbitatradii R =10 M and 12 M .ThesearesummarizedinTable 4-1 Fig. 4-9 showstheconvergenceofourtime-domain( l =2 ;m =2) t -componentof theself-forcetothefrequency-domainresult.Wereachour closestmatchafteratimeof 200 M ,whichisapproximatelyoneorbitalperiod. 106

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1e-10 1e-08 1e-06 0.0001 0.01 1 100 0 20 40 60 80 100 120 140 |(f TD -f FD )/f FD |r /M relative difference between TD and FD 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 6 8 10 12 14 r/M Figure4-8.Relativeerrorbetweentime-domainandfrequen cy-domainresultsfor f 22 ( r ). Excellentagreementisachieved.Relativeerrorsareatwor st 10 6 ,whichis consistentwitherrorexpectedfromastep-size M= 25. R Time-domainFrequency-domainerror @ t R 10 M 3 : 750211 10 5 3 : 750227 10 5 0.000431% @ r R 10 M 1 : 380612 10 5 1 : 378448 10 5 0.157% @ t R 12 M 1 : 747278 10 5 1 : 747254 10 5 0.00139% @ r R 12 M 5 : 715982 10 6 5 : 710205 10 6 0.101% Table4-1.Summaryofself-forceresultsfor R =10 M and R =12 M .Theerroris determinedbyacomparisonwithanaccuratefrequency-doma incalculation [ 50 ]. 4.7Discussion Inthespeciccontextofapointchargeorbitingabackhole, wehaveintroduceda verygeneralapproachsuitableforthetime-domaingenerat ionofwaveformsandalsothe calculationofthebackreactingself-force. Ourinitialtestsareadmittedlyontheveryrestrictivecas eofcircularorbitsof theSchwarzschildgeometry,wherewehavetakenadvantageo fthesphericalsymmetry todecomposethesourceandeldintosphericalharmonics.T hisdecompositionwas employedmainlytoallowustocompareourresultstoavailab lefrequency-domainresults 107

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1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 0 100 200 300 400 500 600 |(f TD -f FD )/f FD |t/M |(f TD -f FD )/f FD | Figure4-9.Relativeerrorinthetime-domaincalculationo f f 22 ,ascomparedwiththe frequency-domaincalculation,versustimeat r 10 M ofveryhighprecision[ 50 60 ].Insomemanner,becauseofouruseofaspherical-harmonic decomposition,ouranalysismightbelikenedtousingspect ralmethods.But,themethod ofeldregularizationinherentlydoesnotrequireamode-d ecompositionandcouldbe implementedwithafull(3+1)numericalcode.Forourtestca seweachieveanextremely accuratecalculationforthetimecomponentoftheself-for ce @ t ~ R ,whichisequivalentto therateofenergylostbyradiation.Notably,inour(1+1)im plementation,theinitialdata settleddowntoprovidethisaccuratecomponentoftheselfforcewithinonlyoneorbitof theparticleasshowninFig. 4-9 .Thismightbecontrastedwithacalculationof dE=dt madefromaruxintegralevaluatedinthewave-zone,which,w ithsimilarlyunspecied initialdata,requiresevolutionoverasubstantialnumber oforbits. Foracircularorbit,theradialcomponentoftheself-force @ r ~ R isconservativeand generallymorediculttocalculate.Wewereabletomatchmo reaccurateanalyses [ 47 50 60 61 ]toabout0 : 1%.Withthesphericalharmonicdecomposition,ouranalysi s wentupto L =39.Thisrelativelyhighnumberisdueprimarilytotheslow polynomial 108

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convergenceresultingfromthemode-decompositionofthes elf-force.Weexpectthistobe endemicinallself-forcecalculationsthatrelyonsomekin dofspectraldecomposition,as itisthepenaltyincurredwhenonerepresentsobjectsoflim iteddierentiabilityinterms ofsmoothfunctions. AtechniquesimilartothatdescribedinRef.[ 50 ]couldpossiblymitigatethis weakness.Forourspecicimplementation,wecouldchooset ocalculateandsummodes onlyupto,say l =15,andthentakeadvantageoftheknownasymptoticfall-o in l showninEq.( 4{30 ).Usingthecomputedmodes,wedeterminethecoecientsint he expectedfall-ofortheself-force,andthenusingthese,a nalyticallycompletethesumto l = 1 .Thisresultsinaslightlymoreaccurateresultfor @ r ~ R .Asimilarprocedureof \tting"toaknownasymptoticfall-omightproveusefulif onechoosestoimplement eldregularizationusingspectralmethods. Weexpecteldregularizationtobebestimplementedona(3+ 1)nite-dierence code,withmeshrenementinthevicinityofthechargetobet terresolvethelimited dierentiabilityofouranalyticallyconstructedsourcef unction.Suchaprocesswill amelioratetheproblemofslowpolynomialconvergenceaili ngtypicalmode-sum prescriptions. FortheEMRIproblemtoday,thereisgreatinterestincalcul atingtherateofenergy beingradiatedforapointmassorbitingarotatingblackhol eandinusingtheresultto modifytheorbitofthemasswithsomeversionofanadiabatic approximation.Fora generalorbit,theenergyruxisnoteasytodetermine.Curre ntmethodsusetheaxial symmetryoftheKerrgeometrytoseparateoutonedimension, andthendealwitha (2+1)Dproblemfortheradiationfromapointmass.Therepre sentationofapointmass onagridistypicallyproblematical.Replacinga -functionsourcebyanarrowGaussian [ 62 63 ]isreasonablebutdoesnotaccuratelyreproducefrequency -domainresults.The recentdistributionofa -functionoveramodestnumberofgridpointsbySundararaja n etal[ 64 ]appearsmorerobust.Thestrategylaidouthereprovidesan aturalremedyto 109

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thisissue.Insteadofdealingwithawaveequationwitha -functionsource,wesolvean equivalent problemwitharegularanddistributedsource.Theresultsd isplayedinFigures 4-7 and 4-8 clearlypointtotheeectivenessandaccuracyofourmethod Beyondthenumericalmodelingof functionsourcesthough,themethodofeld regularizationprovidesdirectaccesstotheself-force,w hichisessentialinafully-consistent treatmentofparticlemotionandwavegeneration.Currentm ethodsunderdevelopment arebaseduponenergyandangularmomentumruxcalculations thatwillcertainlymiss conservativeself-forceeects.Thesemethodsrelyuponru xintegralsevaluatedinthe wave-zoneandsomeorbitaveragingorpost-processingtoe ectthechangeinorbital energyorangularmomentum,whicharediculttoimplementc arefully[ 65 66 ]andto justifyrigorously.Themoredirectapproachoflocallycal culatingtheself-forcetoupdate theparticleorbithasbeenlargelyavoidedbecauseofthepr ohibitivecomputational expenseassociatedwithmode-sumcalculationsoftheselfforce.Ina(3+1)nite dierencingimplementationofeldregularization,calcu latingtheself-forceisnomore expensivethanperforminganumericalderivativeandpossi blyaninterpolation.Assuch, itrepresentsastepforwardtowardsthegoalofecientlypr oducingconsistentnumerical modelsofparticlemotionandradiationincurvedspacetime TherecentproposalofBarack,GolbournandSago[ 67 { 69 ]isclosestinspirittoour methodofeldregularization.Theymodelapointchargewit hadistributedeective sourcederivedinsteadfromtheir\puncturefunction",whi chisquitesimilartoour ~ S Theybasetheirconstructionofthepuncturefunctiononthe `direct'+`tail'decomposition, ratherthanontheGreenfunctiondecompositionin[ 42 ]thatnaturallyprovidesour regularizingsingulareld ~ S .Theircurrentpuncturefunction,however,appearsto preventthemfromcalculatingaself-force.Moreover,inan ticipationofaKerrbackground application,theyenvisageusinga(2+1)code,necessitati ngamode-sumoveramode index m ,whichwillagainfeaturethecharacteristicpolynomialco nvergenceofthis approachtoself-forcecalculation.Thisisdemonstratedi nFig. 4-10 .Usingourresults,we 110

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performpartialsumsover l ,i.e. ( r r R ) m L X l> j m j ( r r R ) lm ; (4{32) togettheresultingfall-oin m .Weobserveafall-ocloseto1 =m 4 inthemodes. Consequently,ifweweretofollowBaracketal's m -modeprescription,theself-forcewould convergeas1 =m 3 1e-09 1e-08 1e-07 10 12 14 16 18 20 22 24 |(F r ) m |m result 1/m 2 fall-off 1/m 4 fall-off Figure4-10.( r r R ) m versus m .Ourresultsshowan m -fall-oclosestto1 =m 4 Tosummarize,ourmethodofeldregularizationappearstoa ddresstwoimportant issuesinthecontextofEMRIsimulations:(a)numericallyr epresenting -functionsources, and(b)calculatingtheself-force.Whatweshowhere,witho urtestofascalarchargeina circularorbitoftheSchwarzschildgeometry,isacarefull ycontrollednumericalexperiment providinguswithdetailedinformationabouttherelations hipbetweentheapproximation for S andtherateofconvergenceoftheself-force.Thetestisele mentarywhencompared tothecaseofapointmassemittinggravitationalwavesfrom agenericorbitoftheKerr 111

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geometry,whichistheappropriatemodelforEMRIs.Neverth eless,ourresultslend condencetotheviabilityofourgeneralprescription. Inthenextchapter,weshalllookathowwellitperformswith (3+1)codes,whichare itsmostnaturalsetting. 112

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CHAPTER5 SELFFORCEWITH(3+1)EVOLUTIONCODES Aself-forcecalculationwith(3+1)evolutioncodesforthe waveequationhasnever beendonebefore.Aswerecall,thisislargelyduetotwocons iderations:(a)representing adeltafunctiononathree-dimensionalgridhasbeendicul t(e.g.pastattemptsthat usenarrowGaussiansasrepresentationshaveyieldedpoorr esults);and(b)theprocessof regularizationwhichisessentialincalculatingaselffor cehashithertobeendonethrough modecomponentsinsteadofthefullelditself,renderinga (3+1)implementationwhich dealswiththefullretardedeldimpractical.Thissituati onisratherunfortunate,given thatthenumericalrelativitycommunityhasbeengoingthro ughasortofGoldenAge, havingmadedramaticbreakthroughsinmodelingthemergers ofblackholebinariesof comparablemass.Ifanything,thisindicatesthatthereexi stspowerful(3+1)codesthat oughttobeutilizedtoshedlightontheextreme-mass-ratio regimeofblackholebinaries aswell.Themethodwehavedevelopedprovidesameansforthi stohappen. Inthischapter,wepresentresultsofapreliminaryimpleme ntationofourtechnique on(3+1)codes.ThisworkwasdoneincollaborationwithW.Ti chyofFloridaAtlantic University,andP.DieneroftheCenterforComputationalTe chnologyatLouisianaState University.Previously,weimplementedourtechniquewith a(1+1)time-domaincode, whichinasense,runscontrarytothemotivationbehindourm ethodintherstplace ofprovidinganeasiermeansforaccessingtheself-forceby avoidingmode-sumsandthe divergentretardedeld.Wearguehoweverthatthe(1+1)exp erimentwasdonemainly toestablishproofofprinciple,andtogaininsightintothe newprescriptionbylinking ittothemorefamiliarmode-sumscheme.Asthisinitialtest hasmetwithencouraging success,wehavegoneaheadandimplementedourmethodasint ended,on(3+1)codes thatdirectlycomputethefulleldandnotcomponentsofit. Thecalculationsinthischapterweredonewithtwoindepend entcodesoriginally designedtosolveEinstein'seldequations.Onewasbasedo nnitedierences(withP. 113

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Diener)andtheotherwasapseudospectralcode(withW.Tich y).Wehaveadaptedboth tosolveoureectivescalarwaveequation,Eq.( 3{11 ),instead. Werevisitthetoyproblemalreadydiscussedintheprevious chapter:ascalarcharge inacircularorbitaroundaSchwarzschildblackhole.Wepro videabitmorediscussion thoughofthesimplephysicsofthissystem,derivingtherel ationshipbetweenthetime componentoftheselfforceandtheenergyruxacrosstheeven thorizonandinnity.In theprocess,wederiveexplicitformulasforthescalarener gyruxthatarenumerically evaluatedwiththe(3+1)codes.Itmustbeemphasizedthatth ecaseofascalarchargein acircularorbitischosenagainnotonthebasisastrophysic alrealism,butbecauseitisthe simpleststrong-eldselfforceproblemthatcanbedone,fo rwhichthereexistsconcrete resultsintheliterature,makingitidealfortestingnewme thods. Inthischapter,wecalculatetheselfforceonthechargeand thescalarenergy ruxacrosstheeventhorizonandinnity.Theagreementweac hievefortheself-force (especiallyforthe r -componentoftheself-force)islimitedprimarilybythere solutionof thegridinthevicinityofthecharge.Weachievegoodagreem entwithresultsarrivedat withmode-sumcalculations: < 0 : 5%for t -componentand 1 : 5%fortheconservative r -component.Moreover,wedemonstratethattheruxacrossth eeventhorizonand innitymatchisconsistentwithwhatisexpectedfromtheco mputed t -component. Thisisencouraging,sincepreviouscalculationsofgravit ationalwaveruxeswithnarrow Gaussiansandmorereneddeltafunctionrepresentationso nagridhaveyieldedasmuch as 29%[ 62 ]andmorerecently 1%accuracies[ 63 64 ],respectively. 5.1Preliminaries Thegoalhereistocomputetheself-forceonascalarchargeb yrstevolvingthe regularizedwaveequation( 3{11 ): r 2 R = S e : (5{1) where R isa C 2 atthelocationofthechargeasaresultofthe C 0 natureof S e (Eq. ( 3{9 )),ascurrentlyconstructed.Werecallthatthislackofdi erentiabilitycanbe 114

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improvedbyfurtherreningourapproximationforthesingu lareld,whichwouldrequire higherorderTHZcoordinates. Duetotheregularityoftheregulareld,wecantakeitsderi vativeatthelocation ofthecharge.Andbecauseofhow S e isconstructedwiththeDetweiler-Whiting decompositionoftheretardedeld,thisderivativegivest heself-force: F a = r a R : (5{2) 5.1.1Kerr-SchildCoordinates TheKerr-SchildformoftheKerrmetrichasbecomeofcommonu seinnumerical relativity.Inparticular,thetwocodesweuseareconvenie ntlyset-upforKerr-Schild coordinates,soweprovideabriefdiscussionofthesecoord inatesinthissection. TheKerrmetricinBoyerandLindquistcoordinatesis ds 2 = 2 [ dt a sin 2 d ] 2 + sin 2 2 [( r 2 + a 2 ) d adt ] 2 + 2 dr 2 + 2 d 2 (5{3) where r 2 2 Mr + a 2 + Q 2 ; 2 r 2 + a 2 cos 2 : (5{4) Here, m isthemassoftheblackhole, aM istheangularmomentumand Q istheelectric charge. InKerr-Schildcoordinates( ~ t;x;y;z ),theKerr-Newmanmetric[ 70 ] g ab = ab + Hk a k b (5{5) where H = 2 Mr Q 2 r 2 + a 2 ( z=r ) 2 (5{6) and k a dx a = r ( xdx + ydy ) r 2 + a 2 + a ( xdy ydx ) r 2 + a 2 z r dz d ~ t (5{7) 115

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isaningoingprinciplenullvector. Kerr-SchildcoordinatesarerelatedtoBoyerandLindquist coordinatesby x + iy =( r + ia ) e i ~ sin ;z = r cos ; ~ t = ~ V r (5{8) and d ~ V = dt +( r 2 + a 2 ) dr= d ~ = d + adr= : (5{9) Also, d ~ t = dt + r 2 + a 2 dr dr = dt + 2 Mr dr: (5{10) FortheSchwarzschildgeometry: ~ = (5{11) and k a dx a = d ~ t + xdx r + ydy r + zdz r = d ~ t + dr = dt + r r 2 M dr (5{12) g ab dx a dx b = d ~ t 2 + dx 2 + dy 2 + dz 2 + H dt + r r 2 M dr 2 = dt + 2 M r 2 M dr 2 + dr 2 + r 2 d n 2 + 2 M r dt + r r 2 M dr 2 = (1 2 M=r ) dt 2 + dr 2 1 2 M=r + dr 2 + r 2 d n 2 : (5{13) TheBoyer-LindquistcoordinatesreducetoSchwarzschildc oordinates f t;r;; g inthe Schwarzschildlimit.Explicitly,thetransformationfrom SchwarzschildtoKerr-Schild 116

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coordinatesissimply: ~ t = t +2 M ln r 2 M 1 (5{14) x = r sin cos (5{15) y = r sin sin (5{16) z = r cos ; (5{17) Fromtheseequationsitisclearthattheoneoftheconvenien cesaordedbythese coordinatesliesintheirbeingMinkowksi-like,exceptfor the r -dependenttimelaginEq. ( 5{14 ). r KS x 2 + y 2 + z 2 equalstheSchwarzschildradialcoordinate r Schw ,butconstant~ t andconstantt surfacesarenotthesame.Tocompute F t and F r Schw withourtwocodes, thecomponentsoftheselfforceinSchwarzschildcoordinat es,whosecorrectvalueswe knowfromtheliterature,wetakeintoaccountthefollowing transformations: F t = F ~ t (5{18) F r = 2 M r KS 2 M F ~ t + x r KS F x + y r KS F y : (5{19) 5.1.2Self-ForceforPerpetualCircularOrbitsofSchwarzs child Thephysicalproblemwedealwithhereisidenticaltothatof Chapter3:aparticleof unitmass( m =1)withunitscalarcharge( q =1)inaperpetualcircularorbit( R =10 M ) aroundaSchwarzschildblackhole.Sincetheparticlehasma ssandscalarchargeits motionresultsintheemissionofgravitationalandscalarr adiation.Inordertokeepthe particleinthisconguration,aforceisnecessaryinorder tocounteractthebackreacting gravitationalandscalarself-forcesonit.Inouranalysis ,thisgravitationalself-forceis ignored,andwefocusonlyonthescalarself-force. Withthechargeinperpetualcircularmotion,thesystemmus tasymptotically approachahelicallysymmetricendstate,intheabsenceofo therexternalsourceswhich mayviolatethissymmetry.Inotherwords,theremustexista helicalKillingvector a 117

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suchthatanyeld G mustasymptoticallyapproach: $ G =0 : (5{20) InSchwarzschildcoordinates,thisKillingvectorissimpl y: a = @ @t a +n @ @ a : (5{21) Thisimpliesthat,forthecircularorbitcasewehavechosen ,thereareonlytwoindependent componentsoftheself-force, F t and F r ,since F isxedby F t accordingto: $ R = @ t R +n @ R = F t +n F =0 ; (5{22) and F =0,byvirtueofthesystem'sbeingrerectionsymmetricabou ttheequator. 5.1.3How F t andtheEnergyFluxareRelated ForascalarchargegoinginacircularorbitaroundaSchwarz schildblackhole,there existsadirectrelationshipbetweenthetime-componentof theself-forceonthescalar chargeandtheenergyruxatspatialinnityandacrosstheev enthorizon. Intheabsenceofexternalelds,themotionofascalarparti cleisgovernedbytheself forceactingonit: ma b = q ( g bc + u b u c ) r c R F a : (5{23) Theenergyperunitmass(i.e.specicenergy)ofaparticlea longageodesicwitha four-velocity u b isjust E = t b u b ,where t b isthetime-translationKillingvectorofthe Schwarzschildspacetime.Therateofchangeinthisspecic energyperproperunittime is E u c r c E = u c u b r c t b t b u c r c u b = t b a b ,since r ( c t b ) =0.InSchwarzschild coordinatesthisisjust E = a t .Evaluatingthisonaparticlemovinginacircularorbit, Eqn.( 5{23 )givesus: E j p = q m ( @ t R + u t u b r b R ) j p (5{24) = q m @ t R j p ; (5{25) 118

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where p signiesthelocationoftheparticle.Thesecondterminthe rstequalityvanishes foracircularorbit: u a r a R j p =( u t @ t R + u @ R ) j p (5{26) =(d t= d )( @ t R +n @ R ) j p (5{27) =(d t= d ) $ R j p (5{28) =0 ; (5{29) where a istheKillingvectorassociatedwiththehelicalsymmetry, sothatthethird equalityvanishesduetoEq.( 5{20 ).Thus F t = m E = q@ t R : (5{30) Thetimecomponentoftheselfforce(inSchwarzschildcoord inates)isthenjustthe amountofenergylostbytheparticleperunitpropertime. Thisenergylossmustobviouslyberelatedtothescalarener gyrux.Weshallnow derivetheserelationshipsandalsotheexplicitexpressio nsfortheseruxesinKerr-Schild coordinates. Thescalareldproducedbythechargeisdeterminedbythee ldequation: g ab r a r b = 4 q Z (4) ( x z ( t )) p g d (5{31) Multiplybothsides t a r a andintegrateover V ,whichwetaketobethe4-volume betweenconstantKerr-Schild t -surfaces t = t i and t = t f ,and t a isthetime-translation KillingvectorofSchwarzschild. Z V t a r a g cd r c r d p g d 4 x = 4 q Z V t a r a Z (4) ( x z ( t )) p g d p g d 4 x (5{32) Thissimpliesto: Z V t a r a r b r b p g d 4 x = 4 q Z t f t i ( t a r a ) j p ( dt=d ) 1 d t (5{33) 119

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Denotethetime-componentoftheself-force F t qt a r a j p .Wenoticealsothatthe integrandontheleftcanbeexpressedas: t a r a r b r b = t a g ab r b r 2 =4 t a g ab r c T bc (5{34) where T bc isthestress-energytensorforthescalareld: T bc = 1 4 r b r c 1 2 g bc r d r d : (5{35) Wethenhave: Z V t a r c T ca p g d 4 x = Z t f t i F t ( dt=d ) 1 d t: (5{36) Since t a isaKillingvector, r ( c t a ) =0,and T ca issymmetricinitsindices,wehave t a r c T ca = r c ( t a T ca ).Thus, Z V r c ( t a T c a ) p g d 4 x = Z t f t i F t ( dt=d ) 1 d t: (5{37) I @ V t a T ca d c = Z t f t i F t ( dt=d ) 1 d t: (5{38) whered c isthedirectionalvolumeelementoftheboundary @ V ,whichiscomprisedof thehypersurfaces t = t i t = t f ,theeventhorizonat r =2 M ,andspatialinnity. t a T a c is nothingbuttheconservedcurrentforthescalareld. Nowwespecializetothecaseofascalarchargeinaperpetual circularorbitof angularvelocityn.Inthisconguration,thereexistsahel icalKillingvector a ,whichis just a = @ @t a +n @ @ a (5{39) WebreakupthelefthandsideofEq.( 5{38 )intothefourhypersurfaceintegrals: I @ V t a T ca d c = Z H ^ k c r 2 d dn+ Z 1 ^ r c r 2 d t dn+ Z t = t f ^ n c p h d 3 x Z t = t i ^ n c p h d 3 x # t a T ca : (5{40) 120

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^ k a isthenullgeneratoroftheeventhorizon,andtherestofthe hattedquantitiesarethe respectiveoutwardunitnormalvectorstotheotherhypersu rfacesmakingup @ V is anarbitraryparameteronthenullgenerators ^ k a oftheeventhorizon.Fromthehelical symmetryoftheproblemitiseasytoseethatthelasttwointe gralsjustcanceleach otherout.Thissimplymeansthatenergycontentineachcons tantt hypersurfaceisthe same.Butwecanshowthisexplicitly.Considerthetimeevol utionofthetotalenergyina t -hypersurface: d d t Z t ^ n c t a T ac p r ( r 2 M ) r 2 d r dn= Z t @ @t (^ n c t a T ac ) p r ( r 2 M ) r 2 d r dn = n Z t @ @ (^ n c t a T ac ) p r ( r 2 M ) r 2 d r dn = n Z d d d ZZ ^ n c t a T ac p r ( r 2 M ) r 2 sin d r d =0 (5{41) wherewehaveusedthehelicalsymmetry $ F = @ @t +n @ @ F =0,forany F .Inother words,timeevolutionisreallyjustaxialrotation. Thus,forthecaseofascalarchargeinacircularorbit r = R ,Eq.( 5{38 )becomes: Z H ^ k c r 2 d dn+ Z 1 ^ r c r 2 d t dn t a T ca = r 1 3 M R Z t f t i F t d t: (5{42) Forconvenience,wemaysetthearbitraryparameter onthehorizontobe t .Ifwe thendierentiatebothsideswithrespectto t ,wenallyget: d E d t r =2 M + d E d t r = 1 = r 1 3 M R F t : (5{43) where d E d t r =2 M = I r =2 M t a T ca ( ^ k c ) r 2 dn(5{44) d E d t r = 1 = I r = 1 t a T ca ^ r c r 2 dn(5{45) 121

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Thesearethegeneralformulasfortheenergyruxatspatiali nnityandtheevent horizon.Inthenextsection,wewritethemoutexplicitlyin termsof anditsderivatives. Noterstthoughthat t inEq.( 5{43 )istheKerr-Schildtime.ButfromEq.( 5{18 ), weseethatthetime-componentsoftheselfforceinKerr-Sch ildandSchwarzschild coordinatesarethesame.Thus,withEq.( 5{30 ),weseethat d E d t r =2 M + d E d t r = 1 = m d E p d t ; (5{46) where E p = t a u a isthespecicenergyofaparticlemovingalongageodesic.T hisis justastatementoftheconservationofenergy:theenergylo stbythechargeperunit Kerr-Schildtimeisalsotheenergyrowingthroughtheevent horizonandinnityperunit Kerr-Schildtime.5.1.4ScalarEnergyFluxinKerr-SchildCoordinates ForconveniencewewriteEqs.( 5{44 )and( 5{45 )intermsof anditsderivatives, inKerr-Schildcoordinates.Theseformulasareessentiall ythesameexceptfortheirunit normals,whereoneisnullandtheotherspacelike. WerstnotethatinKerr-Schildcoordinates,theSchwarzsc hildmetricanditsinverse aresimply: g ab = ab + 2 M r k a k b (5{47) g ab = ab 2 M r k a k b (5{48) k a = 1 ; x i r ; k a = 1 ; x i r (5{49) where r = p x i x i and ab =diag( 1 ; 1 ; 1 ; 1). Webeginrstwiththeenergyruxthroughtheeventhorizon.T heeventhorizon isessentiallyasurfaceofconstantretardedtime u = t ( S ) r ( S ) 2 M ln( r ( S ) = 2 M 1), wherethesubscript S meansthattheseareSchwarzschildcoordinates.InKerr-Sc hild 122

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coordinatesthesesurfacesofconstant u are: t = r +4 M ln( r= 2 M 1)+ C (5{50) where C isjustaconstant.Anyparticularsurfaceinthisfamilycan bedenedparametrically bytheequations: t = (5{51) x = r ( )sin cos (5{52) y = r ( )sin sin (5{53) z = r ( )cos (5{54) where r ( )isdenedimplicitlybytherelation = r +4 M ln( r= 2 M 1) : (5{55) Withthis,thenullgeneratorofthesurface(whichisalsono rmaltoit)is k a @x a @ = 1 ; r 2 M r +2 M x i r : (5{56) Withthestress-energytensorgivenbyEq.( 5{35 )andusingtheexpressionsgivenin Eqs.( 5{47 )-( 5{49 ),asmallamountofalgebrayields: T ab t a k b = 2 + r 2 M r +2 M n i @ i + 1 2 r 2 M r +2 M @ c @ c (5{57) wheretheoverdotmeansaderivativewithrespectto t .At r =2 M ,theenergyruxisthen simplyjust: d E d t r =2 M = 4 M 2 I 2 dn : (5{58) Forthecaseoftheruxatspatialinnity,weconsideraconst ant r surface.The normalone-form a associatedwiththisis a @ a r =(0 ;x i =r ).Thecorresponding 123

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normalizedvectoristhen: r a = g ab b = ab 2 M r k a k b b = r r r 2 M 2 M r ; 1 2 M r x i r (5{59) Thisleadstothefollowing: T ab t a r b = r r r 2 M 2 M r 2 + 1 2 M r @ r (5{60) Takingthelimit r !1 ,thisreducestotheratspacetimecase: T ab t a r b = @ r (5{61) Andso,wehavefortheenergyruxatspatialinnity: d E d t r = 1 =lim R !1 R 2 I R @ r dn : (5{62) Theruxthroughanite r = R (whichiswhatisneededhere)isjust: d E d t r = R = R 2 r R R 2 M I R 2 M R 2 + 1 2 M R @ r : (5{63) OneoftheinternalchecksweperformistoverifythatEq.( 5{43 )holdsbycomputingthe t -componentoftheself-forceandtheruxesgiveninequation s( 5{58 )and( 5{63 ). 5.1.5EectiveSourceandtheNewWindowFunction Forthe(1+1)testdiscussedinthepreviouschapter,awindo wfunctionhavinga Gaussianprolein r wasemployed.[SeeEq.( 4{7 )].Weinitiallyusedthissamewindow inour(3+1)runs.Wequicklydiscoveredthoughthatitwasar atherpoorchoicefora windowfunction.Whileitdidsatisfyalltheconditionsdis cussedinChapter3,itsmain drawbackwasthatitintroducedalotoflarge-amplitudestr uctureawayfromtheparticle. [SeeFig.( 3-1 )].Thiswasnotaproblemforthe(1+1)implementationbecau seofthehigh r -resolutionthatwasused( M= 25).Inusingcoarsergridsforthe3Druns,thelarge structuresduetothewindowwouldtendtodominatetheeect ivesource.Asaresult,our self-forceresultswouldlooklikewidenoiseenvelopescen teredaroundthecorrectvalue. 124

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Inthehopeofnarrowingthesenoiseamplitudes,whileguide dbythepropertiesawindow functionmusthave,weturnedtoonewithafewmoreparameter sthatwecouldtune,and onewhoseimpactontheeectivesourcewasnotasdramaticas ouroriginalone.Like beforethough,weshallchooseasphericallysymmetricwind owfunctioninkeepingwith thesphericalsymmetryofthebackgroundspacetime. Considerthesmoothtransitionfunction f ( r j r 0 ;w;q;s ) = 8>>>>>>><>>>>>>>: 0 ; r r 0 1 2 + 1 2 tanh 24 s 0@ tan 2 w ( r r 0 ) q 2 tan 2 w ( r r 0 ) 1A 35 ;r 0 <>: f ( r j ( R 1 w 1 ) ;w 1 ;q 1 ;s 1 ) r R 1 f ( r j ( R + 2 ) ;w 2 ;q 2 ;s 2 ) r>R (5{65) wheretheregion R 1
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Thissatisesallthekeyrequirementsforawindowfunction (andmore):(a) W ( R )= 1;(b) d n W=dr n j r = R =0,forall n ;(c) W =0if r 2 [0 ; ( R + 2 + w 2 )] [ [ R 1 w 1 ; 1 ) (thusmakingittrulyofcompactsupport);andagain(d) W =1if r 2 [( R 1 ) ; ( R + 2 )]. Fortheactualrunswehaveperformed,wepickedmadethefoll owingchoicesforthese parameters. 1 = 2 =0; q 1 =0 : 6 ;q 2 =1 : 2; s 1 =3 : 6 ;s 2 =1 : 9; w 1 =7 : 9 ;w 2 =20(5{66) Theinnerwidth w 1 waschosensothatthewindowandeectivesourcegotoexactl y zerojustoutsidetheeventhorizon.Therestwerepickedaft erextensivelylookingat manyparametercombinations.Theprimarycriteriaweresim plythattheeective sourcebesucientlysmalleverywhereandthatitdidnotpos sessstructureatextremely smallscales.Asystematicsearchfortheoptimalsetofpara metersvis-a-visitseecton self-forceaccuracywasnotconductedhere,andisleftforf uturework. Akeyfeatureofthenewwindowfunctionisthatwhileitdoesp roducesome unphysicalstructure,forawiderangeofparameterchoices ,thesearesignicantlylesser thanthoseproducedbyouroriginalone.Compareaplotofthe neweectivesourcein Fig.( 5-1 )withtheoriginaloneinFig.( 3-1 ).Onenotesimmediately,thattheadditional structurebroughtbythenewwindowfunctionisalmosttwoor dersofmagnitudesmaller thanthatoftheoriginalone.Moreover,thisstructureisma inlylocatedat r
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-0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 2 4 6 8 10 12 -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.01 0.02 0.03 0.04 S eff x y S eff Figure5-1.Equatorialproleofthenew S e adaptivemesh)forthebestresults.Stillthough,itisanon -singularrepresentationofa pointchargesource,whichmakesitamenabletothe(3+1)cod eswehaveused. 5.2Descriptionofthe(3+1)Codes 5.2.1Multi-BlockCode Wesolvethewaveequationonaxedbackgroundwithasource r r = r ( g r )= S e (5{67) onamulti-blockdomainusinghighordernitedierencing. Thecodeisdescribedin moredetailin[ 71 ],herewewilljustsummarizeitsproperties.Weusetouchin gblocks, wherethenitedierencingoperatorsoneachblocksatise saSummationByParts (SBP)propertyandwherecharacteristicinformationispas sedacrosstheblockboundaries usingpenaltyboundaryconditions.BoththeSBPoperatorsa ndthepenaltyboundary conditionsaredescribedinmoredetailin[ 72 ].Thecodehasbeenextensivelytestedand 127

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-2e-07 0 2e-07 4e-07 6e-07 8e-07 1e-06 1.2e-06 1.4e-06 9.9 9.925 9.95 9.975 10 10.025 10.05 10.075 10.1 -0.1 -0.075 -0.05 -0.025 0 0.025 0.05 0.075 0.1 0 4e-07 8e-07 1.2e-06 1.6e-06 S eff x y S eff Figure5-2. S e zoomedinatthelocationofthecharge. -2e-07 0 2e-07 4e-07 6e-07 8e-07 1e-06 1.2e-06 1.4e-06 S eff at the location of the charge 9.88 9.9 9.92 9.94 9.96 9.98 10 10.02 10.04 10.06 10.08 10.1 x -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 y Figure5-3.Contourplotof S e atthelocationofthecharge. 128

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wasusedin[ 73 ]toperformsimulationsofscalareldinteractingwithKer rblackholes andwasusedtoextractveryaccuratequasinormalmodefrequ encies. Afterthestandard(3+1)split,thewaveequationiswritten in1storderintime,1st orderinspaceformas @ t = ; (5{68) @ t = i @ i + p r @ i p r i + p rH ij v j 2 S e ; (5{69) @ t v i = @ i ; (5{70) where r isthedeterminantofthespatialmetric r ij isthelapsefunction, i istheshift vectorand H ij = r ij i j = 2 Theonlynecessarymodicationstothecode,inordertoappl yittotheproblemat hand,weretheadditionofthesourceterminEq.( 5{69 )andcodetointerpolatethetime andspatialderivatives v i ofthescalareldtothelocationoftheparticle. Boundaryconditionsandinitialdata Thesimulationsbelowwereallperformedusingthe6-blocks ystem,providinga sphericalouterboundaryandsphericalinnerexcisionboun darywithoutanycoordinate singularities.Theinnerradiuswaschosentobe r inner =1 : 8 M andtheouterboundary waschosentobeat r outer =400 M .Attheouterboundaryweuseapenaltyboundary conditiontoenforcezeroincomingcharacteristics.Atthe innerboundary,thegeometry ensuresthatallcharacteristicsleavethecomputationald omain,i.e.therearenoincoming modes. Aswedonot,apriori,knowthecorrecteldconguration,we startthesimulation withzeroscalareld ( t =0)=0,zerotimederivative ( t =0)=0andzerospatial derivatives v i ( t =0)=0,asifthescalarchargesuddenlymaterializesat t =0.Evolving foracoupleoforbits,allowstheinitialtransienttogoawa yandthesystemtoapproacha 129

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helicallysymmetricendstate.Weusedthe8-4diagonalnorm SBPoperatorsandadded someexplicitcompatibleKreiss-Oligerdissipationtoall evolvedvariables. 5.2.2PseudospectralCode Wesolvethewaveequation r r = r ( g r )= (5{71) onaxedSchwarzschildbackgroundwithasource.Forthebac kgroundweuseKerr-Schild coordinatessothat g = +2 Hl l (5{72) where H = M=r and l =( 1 ;x=r;y=r;z=r ).Thecorrespondinglapseandshiftare =1 = p 1+2 H; (5{73) i =2 l i H= (1+2 H ) : (5{74) Inordertonumericallyevolve wealsousetheSGRIDcode[ 74 ].Thiscodeusesa pseudospectralmethodinwhichallevolvedeldsarerepres entedbytheirvaluesatcertain collocationpoints.Fromtheeldvaluesatthesepointsiti salsopossibletoobtainthe coecientsofaspectralexpansion.Asin[ 74 ]weusestandardsphericalcoordinateswith ChebyshevpolynomialsintheradialdirectionandFouriere xpansionsinbothangles. Withinthismethoditisstraightforwardtocomputespatial derivatives.Inordertoobtain asystemofequationsthatisrstorderintimeweintroducet heextravariable = 1 @ t i @ i : (5{75) 130

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Forthetimeintegrationweusea4thorderaccurateRunge-Ku ttascheme.Asin[ 74 ]we ndthatitispossibletoevolvetheresultingsystem @ t = i @ i @ t @ t = i @ i g ij @ i @ j + i @ i g ij @ i @ j + K + (5{76) instablemannerifweuseasinglesphericaldomain,whichex tendsfromsameinner radius R in (chosentobewithintheblackholehorizon)toanmaximumrad ius R out .In thiscaseoneneedsnoboundaryconditionsat R in sinceallmodesaregoingintothe holethereandarethusleavingthenumericaldomain.At R out wehavebothingoing andoutgoingmodes.Weimposeconditionsonlyoningoingmod esanddemandthat theyvanish.However,sinceweneedmoreresolutionnearthe particleitisadvantageous tointroduceseveraladjacentsphericaldomains.Inthatca seonealsoneedsboundary conditionstotransfermodesbetweenadjacentdomains.Wew erenotabletondinter domainboundaryconditionswithwhichwecouldstablyevolv ethesystem( 5{76 ).For thisreasonweintroducethe3additionalelds i = @ i ; (5{77) sothatweneedtoevolvethesystem: @ t = i @ i @ t @ t = i @ i g ij @ i j + i i g ij i @ j + K + @ t i = j @ j i + j @ i j @ i @ i : (5{78) Notethatthissystemisnowsecondorderinbothspaceandtim eanditcanbestably evolvedusingthemethodsdetailedbelow.Alsonoticethatw eevolvetheCartesian 131

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componentsofallelds.Thecharacteristicmodesofthissy stemare u = i i u 0i = i j j i u = : (5{79) Forourshellboundaries i = l i p g ll i = g ij j g ll = g ij l i l j .Theelds u 0i and u have velocity i ,while u havevelocity i i Domainsetup,boundaryconditions,initialdata Wetypicallyuse4adjacentsphericalshellsasournumerica ldomains.Theinnermost shellextendsfrom R in =1 : 9 M to R =10 M .Thenexttwointerdomainboundariesareat 18 : 1 M and27 : 5 M .Theoutermostshellextendsfrom27 : 5 M to R out =210 M Asmentionedabovewedonotimposeanyboundaryconditionsa t R in .At R out we imposeboundaryconditionsinthefollowingway.Firstweco mpute @ t u + fromtheelds @ t @ t and @ t i attheboundary.Thenweimposetheconditions @ t u = + i i : @ t u = =r @ t u 0i =( k i k i ) @ k @ t (5{80) ontheingoingmodes.Finallywerecompute @ t @ t and @ t i from @ t u @ t u 0i and @ t u Fortheinterdomainboundarieswesimplycompute @ t u @ t u 0i and @ t u from @ t @ t and @ t i attheboundaryineachdomain.Ontheleftsideoftheboundar ywe thensetthevaluesoftheleftgoingmodes @ t u @ t u 0i and @ t u equaltothevaluesjust computedontherightsideoftheboundary.Ontherightsideo ftheboundaryweset @ t u + equaltothevaluecomputedontheleftside.Thisalgorithms implytransfersall modesinthedirectioninwhichtheypropagate. 132

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Inordertoobtainastableevolutionweapplyalteralgorit hmintheangular directionsaftereachevolutionstep.Weprojectourdouble Fourierexpansiononto sphericalharmonics.Aftersettingthehighest l modein andtozerowerecompute alleldsatthecollocationpoints.Thislteralgorithmre movesallunphysicalmodesand alsoensuresthat andalwayshaveonelessthanmodethan i Asinitialdatawesimplyuse ==0. 5.3Results Inthissection,wepresentsomeofthekeyresultsofouranal ysis. 5.3.1 F t andtheEnergyFlux Inamode-sumcalculation,the t -componentoftheselfforce, @ t R ,forascalarcharge incircularorbitaroundSchwarzschildiscalculatedextre melyaccurately.Thisisdueto thefactthat @ ret requiresnoregularizationwhenevaluatedatthelocationo fthecharge. Themodesumthenfallsoexponentiallyin l aswasdemonstratedinthetime-domain implementationofChapter4. Initially,weexpectthatthecalculated t -componenttobequiteinaccurateduetobad initialdata.Aftersometimethesetransienteectspropag ateoutofthenumericaldomain andthesystemsettlesdowntoitshelicallysymmetricendst ate.Theplotsbelowallhave thisfeature. TheresultsofFig.( 5-4 )arefromthemulti-blockcode.Itshowsthathelical symmetryisachievedaftertheparticlemakestwofullorbit s( T orb =2 p R 3 =M 200 M ). Thisplotalsofeaturestheeectofgoingtohigherangularr esolution.Foranangular resolutionof20 20angularpointsperpatch,theresultsaredominatedbynoi se.Thisis dramaticallyreducedhoweverwhenthenumberofangularpoi ntsisquadrupled.Curiously though,thisimprovementisnotreplicatedinquadruplingt henumberofangularpoints perpatchfurther.Eachoftheseresultswereforaradialres olutionof r = M= 10; modifyingtheradialresolution(to r = M= 15)didnotsignicantlychangetheresults. 133

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Fig.( 5-5 )plotsthesamefortheresultsoftheSGRIDcode.Thisgures hows thatfortheSGRIDcalculation(unlikethemulti-blockcode ), F t isquiteinsensitiveto increasingangularresolution,forreasonsthatarenotqui teunderstood. N r ;N ,and N arethenumberofcollocationpointsinthe r and directions,respectively.Fortheplot shown N r iskeptconstantat N r =29.Again,itisclearthathelicalsymmetryisreached whentheparticlemakestwofullorbits.Onenotablefeature oftheSGRIDresultsisthe outerboundaryrerectionarrivingattheparticlearound40 0 M .Thisrerectedwaveis miniscule( 10 6 ),andyetitsignicantlyaectstheself-forcewhichitsel fisquitesmall. Themulti-blockcodedoesnotshowthisrerectedwavebecaus eitsouterboundaryis muchfartheraway( R out =400 M ),whereasfortheSGRIDcodethiswasat R out =200 M Thebulkofthebadinitialdatawillfromtheregionwherethe chargesuddenlyappears 10 M ,itisexpectedthenthatfortheSGRIDcode,thererectionsh ouldgettothe particlerightaround400 M (twicethetimefromparticletoouterboundary),whereasth is wouldbearound800 M forthemulti-blockcode. InFig.( 5-6 )wejuxtaposetheresultsfromthetwocodes.Goodagreement is achievedagainuntiltheboundaryrerectionarrivesatthep article. Animportantconsistencycheckfortheseresultsistherela tionbetweenthescalar radiationruxand F t ,asgivenbyEqn.( 5{43 ).Figures( 5-7 )and( 5-8 )aretheresults fromthemulti-blockandSGRIDcodes,respectively.InFig. ( 5-7 )weshowtheresulting fromthe40 40angularresolutionrun,andanextractionradiusof R =150 M .The resultwasconvertedtoaselfforceusingEqn.( 5{43 )foreasycomparison.Alsoplotted aretheactualvaluesfor F t (i.e.computedbytakingthederivativeoftheregularelda t thelocationofthecharge)asafunctionoftime,alsocomput edatthe40 40resolution. Thesearebothcomparedwiththecorrectfrequencydomainre sultrepresentedbythe straightline.computedbyeachcode.Bothshowagreementto within1%. Onenotableobservationcanbedrawnfromtheruxescomputed withthemulti-block code,andthisisthattheruximproveswithlargerextractio nradius.Thisisshownin 134

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3e-05 3.2e-05 3.4e-05 3.6e-05 3.8e-05 4e-05 4.2e-05 4.4e-05 0 100 200 300 400 500 600 F tt 20x20 40x40 60x 60 3.75e-5 Figure5-4. F t computedatdierentangularresolutionsofthemulti-bloc kcode. 3e-05 3.2e-05 3.4e-05 3.6e-05 3.8e-05 4e-05 4.2e-05 4.4e-05 0 100 200 300 400 500 600 F tt Ntheta=64, Nphi=48 Ntheta=80, Nphi=60 Ntheta=96, Nphi=72 3.75e-5 Figure5-5. F t computedatdierentangularresolutionsoftheSGRIDcode. 135

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3e-05 3.2e-05 3.4e-05 3.6e-05 3.8e-05 4e-05 4.2e-05 4.4e-05 0 100 200 300 400 500 600 F tt F t (multi-block) F t (sgrid) 3.75e-5 Figure5-6.Comparing F t resultsfromthemulti-blockandSGRIDcodes. 3e-05 3.2e-05 3.4e-05 3.6e-05 3.8e-05 4e-05 4.2e-05 4.4e-05 200 250 300 350 400 450 500 550 600 Edot, F tt Edot: 40x40, R=300M Extrapolated (4) F t : 40x40 3.75e-5 Figure5-7. E computedatthe40 40angularresolutionofthemulti-blockcode,and outerextractionradiusof R =150 M .Theresultisthenconvertedto F t accordingtoEq.( 5{43 ). 136

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3e-05 3.2e-05 3.4e-05 3.6e-05 3.8e-05 4e-05 4.2e-05 4.4e-05 200 220 240 260 280 300 320 340 360 380 Edot, F tt F t : Ntheta=96, Nphi=72 Edot: Ntheta=96, Nphi=72 3.75e-5 Figure5-8. E computedatthehighestangularresolutionoftheSGRIDcode ,and convertedto F t accordingtoEq.( 5{43 ) Fig.( 5-9 ).Knowingthis,itistemptingtomaketheextractionradius aslargeaspossible. However,howfartheextractionradiuscanbeislimitedbyth efactthattheruxtaken atfartherradiinaturallytakeslongertoequilibriate,si ncethebadinitialdatawaveswill havetopropagatemuchfarther.Moreover,fartherextracti onradiiarereachedearlierby theboundaryrerection.Sobetweenthelateequilibrationt imeoftheruxandtheearly arrivalofthecontaminatingboundaryrerection,therewou ldbeonlyaveryshortinterval withinwhichtheresultsarereliable.InFig.( 5-10 ),onecanseethattheruxcomputed with R =250 M astheextractionradiusdoesnotevenequilibratebeforeit isaectedby thebouncefromtheboundary. Onemaynaturallyaskwhattheruxaccuracycanbeinthelimit ofinnite extraction.Thisisdoneinthenextplot,Fig.( 5-11 ). Inthisplottheresultsfrom R =50 M; 100 M; 150 M; 200 M areusedtoextrapolatethe ruxinthelimitofiniteextractionradius.Firstthough,o nemustaccountforthetime shiftsintheruxes.Obviously,emittedscalarradiationre aches50 M rst,andonlyafter 137

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3.7e-05 3.72e-05 3.74e-05 3.76e-05 3.78e-05 3.8e-05 3.82e-05 3.84e-05 400 405 410 415 420 425 430 435 440 Edott Edot: 40x40, R=50M Edot: 40x40, R=100M Edot: 40x40, R=150M 3.75e-5 Figure5-9.Dependenceofequilibrium E onvariousextractionradii.Fartherextraction radiiisobservedtoyieldbetterresults. 3.74e-05 3.76e-05 3.78e-05 3.8e-05 3.82e-05 3.84e-05 250 300 350 400 450 500 550 Edott Edot: 40x40, R=50M Edot: 40x40, R=100M Edot: 40x40, R=150M Edot: 40x40, R=200M Edot: 40x40, R=250M 3.75e-5 Figure5-10.Evolutionof E usingvariousextractionradii.Notethatfortheextractio n radiusof R =250 M ,arerectedwaveimmediatelycontaminatestherux accuracy. 138

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anintervaloftimewilltheenergycomingoutof R =50 M arriveatthenextextraction radiusat R =100 M .Thiscomplicatestheextrapolationjustabit. Usingthefactthatoutgoingnullgeodesicstravelatcoordi natespeed( r 2 M ) = ( r + 2 M )inKerr-Schildcoordinates,onecanintegrateandndthat thetimedelaybetween thearrivalatvariousradiiareasfollows: IntervalTimeDelay 50 M 100 M 47 : 0351 M 50 M 150 M 95 : 7095 M 50 M 200 M 144 : 572 M Table5-1.Timelags. Shiftingthedatabytheseappropriatetimedelays,weassum eaformfortherux E ( R )atniteextractionradius R givenby: E ( R )= E ( 1 )+ C 1 R + C 2 R 2 + C 3 R 3 : (5{81) Theconstants C 1 ;C 2 ;C 3 and E ( 1 )canthenbesolvedforusingthefoursetsofrux datafromthedierentextractionradii.Theresulting E ( 1 )fromthisprocedureis \Extrapolation(4)"plottedinFig.( 5-11 ).Usingonly C 1 and C 2 ,theextrapolated E ( 1 )isshowninthesameplotas\Extrapolation(3)".Forrefere nce,wealsoinclude thefrequencydomainresultfor F t expressedasaruxwithEq.( 5{43 ).Asexpected, theagreementissignicantlyimproved.Extrapolatingtot heinniteextractionradius, theruxmatchesto 0 : 01%.HavingtheouterboundarysoclosefortheSGRIDruns preventedusfromperformingasimilaranalysisforitsresu lts. 5.3.2TheConservativePiece, F r Theconservativepieceoftheselfforceonascalarchargemo vinginacircularorbit ofSchwarzschildisreallythecrucialquantitytocompute. Thisisthepartofthelocal selfforcethatcannotbeknownbymakingobservationsfaraw ay(unlike F t ,whichis xedbytherux).Inamode-sumscheme,thiswouldbethequant itywhosemodesum convergesas l n ,where n> 1istypicallyasmallnumberdependingonthenumberof 139

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-0.000402 -0.0004 -0.000398 -0.000396 -0.000394 -0.000392 -0.00039 -0.000388 200 250 300 350 400 Fluxt R=50M R=100M R=150M R=200M Extrapolated (4) Extrapolated (3) -0.000394291 Figure5-11. E computedwiththemulti-blockcodeandextrapolatedtoinn iteextraction radius. regularizationparametersonehasaccessto.Inourapproac h,calculating F r (where r is theSchwarzschildradialcoordinate)amountstotakingder ivativesoftheregulareld R andthenusingEq.( 5{19 ). InFigs.( 5-12 )and( 5-13 )aretheevolutionof F r forthemulti-blockandSGRID codes,respectively.Again,onecanseethesensitivityoft heresulttoangularresolutionin themulti-blockcode.Forthelowestangularresolutionuse dinthemulti-blockcode,the F r resultsarecompletelydominatedbynoise.Whenthenumbero fangulardatapoints isquadrupledonce,theresultsimproveconsiderably,butf urtherquadruplingdoesnot provideasdrasticanimprovement.Onetheotherhand,theSG RIDresultsagainappear lesssensitivetoangularresolutionthanthemultiblockco de.Atthehighestangular resolution,themulti-blockcodegivesaresultthatiswith in InFig.( 5-15 )oneseestheSGRIDresultsdramaticallyimprovingwithinc reased radialresolution.Finally,wepresenttheresultsfromthe twocodestogetherinFig. ( 5-16 ).Theagreement 140

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InFigs.( 5-14 )and( 5-15 )weshowtheimpactofincreasingtheradialresolution ontheresultsofeachcode.OneseestheSGRIDresultsdramat icallyimprovingwith increasedradialresolution,whilethemulti-blockremain sfairlyinsensitivetothechange. Finally,wepresenttheresultsfromthetwocodestogetheri nFig.( 5-16 ),butwehavenot includedthepartwheretheSGRIDresultsarealreadycontam inatedbytheboundary rerection. 6e-06 8e-06 1e-05 1.2e-05 1.4e-05 1.6e-05 1.8e-05 2e-05 2.2e-05 0 100 200 300 400 500 600 F rt 20x20 40x40 60x60 1.378448e-5 Figure5-12. F r computedatdierentangularresolutionsonthemulti-bloc kcode. 5.3.3Discussion WesummarizesomeourresultsinTable 5-2 below.Weassesstheaccuracyofour (3+1)resultsbycomparingwiththefrequencydomainresult sof[ 50 ],andseethatover-all weachieveagreementto 1%. Forourruxvaluestoachievethereportedaccuraciesisnote worthy.Thisindicates onceagainthatoureectivesource S e isagoodrepresentationforpointparticles,in placeofdeltafunctionsthatarediculttohandleonagrid. Aspreviouslymentioned, narrowGaussianshavepreviouslybeenemployedforthistas k.In[ 62 ],atime-domain calculationofthegravitationalenergyruxduetoapointma ssorbitingaKerrblackhole 141

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6e-06 8e-06 1e-05 1.2e-05 1.4e-05 1.6e-05 1.8e-05 2e-05 2.2e-05 0 50 100 150 200 250 300 350 F rt Nr=29, Ntheta=64, Nphi=48 Nr=29, Ntheta=80, Nphi=60 Nr=29, Ntheta=96, Nphi=72 1.38e-5 Figure5-13. F r computedatdierentangularresolutionsontheSGRIDcode. 6e-06 8e-06 1e-05 1.2e-05 1.4e-05 1.6e-05 1.8e-05 2e-05 2.2e-05 0 100 200 300 400 500 600 F rt dr=M/10 dr=M/15 1.378448e-5 Figure5-14. F r computedat2dierentradialresolutionsonthemulti-bloc kcode. 142

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6e-06 8e-06 1e-05 1.2e-05 1.4e-05 1.6e-05 1.8e-05 2e-05 2.2e-05 0 50 100 150 200 250 300 350 F rt Nr=29, Ntheta=64, Nphi=48 Nr=41, Ntheta=64, Nphi=48 Nr=53, Ntheta=64, Nphi=48 1.38e-5 Figure5-15. F r computedatdierentradialresolutionsontheSGRIDcode. 6e-06 8e-06 1e-05 1.2e-05 1.4e-05 1.6e-05 1.8e-05 2e-05 2.2e-05 100 150 200 250 300 350 F rt F r (multi-block) F r (sgrid) 1.38e-5 Figure5-16.Comparing F r resultsfromthemulti-blockandSGRIDcodes. 143

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Code(3+1)Frequency-domainerror F t mb(3 : 74 0 : 01) 10 5 3 : 750227 10 5 0.5% F t sg(3 : 739 0 : 001) 10 5 3 : 750227 10 5 0.2% E as F t mb3 : 77 10 5 3 : 750227 10 5 0.6% E as F t mbe*3 : 75 10 5 3 : 750227 10 5 0.02% E as F t sg3 : 749 10 6 3 : 750227 10 6 0.02% F r mb(1 : 398 0 : 001) 10 5 1 : 378448 10 5 1.4% F r sg(1 : 3852 0 : 0002) 10 5 1 : 378448 10 5 0.5% Table5-2.Summaryof(3+1)self-forceresultsfor R =10 M .Theerrorisdeterminedbya comparisonwithanaccuratefrequency-domaincalculation [ 50 ].\ mb "and \ sg "standformulti-blockandsgrid,respectively.\ mbe "istheresultofthe extrapolationtoinniteruxextractionradiusthatwasdon eonresultsofthe multi-blockcode. wasperformed,leadingtoerrorsasmuchas29%.Butbyoptimi zingthenumberofgrid pointsusedtosamplethenarrowGaussianonecanactuallyge tresultsof 1%[ 63 ]. Asmentionedinthepreviouschapter,recentworkbySundara rajan[ 64 ]hasdoneeven betterthanthis,withanoveldiscreterepresentationofth edeltafunction.Errorsof < 1% haveconsistentlybeenachievedwiththistechniqueontime -domaincodesthatsolvethe Teukolsky-Nakamuraequation.Wenotethatourruxresultsa realreadyatcomparable accuracy,albeitonlyforthescalarenergyruxina(3+1)sim ulation.Itisdicultto speculateonhownarrowGaussiansanddiscreterepresentat ionswillperformina(3+1) context. Themainhighlight,however,hastobetheaccuracyofoursel fforceresults,which areall 1%.Itcannotbeemphasizedenoughthatunlikeanyothercalc ulationofself forcethusfar,thesevalueswerecalculatedbymerelytakin gaderivativeoftheregular eldatthelocationofthepointcharge.Limitationsintime andresourceshaveprevented usfromoptimizingourcodestofurtherimproveontheseaccu racies,suchasndingthe bestparametersfortheeectivesourceandexploringthee ectsoffurtherincreasesin radialandangularresolution.Asarstattemptatdoinga(3 +1)selfforcecalculation though,theresultslookencouraging. 144

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Theredoremainsomesurprisingfeaturesthatremainunexpl ained.Itappearsthat theSGRIDcodewithrathermodestrequirementscomparedtot hemultiblockcode achievescomparableorslightlybetteraccuracy.TheSGRID codemakesuseofatmost 53collocationpointsintheradialdirectionfromaround r =2 M to r =210 M ,whilethe multiblockcodemakesuseof 4000pointsuniformlydistributedalongitsradialdomain fromaround r =2 M to r =400 M .Thejudiciousplacementofcollocationpointsby theSGRIDcodeclosetothelocationofthechargeappearstoe nableittorepresentthe eectivesourcebetterasopposedtotheuniformgridthatth emultiblockcodeuses.This seemstosuggeststronglythatdevotingmoreresourcestore solvingtheregionaroundthe charge(likewhatwouldbedonewithadaptivemeshrenement )istherightstrategy. Bothcodesshowconvergencewithrespecttoincreasesinrad ialandangular resolution.Itisclearthatthenitedierentiabilityoft hesourcereducestheorderof convergenceoftheircodesrelativetowhatitwouldbeifone hadasmoothsource.For instance,exponentialconvergenceoughttobeobservedint heSGRIDcode.Wehavenot beenabletoconvinceourselveswhattheorderofconvergenc ein F t and F r asaresultof our C 0 eectivesourceoughttobe. Aseriousissuehasbeenmadeapparentbytheseinitialresul ts.Sincetheselfforceis averysmallquantity,theeectsofimperfectboundarycond itionsbecomeamoreserious causeofconcern.InbothSGRIDandmultiblockruns,boundar yconditionswereproperly implemented.Butwiththemultiblockcodehavingitsouterb oundarysofarout,the smallbouncefromthisboundarydidnotshowinanyoftheresu lts.However,thisbounce isreadilyapparentfromtheSGRIDresults.Notethatthisbo unceisonlyoftheorder 10 6 ,whichwouldbenegligiblebycurrentnumericalrelativity standards.Because ofthesheersmallnessoftheselfforcethough,evenaminusc uleboundaryrerection isenoughtocompletelydominateit.Inasimulationthenwhe reinthecalculatedself forceisusedtoupdatethetrajectory,thiswouldarticial lycausethechargetomakean abruptjump.Thesuccessofourmethodthusreliescrucially onwhetherwecanminimize 145

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boundaryrerectionsevenfurther.Wenotethoughthatanyap proachdesignedtoupdate thetrajectoryofthechargeasitisalsocalculatingthesel fforcewillhavetocontendwith thisconcern. Insummary,withthispreliminarystudy,wehavedemonstrat edhowitispossible tocomputeselfforceswithexisting(3+1)codes.Moreover, ithasbeenshownthateven inthe(3+1)context,oureectivesourceisagoodsmeared-o utalternativetostandard deltafunctionrepresentationsofpointsources.Theruxre sultingfromoureectivesource matchesthatduetoapointchargewithverygoodaccuracy.Th eredoremainsomepoints tobeclaried,likethebenetsofoptimizingthecodes,the reductionoftheconvergence orderduetothenitedierentiabilityoftheeectivesour ce,andthelimitationssetby theeectsofboundaryrerections.Asthisismerelyarstcu tanalysis,weshallleaveall theseforfuturework. 146

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CHAPTER6 SUMMARYANDOUTLOOK Theself-consistentdynamicsofparticlesandeldsinblac kholespacetimeshas becomerelevanttogravitationalwaveastronomyinlightof theusefulnessofpointparticle modelsofextreme-mass-ratioblackholebinaries.Binarie softhiskindareamongthe mostimportantfortheproposedLISAsatellite,whosesucce ssdependscruciallyonthe qualityofourdataanalysistools.Underlyingthesedataan alysisstrategiesistheneedfor accuratewaveformsthatwillbeusedtoextractastrophysic alinformationfromconrmed detectionsofgravitationalwaves.Whiletheredoesexista suiteofapproximateEMRI waveformscurrentlybeingusedinongoingtestsofdataanal ysisstrategies,eachofthese existingEMRIwaveformsinvariablyignoresomepartofthef ulldynamics.Thisbegsthe questionofjusthowimportanttheseignoredpartsmaybe. Theperspectivetakeninthisdissertationisthatitwillbe diculttoassessthe validityofexistingapproximatewaveformswithoutknowle dgeofthecorrectone.This alonejustiestheneedforsimulationsofpointparticlemo tionthatincorporatethefull self-force,whichrepresentstheinteractionofaparticle andtheeldsitgivesriseto. Becausethisentailsquiteatremendouscomputationalexpe nse,itmayverywellturn outthatwaveformsfromself-force-inclusivesimulations enduprelegatedtothetask ofmerelyne-tuningorreducingerrorbarsofestimatedpar ametersinthenalLISA dataanalysispipeline.Butbeforethatcanhappenitwillha vetobemadeclearthat alternativewaveformsconformtomuchofthephysicscontai nedinself-forcewaveforms. Therefore,albeitlessdirectly,LISAdataanalysiswillco ntinuetobeastrongimpetusfor thesubjectofself-forceanalyses,ofwhichthisworkisasm allpart. 6.1ReviewofMainResultsandFutureWork Anovelmethodwasdevelopedinthisworkthatforthersttim ehasallowed thecalculationofaself-forceusinginfrastructureorigi nallydevelopedfornumerical relativityapplications.Themethodwastestedintwosetti ngsandshowninbothto 147

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yieldencouraginglygoodresults.Fromwithinnumericalre lativity,severalsuchpowerful codesexist,andwiththemethodintroducedhere,thereisno waclearavenuethrough whichtheycanbeutilizedtoattackself-forceproblems.In particular,simulationsof self-consistentdynamics,requiringparticletrajectory andeldupdatesinreal-time, appearsmostnaturalina(3+1)context.Thishaslongbeenth eHolyGrailofself-force research,anditistothisendthatthisworkmightbehopedto havesomeimpact. Themainchallengespreventinga(3+1)approachtoself-for cewereasfollows.(1) Distributionalsourcessuchasdeltafunctionsaredicult torepresentonagrid.(2)There didnotexistaclearmethodforcalculatingaself-forcewit houthavingtobreakupelds (andtheirderivatives)intomodecomponents. ThroughtheDetweiler-Whitingdecomposition,thisworksh owedthatthereexists anaturalwaytoavoidbothdiculties.Thisdecompositionc orrectlyidentiesthepiece oftheretardedeldofpointsources(ineitherscalareldt heory,electromagnetism,or linearizedgravity)thatisbothdivergent(andofnitedi erentiability)and,moreover, isirrelevanttotheself-force.Thissimpleideaallowedth econstructionofaneective waveequationfortheremainderoftheretardedeld,onewho sesourceisasmearedout representationofthedeltadistribution,andwhosesoluti ongivestheself-forcewithsimple derivativesevaluatedatthelocationoftheparticle.Thus ,inonefellswoop,wehaveboth issues(1)and(2)avoided. Ofcourse,therestillremainanumberofchallengesahead.I nparticular,thetests herewereperformedonthesimplestnon-trivialtoycase:as calarchargemovingina circularorbitaroundaSchwarzschildblackhole.Fortheca seofscalarcharges,themain challengeremainingisthelackofanecientnumericalpres criptionfordynamically carryingouttheTHZ-\backgroundcoordinate"transformat ionthatwasessentialfor thecalculationoftheeectivesource.Whenthisisathand, afullsimulationofthe self-consistentdynamicsofscalarchargescanbedone(for anybackgroundgeometry). Whileourultimateinterestliesnotwithscalarcharges,in sightwillalreadybegainedfrom 148

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suchanundertaking.Forone,thisislikelytoshedsomeligh tonthevalidityofcertain approximations(e.g.adiabatic)tothefullyself-consist entdynamics.Thisisleftforfuture work. Forthemoreinterestinggravitationalcase,onemayimagin eaprescriptionsimilar tothescalarcase.Justtomakethingsassimilaraspossible ,considerthelinearized rst-orderEinsteinequationinLorenzgauge.Thiseectiv elygivesrisetoacoupledset ofwaveequationsforthetencomponentsofthemetricpertur bation,eachofwhichwill bedistributionallysourced.Followingtheprescriptionf orthescalarcase,thetaskisthen tosmearoutthesesourcesbyregularizingthemwiththesing ularmetricperturbation,an approximationofwhichisalreadyknowninTHZcoordinates[ 54 ].Fromthis,acoupled setofeectivewaveequationsforthecomponentsoftheregu larmetricperturbation willhavetobesolvednumerically,andtheresultofwhichsh allinstructoneonhowto updatethemotionofthepointmass(i.e.withthegeodesiceq uationfortheperturbed spacetime). Ofcourse,therearelikelytobedicultissuesarisingduet othecouplingbetweenthe components,andtheinfamousperennialtroublewithgauge. Eveninthescalarcase,one wouldhavetondacleversolutiontotheperniciousissueof smallboundaryrerectionsso easilydominatingtheminuteself-force.Thesearealldic ultproblems,andweinvitethe intrepidfewinsearchofnewchallengestogivethematry. Inconclusion,therenowexistsaprovenframeworkthroughw hichthetechnology ofnumericalrelativitycanbebroughttobearonself-force problems.Weviewthework presentedinthisdissertationasonlytherststepstoward sachievingself-forcewaveforms. 149

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APPENDIXA THORNE-HARTLE-ZHANGCOORDINATES ATHZcoordinatesystem f t;x;y;z g associatedwithanygivengeodesicofa vacuumspacetimeisanormalcoordinatesystemthatisaddit ionallyharmonic,locally inertialandCartesian.Locally-inertialandCartesianme ansthatthespatialorigin x i =0 isontheworldline,withthemetrichavingtheform: g ab = ab + p (homogeneouspolynomialsin x i ofdegree q )(A{1) where p;q and p + q 2and = p x 2 + y 2 + z 2 Zhang[ 52 ]showedthatthisisachievedbyalternativelyrequiringth efollowing conditions: 1. t isthepropertimealong,and g ab j = ab and @ a g bc j =0.(normalcoordinates) 2. Atlinear,stationaryorder: H ij p gg ij ij = O ( x N +1 ). 3. Thecoordinatesareharmonic: @ a ( p gg ab )= O ( x N ). N herereferstotheorderoftheTHZcoordinates.Howlarge N mustbeisdeterminedby theapplicationinmind.Foramode-sumtypeofcalculationo ftheself-forceforacharge inacircularorbit,itappearsthat N =4yieldsadequateconvergence.Forthecaseof circularorbitsinSchwarzschild,THZcoordinateshavebee nconstructedupto N =4[ 50 ]. A.1ConstructionoftheTHZCoordinates Givenasetofcoordinates f Y A g ,oneeasilysatisescondition(1)withthetransformation X a = A a + B a A ( Y A Y A p )+ 1 2 B a A A BC ( Y B Y B p )( Y C Y C p )+ O (( Y Y p ) 3 )(A{2) A a arearbitraryconstants,and B a A isanarbitraryconstantmatrixexceptforthe requirementthatitbeinvertible.Inthecoordinates f X a g ,onecanshowthat: g ab = g AB @X a @Y A X b Y B = ab + O [( Y Y p ) 2 ] ; (A{3) 150

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andthus, @g ab @X c = O [( Y Y p )] : (A{4) Moreover,wecanchoose A a and B a A sothat X a p = p a t (i.e.,atapoint p along,the onlynonzerocoordinateisthetimecoordinate,withitsval uebeingthepropertimeat p ). Thetaskthenistotransform f X a g intocoordinatesthatsatisfyconditions(2)and (3).Thisisachievedbyperforminginnitesimalgaugetran sformations: x a(new) = X a (old) + a : (A{5) A.1.1SpatialComponentsoftheGaugeVector Weneedtoestablishwhetherornotitisevenpossibletosati sfythegaugeconditions withthefreedominourgaugevector.Withthe f X a g coordinatesabove,wecanwritethe metricdensityas p gg ab = ab H ab = ab H ab ij X i X j + O ( X 3 = R 3 ) ; (A{6) where H ab ij 1 2 @ 2 H ab @X i @X j ; (A{7) whichare known functionsof t Atthelowestlevel( N =2),THZ-coordinatesaresuchthatthespatialcomponentso f H ab are: H ij THZ = O ( x 3 = R 3 ) : (A{8) Following[ 50 ],weachievethiswithasimpleansatzforthegaugevector: a = a ijk ( t ) X i X j X k ; (A{9) sothat x a(new) = X a + a ijk ( t ) X i X j X k ; (A{10) 151

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where a ijk ( t )arefunctionsonlyof t ,whichshallrereferredtoasgaugecoecients. Thischangesthemetricasfollows: H ab new = H ab old + @ a b + @ b a ab @ c c + O ( 2 ) ; (A{11) or,intermsofthegaugecoecients 1 3 H abij new = 1 3 H abij old + abij + baij ab k kij : (A{12) Tosatisfycondition(2)upto N =2,allonedoesissolveEq.( A{12 ).Thisisdone bycleverlyexploitingthesymmetriesintheindicesofthee quation,theresultofwhich canbefoundin[ 50 ].Itisthusclearthatstartingwiththisparticularansatz forthegauge vector,condition(2)canbemetuptoatleast n =2. Aninterestingquestioniswhethertheprescriptionofsolv ingforthegaugevector outlinedabovecanbeimplementedtoprovide N th-orderTHZcoordinates.Herewe demonstratethatthisispossiblesolongasonecanignoreth enonlinearpiecesinthe expansionofthemetric. Supposethatonealreadyhas( N 1)th-orderTHZcoordinates f X a g suchthat: 1. H ij = O ( X N ) 2. @ b H ab = O ( X N 1 ), thetaskistondagaugetransformation, x a = X a + a ; (A{13) thatgivesus 1. H ij = O ( x N +1 ) 2. @ b H ab = O ( x N ). 152

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Followingtheprescriptionintheprevioussections,webeg inwiththefollowingansatzfor thegaugevector: a = a N +1 X N +1 a i 1 i 2 i 3 :::i N +1 X i 1 X i 2 X i 3 X i N +1 : (A{14) Weshallrefertothetime-dependentfunctions a i 1 i 2 i 3 :::i N +1 ( t )asthe gaugecoecients Similarly,withtheassumptionthatwealreadyhave N th-orderTHZcoordinates, H ij canbeexpressedas: H ij = H ij N X N H ij i 1 i 2 i 3 :::i N X i 1 X i 2 X i 3 X i N : (A{15) Weshallalsorefertothetime-dependentfunctions H ab i 1 i 2 i 3 :::i N ( t )asthe metricperturbationcoecients Ageneralgaugetransformationchanges H ab into: H ab new = H ab old + @ a b + @ b a ab @ c c + O ( 2 ) : (A{16) Makinguseofouransatzforthegaugevector,andlookingonl yatthespatialcomponents H ij ,thisgivesrisetoatransformationonthemetricperturbat ioncoecientsgivenby: 1 ( N +1) H ijN new = 1 ( N +1) H ijN old + ijN + ijN ij m m N : (A{17) Thuswehavethefollowingequationsspecifyingourgaugeco ecients: ijN + ijN ij m mN = 1 ( N +1) H old ijN + O ( X N +1 ) : (A{18) Sincethemetricperturbationcoecientsaresymmetricabo utitsrsttwoandlast N indicesseparately,thisisasystemof6 ( N +2)! 2! N =3( N +2)( N +1)equations.On theotherhand,ourgaugecoecientsaresymmetricinitslas t( N +1)indices,sothat wehave3 ( N +3)! 2!( N +1)! = 3 2 ( N +3)( N +2).Forany N> 1,therewillbemoregauge conditionsthanthereareunknowngaugecoecients,whichs eemstosuggestthatwehave anoverdeterminedsystem.(Notethatthelowest-orderTHZc oordinates,afterhaving 153

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satisedcondition(1)ofthedenition,is N =2.)Theexcessingaugeequationsis 3( N +2)( N +1) 3 2 ( N +3)( N +2)= 3 2 ( N 1)( N +2). Thisissueisresolveduponrealizingthatthegaugeconditi ons,whichwerecounted bynaivelyinspectingthenumberofdistinctmetricperturb ationcoecients H old ijN thatare sourcingthegaugeequations,arenotallindependent.Thea ssumptionthatwestartwith asolutiontotheEinsteinequationimpliesthat H ab satisesthelinearizedequation.This leadstodependenciesinthemetricperturbationcoecient s. ThelinearizedEinsteinequationaboutanarbitrarybackgr ound g (B) ab isasfollows: g ab (B) r a r b h cd + g (B) cd r b r a h ab 2 r ( d r a h c ) a +2 R (B) acbd h ab 2 R (B) a ( c h d ) a =0 : (A{19) Wewritethisusingthe( N 1)th-orderTHZcoordinates,with ab asthebackgroundand thetrace-reversedmetricperturbationbeingjust h ab = H ab = O ( X N ).Focusingonlyon thespatialcomponents,wehave: H ijk k ( N 2) + ij H kl kl ( N 2) 2 H ( i j k j k j ) ( N 2) =0 : (A{20) Theseequationsrepresent6 N 2!( N 2)! =3 N ( N 1)equationsrelatingthevariousmetric perturbationcoecients.However,notallareindependent themselves.Atraceoverany upstairsanddownstairsindexmerelyyieldsanidentity: H ilk kl ( N 3) + im H kl klm ( N 3) H ik k l l ( N 3) H lk k i l ( N 3) =0 : (A{21) Thesetraceequationsthusrepresentredundanciesintheli nearizedEinsteinequation. Thereare3 ( N 1)! 2!( N 3)! = 3 2 ( N 1)( N 2)ofthem.Hence,thenumberof independent equationsarisingfromthelinearizedEinsteinequationis just3 N ( N 1) 3 2 ( N 1)( N 2)= 3 2 ( N 1)( N +2).Thisisthenumberofdistinctrelationshipsbetweenou rmetric perturbationcoecients,andisalsoexactlythenumberbyw hichourgaugeequations exceedourgaugecoecients. 154

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Thus,thereisjusttherightnumberofindependentgaugeequ ationsateachorder N toarriveatauniquesolutionforthespatialcomponentsoft hegaugevector. Wehavethusshownthatifcondition(2)issatisedatsomeor der( N 1),thenour ansatzforthegaugevectorissucienttobootstrapconditi on(2)toorder N A.1.2 t -ComponentoftheGaugeVector Intheprecedingsectionweshowedthatitisalwayspossible tosatisfycondition (2)inthedenitionofourTHZ-coordinates.Asshown,condi tion(2)completelyxes thespatialcomponentsofthegaugevector,leavingnofreed omexceptforthetime component, t ,toimposetheharmonicgaugecondition, @ a H ab = O ( x N ). Firstwelookatthespatialcomponentofthiscondition. @ a H ai = H ti + @ j H ji = O ( x N )(A{22) In( N 1)thTHZcoordinates,bothtermswouldbe O ( x N 1 ).Byadjustingthe spatialcomponentsofthegaugevectorthough,wealreadypu shtheorderofthesecond termto O ( x N ).Theorderofthersttermisinferredfromthe( N 1)th-orderharmonic condition,andthisisnotmodiedbychoosingthespatialco mponentsthegaugevector. Itisthusclearthatwewouldneedthelonefreedominthegaug evector t toimpose the N th-orderharmoniccondition.Butthenthesearethreeequat ionsthatneedtobe satisedwithonlyonefreeparametertoadjust.Ingeneralt hen,theansatzfails. Forthelowestcasethough, N =2,thespatialharmonicconditionsareautomatically satised.Fromcondition(1),allthe H ab arealready O ( x 2 ).Choosingthespatial componentsofthegaugevectorappropriately,onecantrans formthespatialcomponents ofthe H ab tobe O ( x 3 ),sothatthesecondtermisautomatically O ( x 2 ).Therstterm howeveristriviallyseentobe O ( x 2 ),byvirtueofstartingwithsomesetofnormal coordinates.Thus,thespatialcomponentsoftheharmonicc onditionareclearlysatised toorder N =2. 155

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Turnnowtothetimecomponentoftheharmoniccondition. @ a H at = @ t H tt + @ i H it = O ( x N ) : (A{23) Obviously,sincethespatialcomponentsoftheharmoniccon ditionalreadyforman overdeterminedsystem,theadditionofanotherconditionj ustmakesthesituationworse. Thenagainthough,the N =2casecanbedone.Sincethelastparameter t was notneededtoxthespatialcomponentsoftheharmoniccondi tion,itcannowbeusedto imposethetimecomponentoftheharmoniccondition.Interm sofgaugecoecients,itis shownin[ 50 ]that t hastobegivenby: t ijk = 1 5 H tp p ( i jk ) : (A{24) Itisthusshownthattheansatzprovidedforthegaugevector ingeneraldoesnot worktobootstraptheorderofone'sTHZcoordinates.Howeve r,iteasilyworksfor the N =2case,whichisthetransitionfromanormalcoordinatesys temtotheTHZ coordinates.Theansatzwaschosenonthebasisofitssimpli city.Onceoneknowstha backgrounggeometryandtheprescribedworldlineofthecha rge,onecanconstruct anormalcoordinatesystem,andthencanconvertthistoaTHZ coordinatesystem bysolvingabunchof algebraic equationssourcedbycomponentsoftheoldmetric. Thismakesitextremelyamenabletoanumericalimplementat ionofthecoordinate transformation.Obvoiuslythough,theansatzwillhavetob eabandonedifoneaimsfor betterapproximationstothesingulareld. A.2THZCoordinatesforaCircularGeodesicaroundSchwarzs child Thefullcoordinatetransformation,whichweuseforourana lysisinboththe(1+1) and(3+1)calculationsmaybefoundinEqs.(B1)-(B9)ofRef. [ 50 ].Belowwegiveonlyan abbreviatedformoftheseequationstogiveasenseofhowthe coordinatetransformation isimplemented.Theformulaebelowgive x y z and t assmoothfunctionsofthe Schwarzschild r and t s .Wecanthendeneafunction = p x 2 + y 2 + z 2 which 156

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hasthepropertythat r a r a (1 = )= 4 ( ~x )+ O (1 = ) : (A{25) Ifwehadused q= astheapproximation ~ s forthesingulareldthentheeectivesource fortheregulareld R inthevicinityofthepointchargewouldbesingular, S e = r a r a ( q= ) 4 q ( ~x )= O (1 = ) ; (A{26) ratherthan O ( ),whichisthecasefortheeectivesourcewhichweactually useas describedinEq.( 3{10 ). Thecoordinateswhichleadto arenowgivenforacirculargeodesicofthe SchwarzschildgeometryatSchwarzschildradius R :Werstdenetwousefulfunctions ~ x = [ r sin cos( n t s ) R ] (1 2 M=R ) 1 = 2 + M R 2 (1 2 M=R ) 1 = 2 ( r R ) 2 2(1 2 M=R ) + R 2 sin 2 sin 2 ( n t s )+ R 2 cos 2 + O ( 3 ) (A{27) and ~ y = r sin sin( n t s ) R 2 M R 3 M 1 = 2 + O ( 3 ) : (A{28) The O ( 3 )termsindicatethatthese(andtheformulaebelow)couldbe modiedbythe additionofarbitrary O ( 3 )termswithoutnecessarilychangingtheusefulnessofthes e coordinates. Intermsofthesetwofunctions,theTHZcoordinates( t; x; y; z )are x =~ x cos(n y t s ) ~ y sin(n y t s )(A{29) and y =~ x sin(n y t s )+~ y cos(n y t s )(A{30) 157

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wheren y =n p 1 3 M=R ,alongwith z = r cos( )+ O ( 3 ) (A{31) and t = t s (1 3 M=R ) 1 = 2 r n R sin sin( n t s ) (1 3 M=R ) 1 = 2 + O ( 3 ) (A{32) Thesetoffunctions( t; ~ x; ~ y; z )formsanon-inertialcoordinatesystemthatco-rotates withtheparticleinthesensethatthe~ x axisalwayslinesupthecenteroftheblackhole andthecenteroftheparticle,the~ y axisisalwaystangenttothespatiallycircularorbit, andthe z axisisalwaysorthogonaltotheorbitalplane. TheTHZcoordinates( t; x; y; z )arelocallyinertialandnon-rotatinginthevicinity ofthecharge,butthesesamecoordinatesappeartoberotati ngwhenviewedfarfrom thechargeasaconsequenceofThomasprecessionasrevealed inthen y t s dependencein Eqs.( A{29 )and( A{30 )above. Thecoordinates( t; x; y; z )givenabovearesaidtobe secondorder THZcoordinates anddierfromtheactual fourthorder onesusedinthemainbodyofthispaperbythe replacementofthe O ( 3 )termsappearingabovebyspecictermswhichscaleas 3 and 4 [ 50 ]andleavetheundeterminedpartsoftheTHZcoordinatesbei ng O ( 5 ). 158

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APPENDIXB SPHERICALHARMONICSANDTHEREGGE-WHEELERGAUGE InthisAppendix,weshowhowanysymmetric(0,2)-tensor,sa ytheperturbation h ab isconvenientlydecomposedintosphericalharmonics.This decompositionallowsoneto takefulladvantageofthesphericalsymmetryintheSchwarz schildspacetime. Werstnotethat,inSchwarzschildcoordinates,theindepe ndentcomponentsofthe metricperturbation h ab transformlikethreescalars,twovectors,andatensoronth eunit 2-sphere: h ab = 0BBBBBBBBBBB@ s 1 s 2 [ V 1 ] s 2 s 3 [ V 2 ] 266664 V 1 377775 266664 V 2 377775 266664 T 377775 1CCCCCCCCCCCA : (B{1) Withmultipoledecomposition,wecanseparateoutthe( ; )-dependencefromthe ( t;r )-dependenceoftheseobjects, ateachmode .Thisisobviousforthescalars: s = X l;m a lm ( r;t ) Y lm ( ; ) ; (B{2) owingtothecompletenessofthescalarharmonics, Y lm ,overthe2-sphere.Forvectors andtensorsonthetwo-sphere,werstestablishsomenotati on.Wecanexpressthe Schwarzschildgeometryas: ds 2 = g ij dx i dx j + r 2 AB d A d B : (B{3) Lower-caseindices i;j;::: runovercoordinates( t;r ),andupper-caseindices A;B;::: runoverthecoordinatesofthetwo-sphere,( ; ). AB representsthemetricontheunit 2-sphere: AB =diag(1 ; sin 2 ).TheLevi-Civitatensorontheunitsphereisdenotedby AB = 0sin sin 0 159

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Anyvectoreld r A onthetwo-spheremayberepresentedbytwoscalarelds r ev and r od : r A = AB r B r ev AB r B r od : (B{4) Thesearescalareldsonthefullmanifold(i.e.notjustont he2-sphere),andarecalled the evenandoddpotentials ,respectively.Decomposingthesescalarpotentialsintur n allowustobreakupthevectorintomodesforwhichtheangula r-dependenceseparates: r A = X lm h lmev ( r;t )( AB r B Y lm )+ h lmod ( r;t )( AB r B Y lm ) : (B{5) Weseethenthat f AB r B Y lm ; AB r B Y lm g spansthespaceofvectorsonatwo-sphere; theyaretheso-called vectorsphericalharmonics Similarly,atensor F AB onthetwo-spheremayberepresentedbyitstracewith respectto AB ,andtwootherpotentials F ev and F od : F AB = 1 2 AB F trace +( r ( A r B ) 1 2 AB r 2 ) F ev ( A C r B ) r C F od ; (B{6) where r 2 istheLaplacianonthetwo-sphere.Hence,weseeagainthatb ydecomposing thetraceandthepotentialsintoscalarsphericalharmonic s,whatresultsisadecomposition ofthetensor F AB : F AB = X lm K lm ( AB Y lm )+ G lm r A r B 1 2 AB r 2 Y lm + h lm(2) ( ( A C r B ) r C Y lm ) (B{7) wherethe( t;r )-dependenceofthefunctions K lm ;G lm ;h lm(2) havebeensuppressedfor notationalclarity.Thus,asbefore,weseethattheset AB Y lm ; r A r B 1 2 AB r 2 Y lm ; ( A C r B ) r C Y lm (B{8) spansthespaceoftensorsonthetwo-sphere.Thelasttwooft heseareanexampleof tensorsphericalharmonics ;thetracepartisreallyjustascalarsphericalharmonic.S ince 160

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r 2 Y lm = l ( l +1) Y lm ,wecaninsteadwrite F AB = X lm K lm ( AB Y lm )+ G lm r A r B + 1 2 AB l ( l +1) Y lm + (2) h lm ( ( A C r B ) r C Y lm ) ; (B{9) whereinourtensorsphericalharmonicsare AB Y lm ; r A r B + 1 2 AB l ( l +1) Y lm ; ( A C r B ) r C Y lm : (B{10) Thesescalar,vector,andtensorharmonics,canbefurtherc lassiedbasedonhowthey transformunderaparitytransformation:( ; + ).Thosethattransform as ( 1) l aresaidtohave evenparity ,whilethosethatgolike ( 1) ( l +1) have oddparity Forcompleteness,welistour odd-parityharmonics : AB r B Y lm ; ( A C r B ) r C Y lm ; (B{11) andour even-parityharmonics Y lm ; AB r B Y lm ; r A r B + 1 2 AB l ( l +1) Y lm : (B{12) Thismachineryallowsustothinkoftheperturbation h ab (oranysymmetric2-tensor, ingeneral)asasumof( l;m )multipoles or modes : h ab = X lm h lm; even ab + X lm h lm; odd ab ; (B{13) eachofwhichhavingaunique,xedangular-dependencethat canbeseparatedfromits accompanying( t;r )-dependence.Ineect,eachmode( l;m )isspeciedbytenfunctions of( t;r ),whichareessentiallythecoecientsinthemultipolarex pansion.Sevenof thesefunctionsbelongtotheeven-paritysector,whilethe remainingthreespecifythe 161

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odd-paritysector.Tobeexplicit,wewrite: h lm; even ab = 0BBBBBBBBBBB@ H 0 H 1 [ j 0 ] H 1 H 2 [ j 1 ] 266664 j 0 377775 266664 j 1 377775 266664 K;G 377775 1CCCCCCCCCCCA (B{14) h lm; odd ab = 0BBBBBBBBBBB@ 00 [ h 0 ] 00 [ h 1 ] 266664 h 0 377775 266664 h 1 377775 266664 h 2 377775 1CCCCCCCCCCCA : (B{15) GaugetransformationsandtheRegge-Wheelergauge Wearefurtherfreetoperformthetransformation( 1{12 )at eachmode inorderto makealltheeven-paritymodeshavetheform: h lm; even ab = 0BBBBBBBBBBB@ H 0 H 1 [0] H 1 H 2 [0] 266664 0 377775 266664 0 377775 266664 K 377775 1CCCCCCCCCCCA ; (B{16) 162

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andtheoddparitymodes: h lm; odd ab = 0BBBBBBBBBBB@ 00 [ h 0 ] 00 [ h 1 ] 266664 h 0 377775 266664 h 1 377775 266664 0 377775 1CCCCCCCCCCCA : (B{17) Thischoiceistheso-calledRegge-Wheelergauge.Allitent ailsisaspecicchoiceof thegaugevector a ,whichcanbefoundbydecomposingthisitselfintoharmonic s,and consideringthegaugetransformationin( 1{12 ).Thisiseasierdonefortheodd-parity sector.The( l;m ),odd-paritypartofageneralgaugevector a willbejust (odd) a = (0 ; 0 ; ( r;t ) AB r B Y lm ),andthetransformationstheseinduceontheodd-parityfu nctions are: h 0 h 00 = h 0 @ @t (B{18) h 1 h 01 = h 1 @ @r + 2 r (B{19) h 2 h 02 = h 2 2 : (B{20) FromtheseitisclearthattheRegge-Wheelergaugeisimpose djustwiththechoiceof = 1 2 h 2 : (B{21) Similarsimplicationsappearintheeven-paritysector.T he( l;m ),even-paritypart ofageneralgaugevector a willbejust (even) a =( 0 ; 1 ; ( r;t ) AB r B Y lm ),whichinduces 163

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transformationsontheeven-parityfunctionsasfollows: H 0 H 0 0 = H 0 2 @ 0 @t +2 ctt c (B{22) H 1 H 0 1 = H 1 @ 0 @r @ 1 @t +2 ctr c (B{23) H 2 H 0 2 = H 2 2 @ 1 @r +2 crr c (B{24) j 0 j 0 0 = j 0 0 @ @t (B{25) j 1 j 0 1 = j 1 1 @ @r + 2 r (B{26) K K 0 = K 2 r 1 2 m r 1 + l ( l +1) r 2 (B{27) G G 0 = G 2 r 2 : (B{28) TheRegge-Wheelergaugeisthenimposedwiththechoices: = 1 2 r 2 G (B{29) 0 = j 0 @ @t (B{30) 1 = j 1 @ @r + 2 r : (B{31) 164

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APPENDIXC PERTURBATIONTHEORYINGENERALRELATIVITY C.1Twometricsonamanifold Inrelativity,thefundamentaleldequationistheEinstei nequation, R ab ( g ) 1 2 g ab R ( g )=8 T ab (C{1) andthefundamentalquantityisthespace-timemetric, g ab ,whichistobeasolutionof such.Perturbationtheoryproceedsfromadecompositionof thisunknownsolution{ thefullmetric{intoaknownexact(usuallyvacuum)solutio n,whichweshallcallthe \background"metric,andaresidualperturbingmetric, h ab ,whichweshalljustcallthe \perturbation".Inthisdiscussion,weshallthebackgroun dmetric, g ( B ) ab .Thus,wehave g ab g ( B ) ab + h ab : (C{2) Itisimportanttonotethatthisrelationinfact denes h ab .Doingthis,wend ourselvesinapeculiarsituationwhereinthereareseveral (therearethree)metricson ourmanifold:twoindependentandtheotherdependentonthe rsttwo.Ifwex g ( B ) ab thenthecorrespondencebetweentheremainingtwomeanstha ttheEinsteinequationis actuallyaeldequationfortheperturbation h ab Usingtheidentities g ab g bc = a c g ( B ) ab g ( B ) bc = a c itiseasytoshowthatuptothirdorderin h g ab isgivenby g ab = g ( B ) ab h ab + h a c h cb h a c h c d h db + O ( h 4 ) : Thiscanalsobewritteninsteadasarecursion: g 0 ab = g ( B ) ab h ab + h a c h cb g 0 ac h cd h d e h eb : 165

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Sincethereareseveralmetricsonourmanifold,thereareco rrespondinglyseveral connections,andthusjustasmanycovariantderivatives.T heonesthatappearinthe Einsteinequationarethosecompatiblewiththeunknownmet ric g ab ,sotechnically, theyarealsounknown,hencemakingtheEinsteinequationin itscurrentformrather inconvenient.Itturnsouthoweverthatwecanexpresstheco variantderivativecompatible with g ab intermsofthosethatarecompatiblewiththebackground. Firstofall,itisshownin[ 75 ]thatthedierencebetweenanytwocovariant derivativesisatype-(1,2)tensor.Weshalldenotethisher eby S c ab .Incomponentsit isgivenby: S r = r ( B ) r ; wherethe r and ( B ) r aretheChristoelsymbolsofthefullandbackground metricrespectively,inaparticularcoordinatesystem.Le t r a and r ( B ) a bethecovariant derivativescompatiblewiththefullandbackgroundmetric srespectively.Thenwehave ( r c r ( B ) c ) g ab = S d ac g db S d bc g ad r ( B ) c g ab = S d ac g db + S d bc g ad r ( B ) c g ab = S bac + S abc : (C{3) Permutingtheindicesinthislastequationleadsto: r ( B ) a g cb = S bca + S cba r ( B ) b g ac = S cab + S acb Takingaccountofthesymmetryof S c ab initslasttwoindices,wecanaddthesetwo equationstoget 2 S cab + S bca + S acb = r ( B ) a g cb + r ( B ) b g ac : (C{4) 166

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Subtracting(3)from(4)thengives 2 S cab = r ( B ) a g cb + r ( B ) b g ac r ( B ) c g ab ; orequivalently S c ab = 1 2 g cd ( r ( B ) a g db + r ( B ) b g da r ( B ) d g ab ) : (C{5) Recallingthat g ab = g ( B ) ab + h ab ,wethenhave S c ab = 1 2 g cd ( r ( B ) a h db + r ( B ) b h da r ( B ) d h ab ) : (C{6) Thisisstillanexactequation.Noapproximationshaveyetb eenmade.Notealsothatthis resultisnotdependentonchoosing g ab asthelowering/raisingoperator. C.1.1 R abc d and R ab WiththiswecannowndexpressionsfortheRiemannandRicci tensorsin termsoftheperturbingmetric.Hereon,wetakethebackgrou ndmetric g ( B ) ab tobeour raising/loweringoperator,andwelet r 0 and r betheconnectionsassociatedwiththefull metricandbackground,respectively.Intermsof S c ab ,[ 75 ]showsthat: ( r 0a r a ) v c = S d ac v d r 0a v c = r a v c S d ac v d ( r 0b r b ) r 0a v c = S d ba r 0d v c S d bc r 0a v d = S d ba ( r d v c S f dc v f ) S d bc ( r a v d S f ad v f ) = S d ba r d v c + S d ba S f dc v f S d bc r a v d + S d bc S f ad v f = S d ba r d v c S d bc r a v d + S d ba S f dc v f + S d bc S f ad v f 167

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r 0b r 0a v c = r b r 0a v c S d ba r d v c S d bc r a v d + S d ba S f dc v f + S d bc S f ad v f = r b ( r a v c S d ac v d ) S d ba r d v c S d bc r a v d + S d ba S f dc v f + S d bc S f ad v f Thus, r 0b r 0a v c = r b r a v c r b S d ac v d S d ac r b v d S d ba r d v c S d bc r a v d + S d ba S f dc v f + S d bc S f ad v f (C{7) andsimilarly, r 0a r 0b v c = r a r b v c r a S d bc v d S d bc r a v d S d ab r d v c S d ac r b v d + S d ab S f dc v f + S d ac S f bd v f (C{8) Subtractingthesetwoequations,wegettheexpressionwewa ntfortheRiemanntensor. Recallthat ( r 0a r 0b r 0b r 0a ) v c R 0 abc d v d : Itthenfollowsthat R 0 abc d v d = R abc d v d +( r b S d ac r a S d bc ) v d +( S d ac S f bd S d bc S f ad ) v f (C{9) Fromthisweget, R 0 abc d = R abc d +( r b S d ac r a S d bc )+( S d bf S f ac S d af S f bc )(C{10) TheperturbedRiccitensoristhengivenby R 0 ab = R ab +( r d S d ab r a S d db )+( S d df S f ab S d af S f bd ) (C{11) Again,theseexpressionsarestillexact,withnoapproxima tionsmade. 168

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Nowwhatwewantistogetfromtheseexplicit2nd-orderexpre ssionsintermsof h ab Ourexpressionfor S c ab ,inequation(6),containsallorders.Upto2nd-orderthisi sjust: S c ab = 1 2 ( g cd h cd )( r a h db + r b h da r d h ab )+ O ( h 3 ) (C{12) NowwecanputthisdirectlyintoEq.( C{11 ). Tokeepthingstidythough,weshallseparatetheperturbedR iccitensorintopieces thatarezeroth-,rst-,andsecond-orderin h ab .Thezeroth-orderpieceistriviallythe Riccitensorofthebackground.Therst-orderpiecewillco mefromtherst-orderpartof S c ab pluggedintothesecondtermof( C{11 ): R (1) ab = r d 1 2 g dc ( r a h bc + r b h ac r c h ab ) r a 1 2 g cd ( r c h bd + r b h cd r d h cb ) (C{13) = 1 2 g dc ( r d r a h bc + r d r b h ac r d r c h ab ) 1 2 g cd ( r a r c h bd + r a r b h cd r a r d h cb )(C{14) andcollectingalltermsofthesameorder,weeventuallyget : R 0 ab = R ab + R (1) ab + R (2) ab + O ( h 3 ) (C{15) where R (1) ab = 1 2 ( r b r a h + r 2 h ab 2 r c r ( a h b ) c )(C{16) and R (2) ab = 1 2 1 2 r a h cd r b h cd + h cd ( r b r a h cd + r d r c h ab 2 r d r ( a h b ) c ) + r d h bc ( r d h a c r c h ad ) ( r d h cd 1 2 r c h )(2 r ( a h b ) c r c h ab ) : (C{17) Theseformulaswillbethebasisofsubsequentcalculations C.1.2 G ab R ab 1 2 R WecontinueinthissectiontowritedowntheEinsteintensor upto2ndorderin h Weshallwrite G ab = G ( B ) ab + G (1)ab + G (2)ab ,andderiveexpressionsfor G (1)ab and G (2)ab 169

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Weshallassumethatthebackgroundmetricisavacuumsoluti on.Thatis,itsatises R ( B ) ab 1 2 g ab R ( B ) =0.Hence,itfollowsthatthezerothordertermintheEinste intensor, G ( B ) ab ,vanishes.Italsofollowsthat R ( B ) ab and R ( B ) vanishaswellsince 0= g ab G ( B ) ab = g ab R ( B ) ab 1 2 g ab g ab R ( B ) = R ( B ) = ) R ( B ) ab =0 Upto2ndorder,theEinsteintensorisjust: G ab = R (1) ab + R (2) ab 1 2 g ab ( R (1) + R (2) ) 1 2 h ab R (1) sothatwehave G (1)ab = R (1) ab 1 2 g ab R (1) (C{18) G (2)ab = R (2) ab 1 2 g ab R (2) 1 2 h ab R (1) (C{19) Toproceed,weneedthescalarcurvature R upto2ndorderin h .Thisisjust R = g ab R ab =( g ab h ab + h a c h cb + O ( h 3 ))( R (1) ab + R (2) ab + O ( h 3 )) = g ab R (1) ab +( g ab R (2) ab h ab R (1) ab )+ O ( h 3 ) R (1) = g ab R (1) ab = 1 2 g ab ( r b r a h + r 2 h ab 2 r c r ( a h b ) c ) = r 2 h + r c r d h cd (C{20) R (2) = g ab R (2) ab h ab R (1) ab = 1 2 [ 1 2 r b h cd r b h cd + h cd ( r 2 h cd + r d r c h 2 r d r a h ca ) r d h bc r c h b d + r d h bc r d h bc ( r d h cd 1 2 r c h )(2 r b h cb r c h )] + 1 2 h ab ( r b r a h + r 2 h ab 2 r c r ( a h b ) c ) (C{21) TheseexpressionsaresucienttowritetheEinsteinequati ontosecondorderin h 170

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BIOGRAPHICALSKETCH IanVegawasborninAugust1980,inCagayandeOroCity,abust lingcitynorth ofthePhilippines'secondlargestisland.Theeldestchild offoursiblingsbyMikeand Maricel,hewasspoiledbyhisparentstherstveyearsofhi slifeandwasalways encouragedtoexplorewhateverinterestedhim.Ian'srsts choolwasonlyafewstepsfrom hishouse.Hedistinctlyremembersitforstrictteacherswh odidn'tquiteappreciatethe humorinpullingchairsfromunderhisclassmatesandsticki ngbubblegumintheirhair. Ashegrewolder,Ianmovedtoasmallelementaryschoolright nexttowherehisMom worked.Itwastherethathestartedlikingmathandbeganhis life-longaairwithbooks. Afterelementaryschool,Ianbecameconvincedthathecould neverbesmarterthan girls.Hardashemighttry,hecouldneverkeephimselfawake enoughforanythingexcept mathandscience.Soforhighschool,Iandecidedtotryhislu cktakingtheentrance examinationofthebesthighschoolinCagayandeOro,aJesui t-run,(once)boys-only, privateschoolmuchtoopriceyforhisparents'wallets.Luc kily,herankedrstinthe exams,earninghimtheJesuits'nodandascholarship. Amajorinruenceinhighschoolwerehisexcellentmathteach ers.FromthemIan learnedthequietpleasureofmullingoverdicultmathprob lems.Inhighschoolhe thrivedinacademiccompetitions,leadingthewayforhissc hoolindebatesandnational levelcontestslikethePhilippineMathOlympiadandtheNat ionalImpromptuSpeaking Contest.Asasophomore,hewassuspendedtwodaysfortalkin gbacktoateacher,which inthePhilippinesmeanshavingtocleanthehallsandtoilet sforthedurationofthe suspension.In1997,despiteinfamousdisciplinaryproble ms,henishedatthetopofhis graduatingclassandwasnamedvaledictorian. Forcollege,Iandecidedtodoublemajorinphysicsandcompu terengineeringatthe AteneodeManilaUniversity,thebestprivateuniversityin thePhilippines,wherehehad beenawardedaprestigiousMeritScholarshipforbeingoneo fthetopentranceexam scorers.Lateron,hetradedcomputerengineeringformathe maticswhenherealizedthat 175

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thedrudgeryofsolderingoncircuitboardsjustwasn'thist hing.Butmuchfurtheron,he reluctantlyhadtodrophisdreamsofamathdegree(justshyo f3classes)duetonancial limitations. OneofIan'sbiggestfortunesincollegewastakingaclassic almechanicsclassby JerroldGarcia,arelativistbytraining,andatrulyinspir ingteacher.Jerroldhasbeenone ofIan'smosttrustedmentorseversince.In2001,Iangradua tedwithaB.S.inPhysicsHonorableMention,nishingsecondinhisphysicsclass. WithJerrold'sconstantencouragement,Ianbecameintento ngoingtograduate schoolintheUnitedStates.Butthathadtowaitbecauseafte rgraduationIansuddenly foundhimselfwithoutascholarship,andthusstrappedforc ash.Soheworkedasan AssistantInstructorfor2yearsinthesamedepartment,whi leveryslowlytryingtosave upforGREsandapplicationfees.Duringthattime,hemetthe mostamazingwomanhe hadeverlaideyeson:Monette.Shehasbecomehisbest-frien deversince. Eventually,Iansavedupenoughtobeginhisjourneyabroad. Heacceptedateaching assistantshipfromtheUniversityofFloridain2003.Ianar rivedinGainesvilleon Augustofthatyear,wherehewasgreetedbyanextremelysupp ortiveFilipinograduate communitytowhichheoweshissanityandpledgeshisuncondi tionalfriendship.In UF,Ianwantedtogetinvolvedinrelativisticastrophysics .Heworkedhardtomakea goodimpressiononSteveDetweiler,whosepastworkongener alrelativityinterested himthemost.Luckily,thehardworkpaido,andhewasaccept edasastudentinthe Fallof2005.Sincethen,hehasbeenlearningfromandworkin gwithSteveonavariety oftopics,thoughmostspecicallyonthesubjectofself-fo rceincurvedspacetime.He feelstrulygratefultohavelearnedimmenselyfromSteve.A lwaysteachingbyexample, StevebecameamentortoIannotonlyinphysics,butalsoinwh atitmeanstobean honest-to-goodnesshard-workingscientist. IntheFallof2005,Ian'sbest-friend,Monette,followedhi mtoFlorida,startingon herownPhDattheUniversityofSouthFloridainTampa.Hepro posedtohersoonafter, 176

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andtheygotmarriedinAugust2007.Theyhavebeenslowlybui ldingtheirdreamsever since. IannishedallhisPhDrequirementsinthespringof2009,bu tgraduatedinAugust ofthatyear,justafewmonthsafteranotherspecialladycam eintohislife{hisbaby daughter,Janna.Togetherwithhistwoladies,Ianwillbehe adingtotheUniversityof Guelphtobeginpostdoctoralworkingravitationalphysics withEricPoisson.Whatlies beyondhecannotpredict,buthe'scertainthatwhatevertha tmaybe,it'sboundtobe anotherexcitingadventure. 177