Numerical treatment of the hyperboloidal initial v

arXiv:gr-qc/9712050v111Dec1997NumericaltreatmentofthehyperboloidalinitialvalueproblemforthevacuumEinsteinequationsI.TheconformalfieldequationsJ¨orgFrauendienerInstitutf¨urTheoretischeAstrophysik,Universit¨atT¨ubingen,AufderMorgenstelle10,D-72076T¨ubingen,GermanyFebruary7,2008AbstractThisisthefirstinaseriesofarticlesonthenumericalsolutionofFriedrich’sconformalfieldequationsforEinstein’stheoryofgravity.Wewilldiscussinthispaperwhyoneshouldbeinterestedinapplyingtheconformalmethodtophysicalproblemsandwhythereisgoodhopethatthismightevenbeagoodideafromthenumericalpointofview.Wedescribeindetailthederivationoftheconformalfieldequationsinthespinorformalismwhichweusefortheimplementationoftheequations,andpresentalltheequationsasareferenceforfuturework.Finally,wediscusstheimplicationsoftheassumptionsofacontinuoussymmetry.1IntroductionMuchofthecurrentworkinnumericalandexperimentalrelativityisdevotedtoobtaininginformationaboutthegravitationalradiationthatisemittedbyastrophysicalprocesseswhicharetakingplaceinouruniverse.Thegoalistoobtainwaveforms,i.e.,the“fingerprints”bywhichdifferentprocessescanbeidentifiedwhenthegravitationalwavesaredetectedbylaserinterferometerslikeLIGOorVIRGO.ThecommonwaytodescribesuchasystemwithinEinstein’stheoryofgravityisbywayofanidealizationwherethesystemisconsideredasbeingsofarawayfromtherestoftheuniversethattheinfluenceofthelattercanbeneglected(cf.[1]foracleardiscussionofwhatisinvolvedinthisprocess).Then,intuitively,thefieldsfarawayfromthesourceshoulddecaysothatthespace-timebecomesasymptoticallyflat.Thedetectorsarethenidealizedasobserverswhicharelocated“atinfinity”wheretheycangatherthegravitationalradiationcomingfromthesystem.Isolatedgravitatingsystemsandthestructureoftheir“farfields”havebeeninvestigatedforalongtimebecauseoftheirimportancefortheinterpretationofmeasurements.Aseriesofarticleswhichheavilyinfluencedthewaywelook11INTRODUCTION2atthesubjecttodaywaspublishedintheearly60’s.Inthesearticlesvari-ousimportantcontributionsweremade:the“peelingproperty”oftheWeyltensor[2],theideaofanalysingthevacuumfieldequationsonoutgoingnullhypersurfacesresultingintheBondimasslossformula[3,4],theinventionoftheNPformalismandtheproposalforconsideringthevacuumBianchiiden-tityasafieldequationfortheWeyltensor[5]andtheasymptoticsolutionoftheEinsteinvacuumequations[6].AssumingthatcertaincomponentsoftheWeylcurvaturefall-offinaspecificwayitwasfoundbyformalpowerseriesanalysisoftheasymptoticcharacteristicinitialvalueproblemthatthefall-offbehaviourofthefieldsalongnulldirectionscouldbecharacterizedintermsofcertainspecialcoordinatesystemswhoseexistencewaspresupposed.Finally,itwasrealizedbyPenrose[7]thatthesefall-offconditionsaswellasthepeelingpropertycouldbeunderstoodinapurelygeometricway.Heintroducedtheno-tionofaconformalextensionbywhichaLorentzmanifold(fM,˜g)isembeddedintoabiggermanifold(M,g)withisomorphicconformalstructurebutwithaLorentzmetricwhichdiffersfrom˜gbyapositivefactorg=Ω2˜g.TheideawastostudytheglobalconformalpropertiesofMinkowskispaceinordertoobtainacriterionforwhatoneshouldcallan“asymptoticallyflat”space-time.GuidedbytheMinkowskisituationPenrosesuggestedthatsuchspace-timesallowtheattachmentofaconformalboundaryJwhichischaracterizedbythevanishingoftheconformalfactorΩ.Thisboundaryisaregularnullhypersurfaceintheambientunphysicalmanifold.Itcanbeinterpretedasthepointswhichareatinfinityforthephysicalmanifoldalongnulldirections.ThequestionarisesastowhatextentthisgeometricpictureiscompatiblewiththeEinsteinequations.Friedrichcouldderiveasystemofequations[8],the“conformalfieldequations”,whicharedefinedonthatlargerunphysicalmanifold.Furthermore,asolutionoftheconformalfieldequationsgivesrisetoasolutionofthestandardfieldequationsonthephysicalspace-time.ThissystemiswrittenintermsofgeometricquantitiesoftheunphysicalmanifoldandtheconformalfactorΩanditisregulareverywhereevenatpointswhereΩvanishes.Inausual3+1decomposition,theconformalfieldequationssplitintoconstraintequationsandevolutionequations.UsingthissystemofequationsFriedrichwasabletoreducetheasymptoticcharacteristicinitialvalueproblemfortheEinsteinequationswheredataaregivenonapartof(past)nullinfinityandaningoingnullhypersurfacewhichintersectsnullinfinityinatwo-dimensionalsurfacetoacharacteristicinitialvalueproblemforasymmetrichyperbolicsystem[8].Inordertodescribeaphysicalsituationonewouldliketoprescribeinitialdatafortheconformalfieldequationsonsomeinitialspace-likehypersurfaceanddeterminefromthemthefutureofthesystem.Ideally,thedatashouldbegivenonanasymptoticallyflatspace-likesurface.Itdoesturnoutthattheinitialdatafortheconformalfieldequationsonsuchahypersurfacearenecessarilysingularbecausetheconformalstructureofspace-timeissingularati0whenevertheADMmassisnon-zero(see[9]foranewapproachtowardsthesolutionofthisproblem).Therefore,theinitialdataaregivenonaspace-likehypersurfacewhichintersectsJtransverselyinatwo-dimensionalsurface.SuchhypersurfacesarecalledhyperboloidalsurfacesbecausetheybehavelikespacesofconstantnegativecurvatureintheneighbourhoodofJ.Friedrich[10]hasshownthattheCauchyproblemfordatagivenonsuchh