The initial value problem in numerical relativity

arXiv:gr-qc/0412002v11Dec2004February7,20083:11WSPC/INSTRUCTIONFILEPfeifferJournalofHyperbolicDifferentialEquationscWorldScientificPublishingCompanyTheinitialvalueprobleminnumericalrelativityHaraldP.PfeifferTheoreticalAstrophysics,CaliforniaInstituteofTechnology,Pasadena,California91125Received(DayMth.Year)Revised(DayMth.Year)Communicatedby[editor]Abstract.TheconformalmethodforconstructinginitialdataforEinstein’sequationsispresentedinboththeHamiltonianandLagrangianpicture(extrinsiccurvaturedecom-positionandconformalthinsandwichformalism,respectively),andadvantagesduetotherecentintroductionofaweight-functionintheextrinsiccurvaturedecompositionarediscussed.Ithendescriberecentprogressinnumericaltechniquestosolvethere-sultingellipticequations,andexploreinnovativeapproachestowardtheconstructionofastrophysicallyrealisticinitialdataforbinaryblackholesimulations.Keywords:Einsteinsequations;initialvalueproblem;numericalrelativity.1.IntroductionNumericalmethodsplayanimportantroleforinvestigationsintothepropertiesofEinstein’sequations.Inparticular,thelatestagesofinspiralandcoalescenceofbinarycompactobjectslikebinaryblackholesarethoughttobeaccessibleonlytonumericalinvestigations.Knowledgeofthefullwaveformofinspiralingbinaryblackholes,includingthehighlynonlinearcoalescencephase,willenhancesensitivityofgravitationalwavedetectorslikeLIGOorGEO600throughcross-correlationoftheobservedsignalwiththeexpectedwaveforms[20].Comparisonoftheobservedsignalswiththepredictionsofgeneralrelativitywilltestgeneralrelativityinthegenuinelynonlinearregime.Besidestheexperimentalurgency,thebinaryblackholeproblemisarguablythemostfundamentaldynamicalproblemingeneralrelativity;however,itremainsunsolved.Initialdataformsthestartingpointforanyevolution.ForEinstein’sequations,themostwidelyusedmethodtoconstructinitialdataistheconformalmethod,pioneeredbyLichnerowicz[30]andextendedtoamoregeneralformbyYorkandcoworkers[49,34,52].Intworecentpapers,York[53]andPfeiffer&York[40]com-pletedtheconformalmethod:ItisnowavailableinaLagrangianandinaHamil-tonianpicture(referredtoastheconformalthinsandwichformalismandtheex-trinsiccurvaturedecomposition,respectively),andbothpicturescompletelyagree.Thetransverse-tracefreepartoftheextrinsiccurvatureisnowdefinedsuchthatit1February7,20083:11WSPC/INSTRUCTIONFILEPfeiffer2HaraldP.Pfeiffervanishesforanystationaryspacetime.Themethodisnowcompletelyinvarianttoconformaltransformationsofthefreedata.Theconformalmethodresultsinasetofcouplednonlinearthree-dimensionalel-lipticpartialdifferentialequations.Overthelastfewyears,numericaltechniquesforsolvingthesecoupledellipticequationswereimprovedtremendously.Constructionofbinaryblackholeinitialdataisnolongerlimitedbynumericalcapabilities,butbytheincompleteunderstandingofthechoiceoffreedataandboundaryconditionsfortheellipticequations.Here,wepresenttheLagrangianandHamiltonianpicturesoftheconformalmethod,includingmanydetailswhichmayhavebeenmentionedinpassingintech-nicalpapers,butwereneverpresentedinacoherentfashion.Inthesecondpartofthispaper,wedescribebrieflynumericalmethodsandthenexplorerecentinno-vativeapproachestotheconstructionofastrophysicallyrealisticbinaryblackholeinitialdata.Throughoutthispaper,emphasisisplacedonphysicalandnumericalissues,ratherthanmathematicalproofs.2.TheinitialvalueproblemUsingthestandard3+1decomposition[1,52]ofEinstein’sequations,wefoliatespacetimewithspaceliket=const.hypersurfaces.Eachsuchhypersurfacehasafuturepointingunit-normalnμ,inducedmetricgμν=(4)gμν+nμnνandextrinsiccurvatureKμν=−12Lngμν.Thespacetimemetriccanbewrittenasds2=−N2dt+gijdxi+βidt
dxj+βjdt
,(2.1)whereNandβidenotethelapsefunctionandshiftvector,respectively.Nmeasurestheproperseparationbetweenneighboringhypersurfacesalongthesurfacenormalsandβidetermineshowthecoordinatelabelsmovebetweenhypersurfaces:Pointsalongtheintegralcurvesofthetime–vectortμ=Nnμ+βμ(whereβμ=[0,βi]),havethesamespatialcoordinatesxi.Einstein’sequationsdecomposeintoevolutionequationsandconstraintequa-tionsforthequantitiesgijandKij.TheevolutionequationsdeterminehowgijandKijarerelatedbetweenneighboringhypersurfaces,∂tgij=−2NKij+∇iβj+∇jβi(2.2)∂tKij=NRij−2KikKkj+KKij−8πGSij+4πGgij(S−ρ)
−∇i∇jN+βk∇kKij+Kik∇jβk+Kkj∇iβk.(2.3)Here,∇iandRarethecovariantderivativeandthescalarcurvature(traceoftheRiccitensor)ofgij,respectively,andK=Kijgijdenotesthemeancurvature.Furthermore,GstandsforNewton’sconstant,ρandSijarematterdensityandstresstensor,respectively,andS=SijgijdenotesthetraceofSij.Theconstraintequationsareconditionswithineachhypersurfacealone,ensur-ingthatthethree-dimensionalsurfacecanbeembeddedintothefour-dimensionalFebruary7,20083:11WSPC/INSTRUCTIONFILEPfeifferTheinitialvalueprobleminnumericalrelativity3spacetime:R+K2−KijKij=16πGρ,(2.4)∇jKij−gijK
=8πGji,(2.5)withjidenotingthemattermomentumdensity.Equation(2.4)iscalledtheHamil-tonianconstraint,andEq.(2.5)isthemomentumconstraint.CauchyinitialdataforEinstein’sequationsconsistsof(gij,Kij)ononehyper-surfacesatisfyingtheconstraintequations(2.4)and(2.5).Afterchoosinglapseandshift(whicharearbitraryandmerelychoose