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1、IntroductiontoIsolatedHorizonsinNumericalRelativityOlafDreyer∗†PerimeterInstituteforTheoreticalPhysics,35KingStreetNorth,Waterloo,OntarioN2J2W9,CanadaBadriKrishnan‡andDeirdreShoemaker§CenterforGravitationalPhysicsandGeometryandCenterforGravitationalWavePh
2、ysics,DepartmentofPhysics,PennStateUniversity,UniversityPark,PA16802,USAErikSchnetter¶†TheoretischeAstrophysik,Universit¨atT¨ubingen,AufderMorgenstelle,72076T¨ubingen,Germany(Dated:February7,2008)Wepresentacoordinate-independentmethodforextractingmass(M∆)
3、andangularmomentum(J∆)ofablackholeinnumericalsimulations.Thismethod,basedontheisolatedhorizonframework,isapplicablebothatlatetimeswhentheblackholehasreachedequilibrium,andatearlytimeswhentheblackholesarewidelyseparated.Assumingthatthespatialhypersurfacesu
4、sedinagivennumericalsimulationaresuchthatapparenthorizonsexistandhavebeenlocatedonthesehypersurfaces,weshowhowJ∆andM∆canbedeterminedintermsofonlythosequantitieswhichareintrinsictotheapparenthorizon.Wealsopresentanumericalmethodforfindingtherotationalsymmet
5、ryvectorfield(requiredtocalculateJ∆)onthehorizon.PACSnumbers:04.25.DmI.INTRODUCTIONnoapparenthorizonsonthespatialslicesusedinagivennumericalsimulationofablackholespacetime1,inprac-Givenanumericalsimulationofaspacetimecontainingtice,thisissuedoesnotarisefor
6、mostnumericalblackholesimulationsthatweareawareof.Weshallthereforeablackhole,oneisfacedwithtwoimmediatequestions:Whereistheblackhole,andwhatarethevaluesofitsdisregardthisissuefortherestofthispaperandonlydealwithpartialCauchysurfaces(inblackholespacetimes)
7、parameters,suchasmassandangularmomentum?forwhichapparenthorizonsdoexistandhavebeenlo-Inanalyticalconsiderations,ablackholeisoftende-cated[2,3,4,5,6,7,8].Anaturalquestionwewantfinedviatheeventhorizon,i.e.thefutureboundaryoftoaskis:canoneunambiguouslyassocia
8、teblackholethecausalpastoffuturenullinfinity.Whilethisnotionparameterstoapparenthorizons?ismathematicallyelegantandhasledtopowerfulresultsinblackholephysics,generallyitisnotdirectlyusefulinOnewaytoattributeamassandan
