Geometrical Hyperbolic Systems for General Relativ

GeometricalHyperbolicSystemsforGeneralRelativityandGaugeTheories[y]AndrewAbrahams,ArlenAnderson,YvonneChoquet-Bruhat[*]andJamesW.York,Jr.DepartmentofPhysicsandAstronomyUniversityofNorthCarolina,ChapelHill27599-3255USAIFP-UNC-516TAR-UNC-053gr-qc/9605014AbstractTheevolutionequationsofEinstein'stheoryandofMaxwell'stheory|thelatterusedasasimplemodeltoillustratetheformer|arewritteningaugecovariantrstordersymmetrichyperbolicformwithonlyphysicallynaturalcharacteristicdirec-tionsandspeedsforthedynamicalvariables.Quantitiesrepresentinggaugedegreesoffreedom[thespatialshiftvectori(t;xj)andthespatialscalarpotential(t;xj),respectively]arenotamongthedynamicalvariables:thegaugeandthephysicalquantitiesintheevolutionequationsareeectivelydecoupled.Forexample,thegaugequantitiescouldbeobtainedasfunctionsof(t;xj)fromsubsidiaryequationsthatarenotpartoftheevolutionequations.Propagationofcertain(\radiative)dynamicalvariablesalongthephysicallightconeisgaugeinvariantwhilethere-mainingdynamicalvariablesaredraggedalongtheaxesorthogonaltothespaceliketimeslicesbythepropagatingvariables.Weobtaintheseresultsby(1)takingafurthertimederivativeoftheequationofmotionofthecanonicalmomentum,and(2)addingacovariantspatialderivativeofthemomentumconstraintsofgeneralrelativity(Lagrangemultiplieri)oroftheGauss'slawconstraintofelectromag-netism(Lagrangemultiplier).Generalrelativityalsorequiresaharmonictime1slicingconditionoraspecicgeneralizationofitthatbringsintheHamiltonianconstraintwhenwepasstorstordersymmetricform.Thedynamicallypropa-gatinggravityeldsstraightforwardlydeterminethe\electricor\tidalpartsoftheRiemanntensor.PACS#:04.20.-q,97.60.Lf21IntroductionWeexaminetheCauchyproblem[1]forgeneralrelativityasthetimehistoryofthetwofundamentalformsofthegeometryofaspacelikehypersurface,itsmetricganditsextrinsiccurvatureK.Byusinga3+1decompositionoftheRiemannandRiccitensors,wesplittheEinsteinequationsintoinitialdataconstraints|equationscontainingonlyg,K,andtheirspacederivatives|andevolutionequationsgivingthetimederivativesofgandKintermsofspacederivativesofthesequantitiesandalsoofthelapseandshift,thatis,ofthevariablesthatxthepropertimeseparationbetweenleavesofthefoliationofspacetimeandthe\timelinesthreadingthefoliation[2,3,4].Theconstraintscanbeposedandsolvedasanellipticsystembyknownmethods.(See,forexample[1,4,5].)However,theequationsofevolutionofgandKareinessencethespatiallycovariantcanonicalHamiltonian[6]equationswithrstordertimederivativesandsecondorderspacederivatives.Castingtheseequationsintohyperbolicform[7,8,9,10,11]isaproblemimportantfortheoreticalanalysisandforpracticalapplications.However,theprocedureusedmayentailalossofspatialcoordinatecovariance,theuseofnon-geometricalvariables,ortheappearanceofunphysicalcharacteristics,thatis,directionswhichareneitheralongthelightconenororthogonaltothetimeslices.Allofthesedicultiescanbeovercome,asweshow,byobtainingrstanonlinearwaveequationfortheevolutionofK.TheevolutionofthespatialmetricgisjustitsdraggingbyKalongtheaxisorthogonaltothespaceliketimeslices.Onemethodforobtainingthisresultrequireschoosingthetimeslicingbyspecifyingthemean(extrinsic)curvatureoftheslices.Itleadstoamixedelliptic-hyperbolicsystemforwhichwehaverecentlyprovenlocalintime,globalinspace,existencetheoremsinthecasesofcompact[12]orasymptoticallyflatslices[13,12,14].Inthispaper,weconcentrateonasecondmethod,whichreliesonaharmonictimeslicingconditionoracertainclassofgeneralizationsofit[12,15,14,16].Weobtainequationsofmotionequivalenttoacovariantrstordersymmetrichyperbolicsystemwithonlyphysicalcharacteristics.Weconstructthissystemexplicitly.Theonlyquanti-3tiespropagatedalongthelightconearecurvatures.Thespacecoordinatesandtheshiftvectorarearbitrary.Inthissense,thesystemisgaugecovariant.2ElectromagnetismMaxwell'stheoryprovidesasimpleexampleforillustratingthemethodweuseintreatingEinstein'stheory.Theideaistoworkwiththedynamicalor3+1formofthetheory,thatis,essentiallythecanonicalform.Byusingafurthertimeandafurtherspacederiva-tiveinadeniteway,oneproceedstoconstructaphysicallynaturalsecondorderwaveequationforthecanonicalmomentum,inthiscasetheelectriceld.Thewaveequationfortheelectriceldcanbewritteninarstordersymmetrichyperbolicform(\FOSHform)thatisalsofluxconservative.Furthermore,thisproceduremaintainsgaugecovari-anceandproducesaFOSHsystemthathasonlythephysicallynaturalcharacteristicdirectionsgivenbynullgeneratorsofthelightconeandtimelikevectorsorthogonaltothespaceliketimeslices.Werefertothesepropertiesas\simplephysicalcharacter-istics.Correspondingly,oneobtains\characteristiceldsthatpropagatealongthelightcone.Bytreatingthissimplecaserst,manystepsindealingwiththetechnicallymoreintricatecaseofgeneralrelativitycanbeabbreviatedbecausetheyfollowthesamepattern.ConsiderMaxwell'stheoryonflatspacelikeslicesofflatMinkowskispacetimewithds2=−dt2+gijdxidxj(1)andprescribedcurrentj(t;xj)=(−;ji)suchthatrj=@0+riji=0(anoverbardenotesaspatialtensororoperator).UsingF=2r[A]andandA=(−;Ai),wedeneRrF,whichleadstotheidentitiesR0−rjEj;(2)Ri−@0E