A New Discretization of Classical and Quantum Gene

arXiv:gr-qc/9304005v210May1994G¨oteborgITP94-5SU-GP-93-4-1CGPG-94/3-3AnewdiscretizationofclassicalandquantumgeneralrelativityOlaBostr¨om,1,aMarkMiller2,bandLeeSmolin3,caInstituteofTheoreticalPhysics,ChalmersUniversityofTechnology,S-41296G¨oteborg,SwedenbDepartmentofPhysics,SyracuseUniversity,Syracuse,U.S.A.,13244cCenterforGravitationalPhysicsandGeometry,PennsylvaniaStateUniversity,UniversityPark,PA16802-6300May9,1994AbstractWeproposeanewdiscreteapproximationtotheEinsteinequations,basedontheCapovilla-Dell-JacobsonformoftheactionfortheAshtekarvariables.ThisformulationisanalogoustotheReggecalculusinthatthecurvaturehassup-portonsetsofmeasurezero.BothaLagrangianandHamiltonianformulationareproposedandwereportpartialresultsabouttheconstraintalgebraoftheHamiltonianformulation.Wefindthatthediscreteversionsofthediffeomor-phismconstraintsdonotcommutewitheachotherorwiththeHamiltonianconstraint.1Emailaddress:[e-mail].2Emailaddress:[e-mail].3Emailaddress:[e-mail].11IntroductionEinstein’sequationsarebeautifulbecausetheycapturethephenomenaofgrav-itationinasimplegeometricalstatement:thevanishingoftheRiccicurvatureinemptyregionsofspacetime.However,whenweusethemtosolveapracticalproblemingravitationwediscovertheyhaveanotherside—theyarecompli-catednonlinearpartialdifferentialequations.Inspecialcases,mosttypicallywhensymmetrieshavebeenimposed,onesometimescanmakeuseofthege-ometrytohelpdiscoverthesolutiontoaphysicalproblem.But,whenoneisstudyingthegenericproblemofconstructingsolutionsto,orevolving,theEin-steinequations,littleofthegeometricbeautycomesoutinthetechniquesweusetotrytosolvethetheory.Thisisespeciallythecasefornumericalapproximationmethods.Suchmeth-odsarecrucialformakingprogresswithimportantastrophysicalquestionssuchasgravitationalwaveproductionbyrealisticsources.Bythetimealloftheelementsnecessaryformakingthenumericalproblemwelldefinedareinplace,includinggaugefixingandfinitedifferencingschemes,verylittleofthegeomet-ricalbeautyoftheequationsremains.ForoverthirtyyearstherehasbeenavailableanalternativetothefinitedifferencingapproachestotheEinsteinequations,whichistheReggecalculus[1].Inthisapproachalargeandgenericsetofsolutionsisconstructedbylimitingsolutionstomanifoldsinwhichthecurvatureisrestrictedtohavesupportonsetsofmeasurezero.Typically,themanifoldisbrokenupintosimplices,andthecurvatureisrestrictedtolieontheboundariesatwhichsimplicesarejoined.TheideaofReggecalculusisthatsuchsimplicialmanifoldscouldapproxi-mateagivensmoothmanifoldarbitrarilywell.Unfortunately,atleastupuntilthistime,Reggecalculushasnotbeendevelopedintoapowerfultoolforuseinrealisticcalculations.Or,atleast,oneshouldsaythatthisisthecaseintheclassicaltheory,asrecentlyaversionofReggecalculushasbeenshowntoyieldaveryeffectivemethodforcalculatingthepathintegralsinquantumgravityintwo[2,3,4,5,6],three[7]andevenfourdimensions[8,9,10]4.InthispaperweintroduceanewdiscretizationofEinsteinequationsthatresultsfromapplyingtheexactEinstein’sequationstoaspecialrestrictedclassofgeometries.Thisnewformulationhastwobasicfeatures.First,itisbasedontheAshtekarformulationofgeneralrelativity[16,17,18],inwhichthedynam-icalvariablesareframefieldsandself-dualconnections.Second,therestrictedclassofgeometrieswestudyarethoseinwhichallthefieldsaredistributional.Theseareconnectedinthefollowingway.AstheAshtekarformalismispolynomialatboththeLagrangianandHamiltonianlevel,thefieldequationsallowawiderclassofsolutionsthantheconventionalformoftheEinsteinequa-tions.Theseincludesolutionsinwhichthedeterminantofthemetricvanishes,4ForotherworkonfourdimensionalquantumgravityusingReggecalculus,see[11,12,13,14,15].2forwhichtheusualrelationsfortheChristoffelsymbolsintermsofthemetriccomponentswouldnotbedefined.Thisispossiblebecauseoneofthebasicfieldsoftheformalismisathreedimensionalframefield,whichischosentohavedensityweightone.(Actually,suchsolutionsareallowedforallfirstor-derformulationsoftheEinstein’sequation,ashasbeenemphasizedrecentlybyHorowitz[19].)Amongsuchdegeneratesolutionsarethoseforwhichthedensitizedframefieldisactuallydistributional.BecauseoftheYang-Millslikegaugeinvarianceofthetheory,itturnsoutthattherearemanysolutionsinwhichtheframefieldshavesupportononedimensionalcurvesintheHamiltonianformalisms.ThisisbecausetheGauss’slawconstraintissolvedbyframefieldswhicharecovariantlydivergencefree.Theseconfigurationsmaybetakentobeoftheform5˜Eaiα(x)=a2Zdsδ3(x,α(s))˙αa(s)eiα(1)Hereαisaclosedoropencurveinthespatialthreemanifold,whichwewilldenoteΣandeiisanelementoftheLiealgebraofSU(2)whichisassociatedwiththatcurve.aisaconstantwithdimensionsoflength,whichisnecessaryif,asisnatural,theLiealgebraelementeiαisdimensionless,sothattheframefieldcanalsobedimensionless.IntheAshtekarformalismitisalwaysimportanttokeeptrackofthedensityweights.Thus,notethattherighthandsideof(1)isnaturallyavectordensity,asisrequired.Wecanconsiderdistributionalgeometriesoftheformof(1)forcomplicatedgraphsorlattices.Forexample,letΓbesomegraphinΣ,withedgesγIwhereI=1,...N.Thenweconsiderdistributionalconfigurationsoftheform˜EaiΓ(x)=a2XIZ