Energy of General Spherically Symmetric Solution i

arXiv:gr-qc/9601044v129Jan1996STUPP–95–141KITASATO–95–1August1995EnergyofGeneralSphericallySymmetricSolutioninTetradTheoryofGravitationTakeshiSHIRAFUJI∗,GamalG.L.NASHED∗†andKenjiHAYASHI∗∗∗PhysicsDepartment,FacultyofScience,SaitamaUniversity,Urawa338∗∗InstituteofPhysics,KitasatoUniversity,Sagamihara228Wefindthemostgeneral,sphericallysymmetricsolutioninaspecialclassoftetradtheoryofgravitation.ThetetradgivestheSchwarzschildmetric.Theenergyiscal-culatedbythesuperpotentialmethodandbytheEuclideancontinuationmethod.Wefindthatunlessthetime-spacecomponentsofthetetradgotozerofasterthan1/√ratinfinity,thetwomethodsgiveresultsdifferentfromeachother,andthattheseresultsdifferfromthegravitationalmassofthecentralgravitatingbody.Thisfactimpliesthatthetime-spacecomponentsofthetetraddescribinganisolatedsphericalbodymustvanishfasterthan1/√ratinfinity.†Permanentaddress:MathematicsDepartment,FacultyofScience,AinShamsUniversity,Cairo,Egypt.1.IntroductionThenotionofabsoluteparallelismwasfirstintroducedinphysicsbyEinstein1)tryingtounifygravitationandelectromagnetisminto16degreesoffreedomofthetetrads.Hisattemptfailed,however,becausetherewasnoSchwarzschildsolutioninhisfieldequation.Mφller2)revivedthetetradtheoryofgravitationandshowedthatatetraddescrip-tionofgravitationalfieldallowsamoresatisfactorytreatmentoftheenergy-momentumcomplexthanthatofgeneralrelativity.TheLagrangianformulationofthetheorywasgivenbyPellegriniandPlebanski.3)IntheseattemptstheadmissibleLagrangianswerelimitedbytheassumptionthattheequationsdeterminingthemetrictensorshouldcoincidewiththeEinsteinequation.Mφller4)abandonedthisassumptionandsug-gestedtolookforawiderclassofLagrangians,allowingforpossibledeviationfromtheEinsteinequationinthecaseofstronggravitationalfields.S´aez5)generalizedMøllertheoryintoascalartetradtheoryofgravitation.Meyer6)showedthatMøllertheoryisaspecialcaseofPoincar´egaugetheory.7),8)Quiteindependently,HayashiandNakano9)formulatedthetetradtheoryofgrav-itationasthegaugetheoryofspace-timetranslationgroup.HayashiandShirafuji10)studiedthegeometricalandobservationalbasisofthetetradtheory,assumingthattheLagrangianbegivenbyaquadraticformoftorsiontensor.Ifinvarianceunderparityoperationisassumed,themostgeneralLagrangianconsistsofthreetermswiththreeunknownparameterstobefixedbyexperiments,besidesacosmologicalterm.Twoofthethreeparametersweredeterminedbycomparingwithsolar-systemexperiments,10)whileonlyanupperboundhasbeenestimatedforthethirdone.10),11)Thenumericalvaluesofthetwoparametersfoundwereverysmallconsistentwithbeingequaltozero.Ifthesetwoparametersareequaltozeroexactly,thetheoryreducestotheoneproposedbyHayashiandNakano9)andMφller,4)whichweshallhererefertoastheHNMtheoryforshort.Thistheorydiffersfromgeneralrelativityonlywhenthetorsiontensorhasnonvanishingaxial-vectorpart.Itwasalsoshown10)thattheBirkhofftheoremcanbeextendedtotheHNMtheory.Namely,forsphericallysymmetriccaseinvacuum,whichisnotnecessarilytimeindependent,theaxial-vectorpartofthetorsiontensorshouldvanishduetotheantisymmetricpartofthefieldequation,andtherefore,withthehelpoftheBirkhofftheorem12)ofgeneralrelativityweseethatthespacetimemetricistheSchwarzschild.Mikhailetal.13)derivedthesuperpotentialoftheenergy-momentumcomplexintheHNMtheoryandappliedittotwosphericallysymmetricsolutions.Itwasfoundthatinoneofthetwosolutionsthegravitationalmassdoesnotcoincidewiththecalculatedenergy.Mikhailetal.14)alsoderivedasphericallysymmetricsolutionoftheHNMtheorystartingfromatetradwhichcontainsthreeunknownfunctionsandfollowingMazumderandRay.15)Thesolutioncontainsonearbitraryfunctionoftheradialco-ordinater,andallprevioussolutionscanbeobtainedfromit.Thephysicalpropertiesofthissolutionhavenotyetbeenexamined,however.WeshowthattheunderlyingmetricoftheirsolutionisjusttheSchwarzschildmetricundercertainconditionsinconsistentwiththeextendedBirkhofftheoremmentionedabove.2Thegeneralformofthetetrad,λμi,havingsphericalsymmetrywasgivenbyRobertson.16)Inthecartesianformitcanbewrittenas∗λ00=iA,λ0a=Cxa,λα0=iDxαλαa=δαaB+Fxaxα+ǫaαβSxβ,(1E1)whereA,C,D,B,F,andSarefunctionsoftandr=(xαxα)1/2,andthezerothvectorλμ0hasthefactori=√−1topreserveLorentzsignature.Weconsideranasymptoticallyflatspace-timeinthispaper,andimposetheboundaryconditionthatforr→∞thetetrad(1E1)approachesthetetradofMinkowskispace-time,λμi=diag(i,δaα).Itistheaimofthepresentworktofindthemostgeneral,asymptoticallyflatsolutionwithsphericalsymmetryintheHNMtheoryandcalculatetheenergyofthatsolution.Wedothisbytwomethodsandcomparetheresults:oneisbyapplyingthesuperpotentialofMikhailetal.,13)andtheotherbasedontheEuclideancontinuationmethodofGibbonsandHawking.17)∼19)Insection2webrieflyreviewthetetradtheoryofgravitation.Insection3wefirststudythegeneral,sphericallysymmetricsolutionwithanonvanishingS-term(see(1E1)),andobtainasolutionwithoneparameter.Thenwestudythegeneral,sphericallysymmetrictetradwithouttheS-term.Alltheremaining,unknownfunctionsareallowedtodependontandr.Wefindthegeneralsolutionwithanarbitraryfunctionoftandr.WealsostudythesolutionofMikhailetal.14)bytransformingittotheisotropic,cartesiancoordinate.