Energy-Momentum Conservation Laws in Affine-Metric

arXiv:gr-qc/9501009v110Jan1995ENERGY-MOMENTUMCONSERVATIONLAWSINAFFINE-METRICGRAVITATIONTHEORY.GennadiASardanashvilyDepartmentofTheoreticalPhysics,PhysicsFaculty,MoscowStateUniversity,117234Moscow,RussiaE-mail:[e-mail]ormulationoffieldtheorydoesnotprovideanyuniversalenergy-momentumconservationlawinordertoanalizethatingravitationtheory.InLa-grangianfieldtheory,wegetdifferentidentitiesinvolvingdifferentstressenergy-momentumtensorswhichhoweverarenotconserved,otherwiseinthecovariantmultimomentumHamiltonianformalism.Intheframeworkofthisformalism,wehavethefundamentalidentitywhoserestrictiontoaconstraintspacecanbetreatedtheenergy-momentumtransformationlaw.Thisidentityremainstruealsoforgrav-ity.Thus,thetoolsareathandtoinvestigatetheenergy-momentumconservationlawsingravitationtheory.Thekeypointconsistsinthefeatureofametricgravi-tationalfieldwhosecanonicalmomentaontheconstraintspaceareequaltozero.1IntroductionInHamiltonianmechanics,thereistheconventionalenergytransformationlawdHdt=∂H∂t(1)onsolutionsoftheHamiltonequations,otherwiseinfieldtheory.ThestandardHamiltonianformalismhasbeenappliedtofieldtheory.Inthestraight-forwardmanner,ittakestheformoftheinstantaneousHamiltonianformalismwhencanonicalvariablesarefieldfunctionsatagiveninstantoftime.Thecorrespondingphasespaceisinfinite-dimensional,sothattheHamiltonequationsinthebracketformarenotthefamiliardifferentialequations,adequatetotheEuler-Lagrangefieldequations.InLagrangianfieldtheory,wehavenoconventionalenergy-momentumtransformationlaw.Onegetsdifferentidentitieswhichinvolvedifferentstressenergy-momentumtensors,inparticular,differentcanonicalenergy-momentumtensors.Moreover,onecannotsayaprioriwhatisreallyconcerved.WefollowthegenerallyacceptedgeometricdescriptionofclassicalfieldsbysectionsoffibredmanifoldsY→X.Theirdynamicsisphrasedintermsofjetspaces[2,7,10,13,15].GivenafibredmanifoldY→X,thek-orderjetspaceJkYofYcomprisestheequivalence1classesjkxs,x∈X,ofsectionssofYidentifiedbythefirst(k+1)termsoftheirTaylorseriesatapointx.Itisafinite-dimensionalsmoothmanifold.Recallthatak-orderdifferentialoperatoronsectionsofafibredmanifoldY,bydefinition,isamorphismofJkYtoavectorbundleoverX.Asaconsequence,thedynamicsoffieldsystemsisplayedoutonfinite-dimensionalconfigurationandphasespaces.Infieldtheory,wecanrestrictourselvestothefirstorderLagrangianformalismwhentheconfigurationspaceisJ1Y.Givenfibredcoordinates(xμ,yi)ofY,thejetspaceJ1Yisendowedwiththeadaptedcoordinates(xμ,yi,yiμ):y′iλ=(∂y′i∂yjyjμ+∂y′i∂xμ)∂xμ∂x′λ.AfirstorderLagrangiandensityontheconfigurationspaceJ1YisrepresentedbyahorizontalexteriordensityL=L(xμ,yi,yiμ)ω,ω=dx1∧...∧dxn,n=dimX.ThecorrespondingfirstorderEuler-LagrangeequationsforsectionssofthefibredjetmanifoldJ1Y→Xread∂λsi=siλ,∂iL−(∂λ+sjλ∂j+∂λsjμ∂μj)∂λiL=0.(2)WeconsidertheLiederivativesofLagrangiandensitiesinordertoobtaindifferentialconservationlaws.Letu=uμ∂μ+ui∂ibeavectorfieldonafibredmanifoldYanduitsjetlift(15)ontothefibredjetmanifoldJ1Y→X.GivenaLagrangiandensityLonJ1Y,letuscomputertheLiederivativeLuL.OnsolutionssofthefirstorderEuler-Lagrangeequations(2),wehavetheequalitys∗LuL=ddxλ[πλi(s)(ui−uμsiμ)+uλL(s)]ω,πμi=∂μiL.(3)Inparticular,ifuisaverticalvectorfieldsuchthatLuL=0,theequality(3)takestheformofthecurrentconservationlawddxλ[uiπλi(s)]=0.(4)Ingaugetheory,thisconservationlawisexemplifiedbytheNoetheridentities.Letτ=τλ∂λ2beavectorfieldonXandu=τΓ=τμ(∂μ+Γiμ∂i)itshorizontalliftontothefibredmanifoldYbyaconnectionΓonY.Inthiscase,theequality(3)takestheforms∗LτΓL=−ddxλ[τμTΓλμ(s)]ω(5)whereTΓλμ(s)=πλi(siμ−Γiμ)−δλμL(6)isthecanonicalenergy-momentumtensorofafieldswithrespecttotheconnectionΓonY.Thetensor(6)istheparticularcaseofthestressenergy-momentumtensors[1,3,6].Inparticular,whenthefibrationY→Xistrivial,onecanchoosethetrivialconnectionΓiμ=0.Inthiscase,thetemsor(6)ispreciselythestandardcanonicalenergy-momentumtensor,andifLτL=0forallvectorfieldsτonX(e.g.,XistheMinkowskispace),theconservationlaw(5)comestothewell-knownconservationlawddxλTλμ(s)=0ofthecanonicalenergy-momentumtensor.Ingeneral,theLiederivativeLτΓLfailstobeequaltozeroasarule,andtheequality(5)isnottheconservationlawofacanonicalenergy-momentumtensor.Forinstance,ingaugetheoryofgaugepotentialsandscalarmatterfieldsinthepresenceofabackgroundworldmetricg,wegetthecovariantconservationlaw∇λtλμ=0(7)ofthemetricenergy-momentumtensor.InEinstein’sGeneralRelativity,thecovariantconservationlaw(7)issuesdirectlyfromgravitationalequations.Butitisconcernedonlywithzero-spinmatterinthepresenceofthegravitationalfieldgeneratedbythismatteritself.Thetotalenergy-momentumconservationlawformatterandgravityisintroducedbyhand.Itreadsddxμ[(−g)N(tλμ+Tgλμ)]=0(8)wheretheenergy-momentumpseudotensorTgλμofametricgravitationalfieldisdefinedtosatisfytherelation(−g)N(tλμ+Tgλμ)=12κ∂σ∂α[(−g)N(gλμgσα−gσμgλα)3onsolutionsoftheEinsteinequations.Theconservationlaw(8)israthersatisfactoryonlyincasesofasymptotic-flatgravitationalfieldsandabackgroundgravitationalfield.Theenergy-momentumconservationlawintheaffine-metricgravitationtheoryandthegaugegravitationtheorywasnotdiscussedwidely[5].Thus,theLagrang