Metric-affine gauge theory of gravity II. Exact so

arXiv:gr-qc/9902076v215Apr1999Metric–affinegaugetheoryofgravityII.ExactsolutionsFriedrichW.Hehl∗andAlfredoMac´ıas†DepartamentodeF´ısicaUniversidadAut´onomaMetropolitana–IztapalapaApartadoPostal55–534,C.P.09340,M´exico,D.F.,MexicoAbstractIncontinuingourseriesonmetric-affinegravity(seeGronwald,IJMPD6(1997)263forPartI),wereviewtheexactsolutionsofthistheory.filemagexac7.tex,1999-04-09TypesetusingREVTEX∗E-mail:[e-mail]:InstituteforTheoreticalPhysics,UniversityofCologne,D–50923K¨oln,Germany†E-mail:[e-mail]–AFFINEGRAVITY(MAG)In1976,anewmetric–affinetheoryofgravitationwaspublished[16].Inthismodel,themetricgijandthelinear(sometimesalsocalledaffine)connectionΓijkwereconsideredtobeindependentgravitationalfieldvariables.Themetriccarries10andtheconnection64independentcomponents.AlthoughnowadaysmoregeneralLagrangiansareconsidered,liketheoneinEq.(10),theoriginalLagrangiandensityofmetric–affinegravityreadsVGR′=√−g2κgijhRicij(Γ,∂Γ)+βQiQji.(1)TheRiccitensorRicijdependsonlyontheconnectionbutnotonthemetric,whereastheWeylcovectorQi:=−gkl∇igkl/4dependsonboth.Here∇irepresentsthecovariantderivativewithrespecttotheconnectionΓijk,furthermoreg=detgkl,κisEinstein’sgravitationalconstant,andβadimensionlesscouplingconstant.Withi,j,k,···=0,1,2,3wedenotecoordinatesindices.Thismodelleadsbacktogeneralrelativity,assoonasthematerialcurrentcoupledtotheconnection,namely√−gΔijk:=δLmat/δΓijk,theso–calledhypermomentum,vanishes.Thus,insuchamodel,thepost–Riemannianpiecesoftheconnectionandthecorrespondingnewinteractionsaretiedtomatter,theydonotpropagate.Asweknowfromtheweakinteraction,acontactinteractionappearstobesuspiciousforcausalityreasons,andonewantstomakeitpropagating,evenifthecarrieroftheinteraction,theintermediategaugeboson,maybecomeveryheavyascomparedtothemassoftheproton,e.g..However,beforewereportonthemoregeneralgaugeLagrangiansthathavebeenused,weturnbacktothegeometryofspacetime.MAGrepresentsagaugetheoryofthe4–dimensionalaffinegroupenrichedbytheexis-tenceofametric.Asagaugetheory,itfindsitsappropriateformifexpressedwithrespecttoarbitraryframesorcoframes.Therefore,theapparatusofMAGwasreformulatedinthecalculusofexteriordifferentialforms,theresultofwhichcanbefoundinthereviewpaper[14],seealso[19]and[15].Ofcourse,MAGcouldhavebeenalternativelyreformulatedin2tensorcalculusbyemployinganarbitrary(anholonomic)frame(tetradorvierbeinformal-ism),butexteriorcalculus,basicallyinaversionwhichwasadvancedbyTrautman[42]andothers[31,4],seemstobemorecompact.Inthenewformalism,wehavethenthemetricgαβ,thecoframeϑα,andtheconnection1–formΓαβ(withvaluesintheLiealgebraofthe4–dimensionallineargroupGL(4,R))asnewindependentfieldvariables.Hereα,β,γ,···=0,1,2,3denote(anholonomic)frameindices.Fortheformalism,includingtheconventions,whichwewillbeusinginthispaper,wereferto[19].AfirstorderLagrangianformalismforamatterfieldΨminimallycoupledtothegravi-tationalpotentialsgαβ,ϑα,Γαβhasbeensetupin[19].Spacetimeisdescribedbyametric–affinegeometrywiththegravitationalfieldstrengthsnonmetricityQαβ:=−Dgαβ,torsionTα:=Dϑα,andcurvatureRαβ:=dΓαβ−Γαγ∧Γγβ.ThegravitationalfieldequationsDHα−Eα=Σα,(2)DHαβ−Eαβ=Δαβ,(3)linkthematerialsources,thematerialenergy–momentumcurrentΣαandthematerialhypermomentumcurrentΔαβ,tothegaugefieldexcitationsHαandHαβinaYang–Millslikemanner.In[19]itisshownthatthefieldequationcorrespondingtothevariablegαβisredundantif(2)aswellas(3)arefulfilled.IfthegaugeLagrangian4–formV=Vgαβ,ϑα,Qαβ,Tα,Rαβ(4)isgiven,thentheexcitationscanbecalculatedbypartialdifferentiation,Hα=−∂V∂Tα,Hαβ=−∂V∂Rαβ,Mαβ=−2∂V∂Qαβ,(5)whereasthegaugefieldcurrentsofenergy–momentumandhypermomentum,respectively,turnouttobelinearintheLagrangianandintheexcitations,3Eα:=∂V∂ϑα=eα⌋V+(eα⌋Tβ)∧Hβ+(eα⌋Rβγ)∧Hβγ+12(eα⌋Qβγ)Mβγ,(6)Eαβ:=∂V∂Γαβ=−ϑα∧Hβ−gβγMαγ.(7)Hereeαrepresentstheframeand⌋theinteriorproductsign,fordetailssee[19].II.THEQUADRATICGAUGELAGRANGIANOFMAGThegaugeLagrangian(1),inthenewformalism,isa4–formandreads[27]VGR′=12κ−Rαβ∧ηαβ+βQ∧∗Q.(8)Hereηαβ:=∗(ϑα∧ϑβ),∗denotestheHodgestar.BesidesEinsteingravity,itencompassesadditionallycontactinteractions.ItisobviousofhowtomakeQapropagatingfield:Oneadds,tothemassiveβ–term,akineticterm[17,36]−αdQ∧∗dQ/2.SincedQ=Rγγ/2,thekinetictermcanalternativelybewrittenas−α8Rββ∧∗Rγγ.(9)Thisterm,withtheappearanceofoneHodgestar,displaysatypicalYang–Millsstructure.Moregenerally,propagatingpost–RiemanniangaugeinteractionsinMAGcanbeconsis-tentlyconstructedbyaddingtermsquadraticinQαβ,Tα,RαβtotheHilbert-EinsteintypeLagrangianandthetermwiththecosmologicalconstant.Inthefirstorderformalismweareusing,higherorderterms,i.e.cubicandquarticonesetc.wouldpreservethesecondorderofthefieldequations.However,thequasilinearityofthegaugefieldequationswouldbedestroyedand,inturn,theCauchyproblemwouldbeexpectedtobeill–posed.ThereforewedonotgobeyondagaugeLagrangianwhichisquadraticinthegaugefieldstrengthsQαβ,Tα,Rαβ.Incidentally,aquadraticLagrangianisalreadysomessythatitwouldbehardtohandleastillmorecomplexoneanyway.Differentgroupshavealreadyadded,withinametric–affineframework,differentquadraticpiecestotheHilbert–Ein