A Solution to Symmetric Teleparallel Gravity(2)

Teleparallel gravity models, in which the curvature and the nonmetricity of spacetime are both set zero, are widely studied in the literature. We work a different teleparallel theory, in which the curvature and the torsion of spacetime are both constrained

however,adoptdi erentcriteriainordertodeterminewhatpropertiesatheoryshouldpossessinorderforittoqualifyasagaugetheory.WetakethegravitationalgaugegrouptobethelocalLorentzgroup[7].

Inthispaperwewillstudyagravitymodelinaspacetimewhosecurvatureandtorsionarebothzero,butthenonmetricityisnonzero.Thereisafewworkintheliteratureaboutgravitymodelsinthiskindofspacetimes;theso-calledsymmetricteleparallelgravity[8].

2Mathematicalpreliminaries

Spacetimeisdenotedbythetriple{M,g, }whereMisa4-dimensionaldi erentiablemani-fold,equippedwithaLorentzianmetricgwhichisasecondrank,covariant,symmetric,non-degeneratetensorand isalinearconnectionwhichde nesparalleltransportofvectors(ormoregenerallytensorsandspinors).Withanorthonormalbasis{Xa},

g=ηabea eb,a,b,···=0,1,2,3(1)

whereηab=( ,+,+,+)istheMinkowskimetricand{ea}istheorthonormalco-frame.Thelocalorthonormalframe{Xa}isdualtotheco-frame{ea};

beb(Xa)=δa.(2)

ThemanifoldMisorientedwiththevolume4-form

1=e0∧e1∧e2∧e3(3)

where denotestheHodgemapanditisconvenienttoemployinthefollowingthegradedinterioroperator Xa≡ a:

b aeb=δa.(4)

Inaddition,theconnection isspeci edbyasetofconnection1-formsΛab.Inthegaugeapproachtogravityηab,ea,Λabareinterpretedasthegeneralizedgaugepotentials,whilethecorresponding eldstrengths;thenonmetricity1-forms,torsion2-formsandcurvature2-formsarede nedthroughtheCartanstructureequations

2Qab:= Dηab=Λab+Λba,

Ta:=Dea=dea+Λab∧eb,

Rab:=DΛab:=dΛab+Λac∧Λcb(5)(6)(7)

wheredandDdenotetheexteriorderivativeandthecovariantexteriorderivative,respectively.These eldstrengthssatisfytheBianchiidentities1

DQab=1

1SinceQab=1

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说教育文库A Solution to Symmetric Teleparallel Gravity(2)在线全文阅读。