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