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
Thelinearconnection1-formscanbedecomposeduniquelyasfollows[9],[10]
Λab=ωab+Kab+qab+Qab
whereωabaretheLevi-Civitaconnection1-formsthatsatisfy
dea+ωab∧eb=0,
Kabarethecontortion1-formssuchthat
Kab∧eb=Ta,
andqabaretheanti-symmetrictensor1-formsde nedby
qab= ( aQbc)∧ec+( bQac)∧ec.
Intheabovedecompositionthesymmetricpart
Λ(ab)=Qab
whiletheanti-symmetricpart
Λ[ab]=ωab+Kab+qab.(16)(15)(14)(13)(12)(11)
Itiscumbersometotakeintoaccountallcomponentsofnonmetricityingravitationalmodels.Thereforewewillbecontentwithdealingonlywithcertainirreduciblepartsofittogainphysicalinsight.TheirreducibledecompositionsofnonmetricityinvariantundertheLorentzgrouparesummarilygivenbelow[10].Thenonmetricity1-formsQabcanbesplitintotheirtrace-free
Qab+1
Qab=0.Letusde ne
Λb:= a
Qab∧ea),Θ:=eb∧Θb, a:=Θa 1
(3)
(4)Qab=Qab=321(ea∧ b+eb∧ a)2ηabΛ)(20)(21)