A Solution to Symmetric Teleparallel Gravity(3)

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)

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