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
3Symmetricteleparallelgravity
Inthesymmetricteleparallelgravity(STPG)[8],wehavetwogeometricalconstraints
Rab=dΛab+Λac∧Λcb=0(24)
Ta=dea+Λab∧eb=0.(25)
Theseequationsmeanthatthereisadistantparallelism,buttheanglesandlengthsmaychangeduringaparalleltransport.
Intheliteraturetherearemanyworksonteleparallelgravitymodels[2]-[6]inwhichcon-straintsaregiven
Rab=0,Qab=0.(26)
Onetrivialsolutionto(26)isηab=( ,+,+,+)andΛab=0.Thentheorthonormalco-frame{ea}isleftoverastheonlydynamicalvariable.WecallsuchachoiceWeitzenb¨okgauge.ThisgaugecannotbeasolutiontoSTPGbecauseofequations(24)and(25)sincewhenwesetηab=( ,+,+,+)andΛab=0thisgiveriseidenticallytoea=dx a:theso-calledMinkowskigauge[8].
NowwegiveabriefoutlineofGR.GRiswrittenin(pseudo-)Riemannianspacetimeinwhichtorsionandnonmetricityarebothzero,i.e.,connectionisLevi-Civita.Einsteinequationcanbewritteninthefollowingform
Ga:= 1
2Rea=κ τa(28)
whereGaisEinsteintensor3-form,Rab(ω)isRiemanniancurvature2-form,(Ric)a= bRba(ω)isRiccicurvature1-form,R= a(Ric)aisscalarcurvature,τaisenergy-momentum3-formandκiscouplingconstant.
ForthesymmetricteleparallelequivalentofEinsteinequationwe rstdecomposenon-Riemanniancurvature2-form(7)via(11)asfollows,withKab=0
Rab(Λ)=Rab(ω)+D(ω)(qab+Qab)+(qac+Qac)∧(qcb+Qcb)(29)whereD(ω)isthecovariantexteriorderivativewiththeLevi-Civitaconnection.AftersettingRab(Λ)=0weobtainthesymmetricteleparallelequivalentof(27)
Ga:=1