A Solution to Symmetric Teleparallel Gravity(4)

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

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