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
equationseasily,theinvariantdescriptionprovidesthecorrectunderstandingofthephysicalcontentsofasolution.
Sincemetricandconnectionareindependentquantitiesinnon-Riemannianspacetimes,wehavetopredictseparatelyappropriatecandidatesforthem.Thereforewe rstwritealineelementinordertodeterminethemetric.Wenaturallystartdealingwiththecaseofsphericalsymmetryforrealisticsimplicity,
g= F2dt2+G2dr2+r2dθ2+r2sin2θd 2
whereF=F(r)andG=G(r).Aconvenientchoiceforatetradreads
e0=Fdt,
1
1
Gr)e,1(31)(32)cotθe1=Gdr,e2=rdθ,e3=rsinθd .Inaddition,forthenon-RiemannianconnectionwechooseΛ12= Λ21=
FG1e,1e3,1Λ23= Λ32= G)e1,Λ11=
)e1,Λ22=others=0.G
Thesegaugecon gurations(32)and(33)satisfytheconstraintequationsRab(Λ)=0,0.OnecancertainlyperformalocallyLorentztransformationΛ33=
ea→Labeb,Λab→LacΛcdL 1db(33)Ta(Λ)=(34)+LacdL 1cb
whichyieldstheMinkowskigaugeΛab=0.Thismaymeanthatweproposeasetofconnectioncomponentsinaspecialframeandcoordinatewhichseemscontrarytothespiritofrelativitytheory.Howeverinphysicallynaturalsituationswecanchooseareferenceandcoordinatesystematourbestconvenience.
Wededucefromequations(32)-(33)
ω01=
Q00=FF′′rG
r1(1 e2,1ω13= 1(1 1re3r(1 1
(35)
whosecomponentsreadexplicitly
Zerothcomponent
Firstcomponent
Secondcomponentothers=0.FGGGWhenweput(35)into(30)weobtain,withτa=0 bc bfcbfcdq+2ωf∧q+qf∧q∧(ea∧eb∧ec)=0 2(G 1)′ r2G22F′r2G2rFG2
F′+(G 1)′ e1∧e2∧e3=0e0∧e2∧e3=0e0,q12= 1)e2,q13=1r 1)e3,(36)(37)(38) (F′G 1)′
+FGrGe0∧e1∧e2=0.(40)