A Solution to Symmetric Teleparallel Gravity

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

ASolutiontoSymmetricTeleparallelGravity

Muza erADAK

DepartmentofPhysics,FacultyofArtsandSciences,

PamukkaleUniversity,

arXiv:gr-qc/0412007v1 2 Dec 200420100Denizli,Turkeymadak@pamukkale.edu.tr¨OzcanSERTDepartmentofPhysics,FacultyofArtsandSciences,PamukkaleUniversity,20100Denizli,Turkeyosert@pamukkale.edu.trFebruary7,2008AbstractTeleparallelgravitymodels,inwhichthecurvatureandthenonmetricityofspacetimearebothsetzero,arewidelystudiedintheliterature.Weworkadi erentteleparallelthe-ory,inwhichthecurvatureandthetorsionofspacetimearebothconstrainedtozero,butthenonmetricityisnonzero.Afterreformulatingthegeneralrelativityinthisspacetimewe ndasolutionandinvestigateitssingularitystructure.

1Introduction

Einstein’sgeneralrelativityprovidesanelegant(pseudo-)Riemannianformulationofgravitationintheabsenceofmatter.Inthevariationalapproach,Einstein’s eldequationsareobtainedbyconsideringvariationsoftheEinstein-HilbertactionwithrespecttothemetricanditsassociatedLevi-Civitaconnectionofthespacetime.Thatis,theabsenceofmattermeansthattheconnectionismetriccompatibleandtorsionfree,asituationwhichisnaturalbutnotalwaysconvenient.AnumberofdevelopmentsinphysicsinrecentyearssuggestthepossibilitythatthetreatmentofspacetimemightinvolvemorethanaRiemannianstructureTheoriesofgravitybasedonthegeometryofdistantparallelismarecommonlycon-sideredastheclosestalternativetothegeneralrelativity(GR)theory.Teleparallelgravitymodelspossessanumberofattractivefeaturesbothfromthegeometricalandphysicalview-points.TeleparallelismnaturallyariseswithintheframeworkofthegaugetheoryofthegroupofgeneralcoordinatetransformationswhichunderliesGR.Accordingly,theenergy-momentumcurrentrepresentsthemattersourceinthe eldequationsoftheteleparallelgravity.

Sincegaugetheoriesseemimportantforthedescriptionoffundamentalinteractionsitap-pearsnaturaltoexploitanygaugestructurepresentintheoriesofgravity.Di erentauthors,

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

however,adoptdi erentcriteriainordertodeterminewhatpropertiesatheoryshouldpossessinorderforittoqualifyasagaugetheory.WetakethegravitationalgaugegrouptobethelocalLorentzgroup[7].

Inthispaperwewillstudyagravitymodelinaspacetimewhosecurvatureandtorsionarebothzero,butthenonmetricityisnonzero.Thereisafewworkintheliteratureaboutgravitymodelsinthiskindofspacetimes;theso-calledsymmetricteleparallelgravity[8].

2Mathematicalpreliminaries

Spacetimeisdenotedbythetriple{M,g, }whereMisa4-dimensionaldi erentiablemani-fold,equippedwithaLorentzianmetricgwhichisasecondrank,covariant,symmetric,non-degeneratetensorand isalinearconnectionwhichde nesparalleltransportofvectors(ormoregenerallytensorsandspinors).Withanorthonormalbasis{Xa},

g=ηabea eb,a,b,···=0,1,2,3(1)

whereηab=( ,+,+,+)istheMinkowskimetricand{ea}istheorthonormalco-frame.Thelocalorthonormalframe{Xa}isdualtotheco-frame{ea};

beb(Xa)=δa.(2)

ThemanifoldMisorientedwiththevolume4-form

1=e0∧e1∧e2∧e3(3)

where denotestheHodgemapanditisconvenienttoemployinthefollowingthegradedinterioroperator Xa≡ a:

b aeb=δa.(4)

Inaddition,theconnection isspeci edbyasetofconnection1-formsΛab.Inthegaugeapproachtogravityηab,ea,Λabareinterpretedasthegeneralizedgaugepotentials,whilethecorresponding eldstrengths;thenonmetricity1-forms,torsion2-formsandcurvature2-formsarede nedthroughtheCartanstructureequations

2Qab:= Dηab=Λab+Λba,

Ta:=Dea=dea+Λab∧eb,

Rab:=DΛab:=dΛab+Λac∧Λcb(5)(6)(7)

wheredandDdenotetheexteriorderivativeandthecovariantexteriorderivative,respectively.These eldstrengthssatisfytheBianchiidentities1

DQab=1

1SinceQab=1

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)

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

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)

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

Thenfrom(37)and(38)

G(r)=1/F(r)(41)

andfrom(39)and(40)

F2(r)=1 C

e10,R02(ω)=F′

rFG2e30,

R12(ω)(G 1F)′G=rGe31,R23(ω)=1

G2)e32.(43)

ThusthequadraticinvariantoftheRiemanniancurvaturereads

R)∧ Rab(ω)= 2 (F′G 1)′ 2 (G 1)′

ab(ωrFG2+4r2 1 1

r61(44)

andthespacetimegeometryisnaturallycharacterizedbythequadraticinvariantofthenon-metricity

Qab∧ Qab= F′

r 1 1

4r3(r C) 3C

r2 1 1 C

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

r=C.Sincewearedealingwithsymmetricteleparallelgravity,itisnecessaryalsotoanalyzethebehaviorofnonmetricity.Asseenfrom(45),thenonmetricityinvariantdivergesnotonlyattheoriginr=0,butalsoattheSchwarzschildhorizonr=C.ThehorizonisaregularsurfacefromtheviewpointoftheRiemanniangeometry,butitissingularfromtheviewpointofsymmetricteleparallelgravity.WeintendtoclarifythegeometricalandphysicalmeaningofthesingularitiesinSTPGbyinvestigatingmattercouplingtoSTPGinaseparatepaper.Acknowledgement

ThisworkissupportedbytheScienti cResearchProject(BAP)2002FEF007,PamukkaleUniversity,Denizli,Turkey.

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