Gravitional coupling constant in higher dimensions

Assuming the equivalence of FRW-cosmological models and their Newtonian counterparts, we propose using the Gauss law in arbitrary dimension a general relation between the Newtonian gravitational constant G and the gravitational coupling constant \kappa.

Mon.Not.R.Astron.Soc.000,000–000(0000)Printed7February2008A(MNLTEXstyle lev1.4)

GravitationalcouplingconstantinhigherdimensionsRezaMansouri1,2 andAliNayeri3

forStudiesinTheoreticalPhysicsandMathematics,P.O.Box5746,19395Tehran,Iran.

ofPhysics,SharifUniversityofTechnology,P.O.Box11365-9161,Tehran,Iran.

3Inter-UniversityCentreforAstronomyandAstrophysics,PostBag4,Ganeshkhind,Pune411007,India2Department1Institute

ABSTRACT

arXiv:gr-qc/9609061v1 26 Sep 1996AssumingtheequivalenceofFRW-cosmologicalmodelsandtheirNewtoniancoun-terparts,weproposeusingtheGausslawinarbitrarydimensionageneralrelationbetweentheNewtoniangravitationalconstantGandthegravitationalcouplingcon-stantκ.Keywords:cosmology:theory.1INTRODUCTIONItisnowacommonbeliefamongcosmologistsandrela-tiviststhatalthoughspacetimeappearssmooth,nearly at,andfourdimensionalonlargescales,atsu cientlysmalldistancesandearlytimes,itishighlycurvedwithallpossi-bletopologiesandofarbitrarydimensions.TheinitialideaofKaluza-Kleinhasbeenextensivelyusedinuni edtheo-riesofinteractions;e.g.Green,Shwarz,&Wit-ten(1987).Theretheextradimensionsareassumedtobecompacti edtoPlanckiansize,andthereforedonotdisplaythemselvesinmacroscopicprocesses.Themultidimensionalcosmologiesbasedontheseideasbeenextensivelystud-iedinthelastyears[seee.g.Cho(1990),Kirillov&MelnikovWessonUsuallyincosmologicalmodelsbasedonhigherdimensionstheproblemofthedimensionalityofthegravitationalcou-plingconstantisnottackledon,beingtacitlyassumedtobe

κ=8πG,(1)

whereGistheNewtoniangravitationalconstant.Thisisof-

coursearelationbeingderivedinfourdimension.Evenin

sometextbooksthishasnotbeendi erentiated[seee.g.Kolb

&TurnerThee ectivegravitationalconstantinthe

mutidimensionalcosmologiesisde nedthroughthemulti-

plicationofthisfour-dimensionalvaluewithsomevolume

oftheinternalspacewhichcouldevenbein nite[Rainer&

ZhukTherearehowevercosmologicalmodelsinhigherdimen-

sionwherethispicturedonotworkandoneshouldbecu-

siousaboutthevalueofthecouplingconstantindimen-

sionshigherthanfour[seee.g.ChatterjeeChaterjee

&Bhui(1990),Khorrami,et.al(1995),Khorrami,Mansouri

email:mansouri@netware2.ipm.ac.ir email:ali@iucaa.ernet.in

c0000RAS &MohazzabInfact,Chatterjee(1992),Chaterjee&Bhui(1990)calculateahomogeneousmodelinahigherdimensionalgravityusingthesame4-dimensionalrelationasin(1).Weproposetoshowsomede cienciesofthismisuseandpro-poseageneralizationforthegravitationalcouplingconstantforanyarbitrarydimension.Anyde nitionofacouplingconstantinhigherdimensionwillin uencethetimedepen-denceofGandtherforecouldhavedirectobservationalcon-sequences;e.g.BarrowDegl’Innocenti,et.alRamero&Melnikov(1995).Inthisnotewecon neourselvestoameregeneralizationoftheκ,usingdiscrepanciesincomparingrelativisticandNewtoniancosmologiesinhigherdimensions.ItbelongstothefolkloreofthetheoriesofgravitationthattheirweaklimitmustbetheNewtoniantheoryofGravitation.More-over,onaccountoftherelation(1),thereisaNewtonianderivationoftheFRWcosmologies.Wewillshowthatthisderivation,usingthecouplingconstant(1)isjustvalidin(3+1)-dimension,andhastobechangedforhigherdiemen-sionaltheories.OnaccountoftheGausstheoremwegiveageneralizationofit,whichmakestheNewtonianderivationvalidinarbitrarydimensions.2FRIEDMANNMODELSANDNEWTONIANCOSMOLOGYINDDIMENSIONConsiderthefollowingHilbert-EinsteinactioninD+1di-mensionalspace-timewithDasthe xeddimensionofspace:Sg= 1 gdD+1x+1 gdD+1x,(2)whereκisthegravitationalconstant,again.Forsimplic-ity,weconsiderthek=0FRWmodel.Inanarbitrary xdimensionweontainthefollowingFriedmannequation,(Khorrami,et.al.1995)

Assuming the equivalence of FRW-cosmological models and their Newtonian counterparts, we propose using the Gauss law in arbitrary dimension a general relation between the Newtonian gravitational constant G and the gravitational coupling constant \kappa.

2RezaMansouriandAliNayeri(a˙

D(D 1)ρ,(3)whichleadstothefamiliarFriedmannequationinD=3:(a˙

3ρ=8πG

=8πG

a)2

(D/2 1)!ρ.(8)OntheotherhandwegetforarbitraryD

RD 200=(

(D 2)(D/2 1)!.(10)ThecorrectformofFriedmannequationinanyarbitrary xeddimensionisnowderivedtobe

(a˙

D(D 2)(D/2 1)!ρ.(11)Asitiseasilyseen,theaboverelationisincompleteagree-mentwithitsNewtoniancounterpartinalldimensions.4CONCLUSION

Thedimensionaldependenceofthegravitationalconstantκmayhaveverydi erentandserious eldtheoreticandastrophysicalconsequenceshithertounnoticed.Itwouldbeinteresting,e.g.,toseewhichchangesaretobeexpectediftheresultsofthisnotearecombinedwithKaluza-Kleinparadigm.TheconsequencesontimevariationofGwithinvariousmodelsisanotherveryinterestingcosmologicalissuewhichisunderinvestigation.ACKNOWLEDGEMENTA.N.wishestothankProf.Padmanabhanformanyusefuldiscussions.A.N.was nanciallysupportedbytheCouncilofScienti candIndusterialResearch,India.REFERENCESBarrow,J.D.1978,MNRAS,184,677.Chatterjee,S.,1992.Ann.Phys.,218,121.Chatterjee,S.&Bhui,B.,1990.Mon.Not.R.astr.Soc.,247,57.Cho,Y.M.,1990.Phys.Rev.D.,41,2462.Degl’Innocenti,S.,Fiorentini,G.,Ra elt,G.G.,Ricci,B.&Weiss,A.,1996.Astron.Astrophys.,312,345.Green,M.B.,Schwarz,J.H,&Witten,E.,1987.SuperstringThe-ory,CambridgeUniversityPress,Cambridge.Kirillov,A.A.,&Melnikov,V.N.,1995.Phys.Rev.D.,52,723.Kolb,E.W.,&Turner,M.S.,1990.Theearlyuniverse,Addison-Wesleypublishingcompany,NewYork.

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