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.