Cosmological Relativity A General-Relativistic The

arXiv:astro-ph/0008337v122Aug2000COSMOLOGICALRELATIVITY:AGENERAL-RELATIVISTICTHEORYFORTHEACCELERATINGEXPANDINGUNIVERSE⋆MosheCarmeliandSilviaBeharDepartmentofPhysics,BenGurionUniversity,BeerSheva84105,Israel(E-mail:[e-mail]@bgumail.bgu.ac.il)ABSTRACTRecentobservationsofdistantsupernovaeimply,indefianceofexpectations,thattheuni-versegrowthisaccelerating,contrarytowhathasalwaysbeenassumedthattheexpansionisslowingdownduetogravity.Inthispaperageneral-relativisticcosmologicaltheorythatgivesadirectrelationshipbetweendistancesandredshiftsinanexpandinguniverseispre-sented.ThetheoryisactuallyageneralizationofHubble’slawtakinggravityintoaccountbymeansofEinstein’stheoryofgeneralrelativity.Thetheorypredictsthattheuniversecanhavethreephasesofexpansion,decelerating,constantandaccelerating,butitisshownthatatpresentthefirsttwocasesareexcluded,althoughinthepastithadexperiencedthem.Ourtheoryshowsthattheuniversenowisdefinitelyinastageofacceleratingexpansion,confirmingtherecentexperimentalresults.⋆PaperdedicatedtoProfessorSirHermannBondiontheoccasionofhis80thbirthday.Thegeneral-relativistictheoryofcosmologystartedin1922withtheremarkableworkofA.Friedmann[1,2],whosolvedtheEinsteingravitationalfieldequationsandfoundthattheyadmitnon-staticcosmologicalsolutionspresentinganexpandeinguniverse.Einstein,believingthattheuniverseshouldbestaticandunchangedforever,suggestedamodificationtohisgravitationalfieldequationsbyaddingtothemtheso-calledcosmologicaltermwhichcanstoptheexpansion.SoonafterthatE.Hubble[3,4]foundexperimentallythatthefar-awaygalaxiesarerecedingfromus,andthatthefartherthegalaxythebiggeritsvelocity.Insimplewords,theuniverseisindeedexpandingaccordingtoasimplephysicallawthatgivestherelationshipbetweentherecedingvelocityandthedistance,v=H0R.(1)Equation(1)isusuallyreferredtoastheHubblelaw,andH0iscalledtheHubbleconstant.Itistacitlyassumedthatthevelocityisproportionaltotheactualmeasurementoftheredshiftzoftherecedingobjectsbyusingthenon-relativisticrelationz=v/c,wherecisthespeedoflightinvacuum.TheHubblelawdoesnotresemblestandarddynamicalphysicallawsthatarefamiliarinphysics.Rather,itisacosmologicalequationofstateofthekindonehasinthermodynamicsthatrelatesthepressure,volumeandtemperature,pV=RT[5].ItisthisHubble’sequationofstatethatwillbeextendedsoastoincludegravitybyuseofthefullEinsteintheoryofgeneralrelativity.Theobtainedresultswillbeverysimple,expressingdistancesintermsofredshifts;dependingonthevalueofΩ=ρ/ρcwewillhaveaccelerating,constantanddeceleratingexpansions,correspondingtoΩ1,Ω=1andΩ1,respectively.Butthelasttwocaseswillbeshowntobeexcludedonphysicalevidence,althoughtheuniversehaddeceleratingandconstantexpansionsbeforeitreacheditspresentacceleratingexpansionstage.Asiswellknown,thestandardtheorydoesnotdealwiththisproblem.Beforepresentingourtheory,andinordertofixthenotation,weverybrieflyreviewtheexistingtheory[6,7].Inthefour-dimensionalcurvedspace-timedescribingtheuniverse,our2spatialthree-dimensionalspaceisassumedtobeisotropicandhomogeneous.Co-movingcoordinates,inwhichg00=1andg0k=0,areemployed[8,9].Here,andthroughoutourpaper,low-caseLatinindicestakethevalues1,2,3whereasGreekindiceswilltakethevalues0,1,2,3,andthesignaturewillbe(+−−−).Thefour-dimensionalspace-timeissplitinto1L3parts,andtheline-elementissubsequentlywrittenasds2=dt2−dl2,wheredl2=(3)gkldxkdxl=−gkldxkdxl,andthe3×3tensor(3)gkl≡−gkldescribesthegeometryofthethree-dimensionalspaceatagiveninstantoftime.Intheaboveequationsthespeedoflightcwastakenasunity.Becauseoftheisotropyandhomogeneityofthethree-geometry,itfollowsthatthecur-vaturetensormusthavetheform(3)Rmnsk=Kh(3)gms(3)gnk−(3)gmk(3)gnsi,(2)whereKisaconstant,thecurvatureofthethree-dimensionalspace,whichisrelatedtotheRicciscalarby(3)R=−6K[10].Bysimplegeometricalargumentsonethenfindsthatdl2=1−r2/R2−1dr2+r2dθ2+sin2θdφ2,(3)whererR.Furthermore,thecurvaturetensorcorrespondingtothemetric(3)satisfiesEq.(2)withK=1/R2.Inthe“spherical”coordinates(t,r,θ,φ)wethushaveg11=−1−r2/R2−1.(4)Riscalledtheradiusofcurvatureoftheuniverse(ortheexpansionparameter)anditsvalueisdeterminedbytheEinsteinfieldequations.Onethenhasthreecases:(1)auniversewithpositivecurvatureforwhichK=1/R2;(2)auniversewithnegativecurvature,K=−1/R2;and(3)auniversewithzerocurvature,K=0.Theg11componentforthenegative-curvatureuniverseisgivenbyg11=−1+r2/R2−1.(5)Forthezero-curvatureuniverseoneletsR→∞.3Althoughgeneralrelativitytheoryassertsthatallcoordinatesystemsareequallyvalid,inthistheoryonehastochangevariablesinordertogetthe“right”solutionsoftheEinsteinfieldequationsaccordingtothetypeoftheuniverse.Accordingly,onemakesthesubstitutionr=Rsinχforthepositive-curvatureuniverse,andr=Rsinhχforthenegative-curvatureuniverse.Notonlythat,butthetime-likecoordinateisalsochangedintoanotheroneηbythetransformationdt=Rdη.Thecorrespondinglineelementsthenbecome:ds2=R2(η)hdη2−dχ2−sin2χdθ2+sin2θdφ2i(6a)forthepositive-curvatureuniverse,ds2=R2(η)hdη2−dχ2−sinh2χdθ2+sin2θdφ2i(6b)forthenegative-curvatureuniverse,andds2=R2(η)hdη2−dr2−r2dθ2+sin2θdφ2i(6c)fortheflatthree-dimensional