Cosmological implications of the new metric for an

COSMOLOGICALIMPLICATIONSOFTHENEWMETRICFORANACCELERATINGEXPANDING,ANDDOUBLYROTATINGUNIVERSEEvangelosChaliasos365ThebesStreetGR-12241AegaleoAthens,GreeceAbstractWegivethecorrectinterpretationofthenewmetric,foundlatelybytheauthor.ThismetricresultsasanexactsolutionoftheEinsteinfieldequations,withoutthecosmologicalconstant.Thenewfeatureistheintroduction,fromthebeginning,oftworotationsattributedtotheUniverse.Asaresultalsoanexpanding,andindeedinacceleratingrate,Universeisobtained.Thus“darkenergy”isnotnecessaryatall.WeexploresomepropertiesofthisUniverseinthepresentpaper.Inparticular,alloldpropertiesoftheUniverseconcerningtheHubblelawremaininamodifiedway,implyinganisotropy.Besides,weexplorethevalidityoftheRyle-Clarkeeffect,whichisverified,andwefindthecorrectageoftheUniverse.Finally,theresultingshapeoftheUniverseisexamined,anditisfoundtobehypertoroidal.1.IntroductionInourpreviouspaper[1]weconsideredourthree-dimensionalspaceembeddedinafictitiousfour-dimensionalspace,whichweallowedtoperformtwoindependentrotations,eachonacoordinateplane,thesetwoplaneshavingincommononlytheorigin.Thenwewrotedowntheappropriateforthiscaseformofthemetricinthefour-dimensionalspace-time.Afteralotofintermediatecalculations,wewrotedowntheresultingEinsteinfieldequations,whichwesolvedanalyticallyafteralotofefforttoo.Wethusfoundthatthemetric,incomovingcoordinates,isdescribingadifferentialrotationofthespacewithangularvelocityplusarigidrotationofthespacewithangularvelocityinanexpandingUniversewithscalefactorwhereΘ,Ε,ξ,ηandGareconstantscomingfromintegrations.Weseeatoncefrom(1.4)thattheexpansionisaccelerated.ThecoordinatesysteminwhichtheabovesolutionofEinstein´sfieldequationsisdescribedisnot“inertial”.Wecanthoughtakefromitan“inertial”coordinatesystemperformingthecoordinatetransformation2222222221122(){()()},(1.1)dscdtUtdrddtrddtφωφω=-++++313(),(1.2)GtcteEωξ-=323(),(1.3)GtcteEωη-=(),(1.4)GtUte=ΘΕ31313223.(1.5)33GtGtttrrceGEceGEξϕφφφηφ--⎧⎫⎪⎪⎧⎫⎪⎪⎪⎪⎪⎪⎪⎪→⎨⎬⎨⎬=+⎪⎪⎪⎪⎪⎪⎪⎪⎩⎭Φ=+⎪⎪⎩⎭Describedinthenewcoordinatesystem,themetricbecomesThismetricresemblestheflatFriedmannmodel(spaceflat–spacetimecurved),butofcoursethisisnotthatmodel(lookateq.1.4).Notethatthe“inertial”coordinatesystemissynchronous,andthusthepropertimeτ=t.Thisisnottrueforthecomovingcoordinatesystem,aswewillseeinsection2.InthecomovingcoordinatesystemwehaveforamaterialpointofcourseThusfromthemetric(1.6)wegetthatinthe“inertial”coordinatesystemthematerialpointperformstworotations,withangularvelocitiesthefirstbeingadifferentialrotationandthesecondbeingarigidrotation,or,aswemaysay,theparticle(ortheUniverse,sinceitconsistsofallparticles)isdoublyrotating.Inref.[1],wehavefoundalsothat,fortheenergydensity,ε=constant×G.ThatiswehadobtainedasteadystatefortheUniverse.Then,wehadcommentedasfollows:since1)fromtheRyle-Clarkeeffectasteadystatehasbeenexcluded(forε≠0ofcourse),and2)fromthetime-symmetrictheoryoftheauthor(forabriefaccountofitseee-AppendixBinref.[1])wehavetohaveε=0,wewereobligedtotakeG=0,withtheresulttoobtaininadditionaconstantgravitationalfield,thatisnoexpansionatall,andbesides,becauseoftheformofthemetric(1.6),merelyaflat(Minkowskian)space-time(wehadattributedthegalacticrecession,andespeciallyinacceleratingrate,tootherreasons,asitcanbeseeninref.[1]afterp.55).3120.(1.7)dddtdtφφ==12&,(1.8)dddtdtϕωωΦ=={}22222222().(1.6)dscdtUtdrdrdϕ=-++ΦInref.[1]thecoordinaterwasnamedz.Andsinceitwasacomovingcoordinate,amaterialparticlehadtomoveaccelerating,inthe“inertial”coordinatesystem,onaz=constantplane.Itsvelocitywasproportionaltoitsdistance(there)rfromthez-axis,andithadtogotoamaximumdistancer=z,whereitsvelocitybecameequaltoc.Itcouldofcoursestartfromr=0,thatisfromthez-axis.Andsinceitwasvalidforallz,thebig-banghadtobethewholez-axis.Butthebig-bangisverywellestablisheduptonowtobeapointratherthananaxis.Thuswehavetoabandonthatpicture,andacceptG≠0,inordertorecovertheexpansion(fromapointofcourse,theorigin).Butthenbesidesthedesiredexpansion(andindeedaccelerating),wehavetonowacceptε=constant≠0,thatisasteadystate.Wewillseethattheauthor´stimesymmetrictheorymentionedbeforeeventuallyisnotviolatedinsection6.ConcerningtheRyle-Clarkeeffectwewilltrytoexploreitsvalidity,andwesurprisinglyseethat,becauseofthenewgeometry,itisalsofulfilled,afteraderivationinsection3.Insections4&5wewillseethatHubble´slawcontinuestohold,ifweneglecttheangularvelocitiesω1&ω2.And,insection6,wewillseethatifwetakethemintoaccount,ananisotropyintheHubbleflowresults,eventothefirstorderbothinω12&ω22,andinthedistancefromthesource,holdingthusforsmalldistancesandsmallω12&ω22,too.Andafewwordsontheinterpretationofthenewmetric,inparticularintheform(1.6).Atfirst,weseethatitdescribesarigidrotationabouttheoriginonthe(r,Φ)planewithangularvelocityω2.Thenthewhole(r,Φ)planerotatesdifferentiallywithangularvelocityω1aboutanaxis,sayZ,perpendiculartotheplaneofφ´s,whichaxiswecantakeasthecommonoriginalaxisformeasuringΦ´sonthe(r,Φ)planes(Fig.1)4Alternatively,wecanrename(becauseoftheabsenceofcoefficientindφ2)φtoz,sothatitisrecognizedthatthemetricinsidethecurlybracketsin(1.6)isjustthelineelement,