Time at the origin of the Universe fluctuations between two possibilities
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arXiv:gr-qc/0206058v1 20 Jun 2002TIMEATTHEORIGINOFTHEUNIVERSE:FLUCTUATIONSBETWEENTWOPOSSIBILITIES
V.DZHUNUSHALIEV1AffiliationKyrgyz-RussianSlavicUniversity,KievskayaStr,44,720000,Bishkek,Kyrgyzstane-mail:dzhun@hotmail.kg
Abstract.AvariationofHawking’sideaaboutEuclideanoriginofanon-singularbirthoftheUniverseisconsidered.Itisassumedthatneartozeromomentt=0fluctuationsofametricsignaturearepossible.
1.IntroductionThetimeinthemodernLorentz-invariantphysicsisconnectedwithametricsignature.Thespacetimemetricforanyspacecanbewrittenas
ds2=gµνdxµdxν=e¯ae¯bη¯a¯b=h¯aµdxµh¯bνdxνη¯a¯b(1)wheregµν=h¯aµhbνη¯a¯bisthemetric,e¯a=h¯aµdxµis1-form,h¯aµisvier-bein,aisvier-beinindex,µisthecoordinateindexandη¯a¯bisthemetricsignatureη¯a¯b=diag{σ,1,1,1}.Anundefinednumberσcanbe+1fortheEuclideanspaceand−1fortheLorentzianspacetime.WeseethatthedifferencebetweenEuclideanandLorentzianspacetimesisconnectedwiththesignofσ=η¯0¯0=±1.Forη¯0¯0=+1wesaythatthereistheEuclideanspaceandforη¯0¯0=−1theLorentzianspacetime.Iteasytoseethatthemetricgµνhastwodifferentdegreesoffree-dom:vier-beinh¯aµandthemetricsignatureη¯a¯b.h¯aµcanbedeterminedfromEinstein’sequationsbutforthemetricsignatureη¯a¯bwehavenotanydynamicalequations.Weputinthemetricsignatureη¯a¯bbyhandintoEin-stein’sequations(inthevier-beinformalism).Ofcoarse,wecandeterminethetruevalueofη¯a¯bfromexperiments:inourUniverseη¯a¯b=(−1,1,1,1).Butonthequantumlevel(onthePlancklevel)wecanassumethatη¯a¯bis2Euclidean regionLorentzian regionnot any singularityt = 0Figure1.Hawking’snonsingularUniversewiththeEuclideanregionattheorigin.Lorentzian regionnot any singularity region withfluctuating signature
t = 0
Figure2.AttheoriginofUniversethereisaregionwithfluctuationsofthemetricsignature.
fluctuatingquantitydiag(−1,1,1,1)⇆diag(+1,1,1,1).(2)CosmologicalsolutionsofEinstein’sequationswithordinarymatterhavealmostwithoutexceptionacosmologicalsingularity.Theexistenceofsuchsingularityisoneofthemostsignificantprobleminthemodernphysics.Therearevariousapproachestothesolutionofthisproblem.Hawking’spointofview[1]isthatattheoriginoftimeanEuclideanspaceemergedfromNothingandthemetricsignatureneartozeromomentt=0hastheEuclideanvalueηab=diag(+1,1,1,1)butaftershorttimeinterval(≈tPl)thesignatureundergoesaquantumjumptotheLorentzianvalueηab→diag(−1,1,1,1)Anotherwords:inordertokillsingularitywemustkillthetime.ThuswehavethefollowingpictureforHawking’snonsingularUniverse(see,Fig.1).TheideapresentedhereisthatneartozeromomenttherearequantumfluctuationsbetweenEuclideanandLorentziansigna-turesandaftershorttime(≈tPl)thefluctuationsceaseandthemetriccomestothestatewiththedefinite(Lorentzian)metricsignature(see,Fig.2).Wecanassumethatinquantumgravitycanbevarioussortofquantumfluctuations1.Thefluctuationsofthemetric.2.Thetopologyfluctuations.Thisphenomenonisknownasahypothe-sizedspacetimefoam.3.Thefluctuationsofthemetricsignature.4....HereIwillconsideronlythethirdkindofquantumgravitationalfluctu-ations.Onecansaythattwodifferentapproachestotheproblemofthemetricsignaturefluctuationsarepossible:1.OnecansolvetheEinstein’sequationswithundefinedσ=±1andthenwewillhavefluctuatingquantityσinthesolution.3fixed algorithmalgorithm with quantum fluctuationsσ = + 1
σ = −1quantum fluctuations
Figure3.Afluctuatingalgorithm.
2.AnotherapproachisthateveryEinstein’sequation(forexample,R00−14l(P)islengthoftheprogramP;A(P,y)isthealgorithmforcalculatinganobjectx,usingtheprogramP,whentheobjectyisgiven.
3.The5DFluctuatingUniverseHereIwouldliketoconsiderthescenariowhereattheoriginoftheUniversefluctuationsbetweenEuclideanandLorentzianmetricsoccur[2][3].Westartwithavacuum5DUniversewiththemetric
ds2(5)=σdt2+b(t)(dξ+cosθdϕ)2+a(r)dΩ22+r20e2ψ(t)[dχ−ω(t)(dξ+cosθdϕ)]2(4)
hereσ=±1fortheEuclideanandLorentziansignaturesrespectively.Ac-cordingtotheappropriatetheoremthemultidimensionalmetricinEq.(4)hasthefollowingelectromagneticpotential
A=ω(t)(dξ+cosθdϕ)=ωbe¯1(5)whichyieldsanelectricalfieldE¯1andamagneticfieldH¯1likeE¯1=F¯0¯1=˙ωb(6)H¯1=1a(7)The5D,vacuumEinsteinequationsresultingfromEq.(4)areG¯0¯0∝2˙b˙ψa+2˙a˙ba2+σba+r20e2ψσH2¯1−E2¯1=0,(8)G¯1¯1∝4¨ψ+4˙ψ2+4¨aa+σ3ba−˙a2
b+2˙b˙ψb2+2¨a
a+˙a˙ba2−σb
a+˙b˙ψ2e2ψσH2¯1+E2¯1=0,(11)R¯2¯5∝¨ω+˙ω˙a2b+3˙ψ−σb