Casimir energy density in closed hyperbolic universes

arXiv:gr-qc/0205050v2 5 Jun 2002Casimirenergydensityinclosedhyperbolic

universes∗

DanielM¨uller

InstitutodeF´isica,UniversidadedeBras´ilia

Bras´ilia,DF,Brazil

E-mail:muller@fis.unb.br

and

HelioV.Fagundes

InstitutodeF´isicaTe´orica,UniversidadeEstadualPaulista

S˜aoPaulo,SP,Brazil

E-mail:helio@ift.unesp.br

February7,2008

Abstract

TheoriginalCasimireffectresultsfromthedifferenceinthevac-uumenergiesoftheelectromagneticfield,betweenthatinaregionofspacewithboundaryconditionsandthatinthesameregionwithoutboundaryconditions.Inthispaperwedevelopthetheoryofasimilarsituation,involvingascalarfieldinspacetimeswithnegativespatialcurvature.

1INTRODUCTION

Inapreviouswork[1]theCasimirenergydensitywasobtainedforaRobertson-

Walker(RW)cosmologicalmodelwithconstant,negativespatialcurvature.ItsspatialsectionwasWeeksmanifold,whichisthehyperbolic3-manifold

withthesmallestvolume(normalizedtoK=−1curvature)intheSnapPea

census[2].Herewefurtherdevelopandclarifythetheoreticalformalismofthat

paper.

OursignconventionsforgeneralrelativityarethoseofBirrellandDavies

[3]:metricsignature(+−−−),RiemanntensorRαβγδ=∂δΓαβγ−...,Ricci

tensorRµν=Rαµαν.

2THEORIGINALCASIMIREFFECT

TheoriginaleffectwascalculatedbyCasimir[4].Briefly,onesetstwometal-

lic,unchargedparallelplates,separatedbyasmalldistancea.Betweenthem

theelectromagneticfieldwavenumbersnormaltotheplatesareconstrained

bytheboundaries.SothereisadifferenceδEbetweenthevacuumenergy

forthisconfigurationandthevacuumenergyforunboundedspace.IfAis

theareaofeachplate,onehas(see,forexample,[5],[6],[7])

δE

2󰀇󰀇dkxdkyk2x+k2y+(πn/a)2−2a󰀇dkzk2x+k2y+k2z󰀄

,

whereweomitteddampingfactorsneededtoavoidinfinities.Theresultsis

δE(a)=−π2󰀂c

240a4A

fortheattractiveforcebetweentheplates.

3CASIMIRENERGY(CE)INCOSMOL-

OGYWITHNONTRIVIALTOPOLOGY

Thereisnoboundaryforauniversemodelwithclosed(i.e.,compactand

boundless)spatialsections.Butafieldinthesemodelshasperiodicities,

2whichleadstoaneffectsimilartotheaboveone,thatmayalsobecalleda

Casimireffect.

Asimpleexample,takenfromBirrellandDavis[3],isthatofascalar

fieldφ(t,x)inspacetimeR1×S1,withoneclosedspacedirection.IfS1has

lengthLthen

φ(t,x+L)=φ(t,x),

andthevacuumenergydensityis

ρ=−π󰀂c/6L2.

AnanalyticalexpressionfortheCEinaclassofclosedhyperbolicuni-

verses(CHUs)wasobtainedbyGoncharovandBytsenko[8].

Herewedevelopaformalismsuccintlydescribedin[1],forthenumerical

calculationoftheCEdensityofclosedhyperbolicuniverses.

Ournotation:i,j,...=1−3;α,µ=0−3;x=(xi);x=(xµ)=(t,x).

Signconventionsarethoseof[3]:metricsignature(+−−−),Riemann

tensorRαβγδ=∂δΓαβγ−...,Riccitensor:Rµν=Rαµαν.

4SCALARFIELDφ(x)INCURVEDSPACE-

TIME

Theactionforascalarfieldinacurvedspacetimeofmetricgµνandmassm

is

S=󰀇

L(x)d4x,

with

L=1−g󰀂gµνφ;µφ;ν−(m2+ξR)φ2󰀅,

whereRisscalarcurvatureofspacetime,g=det(gµν),andξisaconstant.

3Withξ=1/6(“conformal”value)wegettheequationforφ(x):

δS

6R)φ=0,

where󰀃isthegeneralizedd’Alembertian:

󰀃φ=gµν▽µ▽νφ=(−g)−1/2∂µ󰀂(−g)1/2gµν∂νφ󰀅.

Theenergy-momentumtensoris(cf.[3])

Tµν=2(−g)−1/2δS/δgµν

=2

6gµνφ;σφ;σ−1

12gµνφ󰀃φ

−1

24gµνRφ2+16STATICMODELSOFNEGATIVESPA-

TIALCURVATURE

TheRobertson-WalkermetricforspatialcurvatureK=−1/a2is

ds2=dt2−a2(dχ2+sinh2χdΩ2)

=dt2−a2󰀊

δij−xixj

a󰀌2=1

3ρ+Λ

a=−4πG(ρ+3P)+Λ.

Assuming˙a=¨a=0andP=ρ/3wegeta2=−3/2Λ,henceΛ<0,and

a=󰀆Eachcellγ(FP)isacopyofFP,hencewehaveperiodicityoffunctions

onaCHM,andthepossibilityofacosmologicalCasimireffect.

Face-pairinggeneratorsγk,k=1−n,satisfy

FP∩γk(FP)=facekofFP.

Withthesegeneratorstherelationsalsohaveacleargeometricalmeaning:

theycorrespondtothecyclesofcellsaroundtheedgesofFP.

ThesoftwareSnapPea[2]includesa“census”ofabout11,000orientable

CHMs,withnormalizedvolumesfrom0.94270736to6.45352885.Foreachof

thesetheFPcenteredonaspecialbasepointOisgiven,aswellastheface-

pairinggeneratorsinboththeSL(2,C)andtheSO(1,3)representations.

Analgorithm[12]tofindasetofcellsγ(FP)thatcompletelycovera

ballofradiusrreducesthisproblemtooneoffindingallmotionsγ∈Γ,such

that

distance[O,γ(O)]

ForastudyofCHMsfromacosmologicalviewpoint,seeforexample[9]

andreferencestherein.Fornumericaldataonacoupleofthem,see[10],[11].

8CLOSEDHYPERBOLICUNIVERSES

WeareconsideringstaticCHUs.AsobtainedinSec.6,themetricis

ds2=dt2−3

1+x2󰀌

dxidxj.

Thespacetimeshavenontrivialtopology:

M4=R1×Σ,

whereR1isthetimeaxisandΣ=H3/ΓisaCHM.

Asfoundabove,thesemodelshavenegativeenergydensity,ρ=Λ/8πG,

whichhasnoobviousphysicalmeaning,andviolatestheenergycondition

Tµνuµuν≥0.Butwearedealingwiththeveryearlyuniverse,whereone

feelsfreertospeculate.AndarecentpaperbyOlum[13]castsdoubtonthe

universalityofthiscondition.

6