Phantom stars and topology change

19 Pages·2013·0.85 MB·English

Phantom starsand topology change
Andrew DeBenedictis(cid:3)
Paci(cid:12)cInstitute forthe MathematicalSciences,Simon FraserUniversity Site and
DepartmentofPhysics,Simon FraserUniversity Burnaby,British Columbia,V5A IS6,Canada
Remo Garattiniy
Universita(cid:18) degliStudidiBergamo,Facolt(cid:18)a diIngegneria,
Viale Marconi5,24044 Dalmine (Bergamo) ITALY and
INFN -sezione diMilano,Via Celoria 16,Milan,Italy.
Francisco S.N.Loboz
InstituteofCosmology & Gravitation,University ofPortsmouth,Portsmouth PO1 2EG,UK and
Centro de Astronomia e Astrof(cid:19)(cid:16)sica da Universidade de Lisboa,
8
Campo Grande,Ed. C8 1749-016 Lisboa,Portugal
0
(Dated:April21,2013)
0
2 In thiswork,weconsidertime-dependentdark energy starmodels,with an evolvingparameter
v ! crossing thephantom divide,! = (cid:0)1. Oncein thephantom regime,thenullenergy condition
o isviolated,which physically impliesthatthenegativeradialpressureexceedstheenergy density.
N Therefore,anenormousnegativepressureinthecentermay,inprinciple,implyatopologychange,
consequentlyopeningupatunnelandconvertingthedarkenergystarintoawormhole.Thecriteria

6 forthistopology changearediscussed and,in particular,weconsideraCasimirenergy approach
involvingquasi-localenergy di(cid:11)erencecalculationsthatmay re(cid:13)ectormeasuretheoccurrenceof

] a topology change. W e denote these exotic geometries consisting ofdark energy stars(in the
c
phantom regime)and phantom wormholesasphantom stars.The(cid:12)nalproductofthistopological
q
change,namely,phantom wormholes,havefar-reachingphysicalandcosmologicalimplications,asin
-
r additiontobeingusedforinterstellarshortcuts,anabsurdlyadvancedcivilization maymanipulate
g
thesegeometriestoinduceclosedtimelikecurves,consequentlyviolatingcausality.
[

3 PACSnumbers:04.20.Jb,04.40.Dg,97.10.-q
v
9
3
8
0 I. INTRODUCTION
.
8
0 Recenthigh-precision observationaldata havecon(cid:12)rmed thattheUniverseisundergoing a phaseofaccelerated
8 expansion[1].Severalcandidates,responsibleforthisexpansion,havebeen proposedin theliterature,inparticular,
0 dark energy models(see Ref. [2]fora review)and modi(cid:12)ed gravity (e.g.,see Refs. [3]forrecentreviews). In
v: particular,theformermodelsarefundamentalcandidates,in which a simpleway to parameterizethedark energy
i isby an equation ofstateoftheform ! (cid:17) p=(cid:26),wherep isthespatially homogeneouspressureand (cid:26) isthedark
X
energy density. A valueof! < (cid:0) 1=3isrequired forcosmicexpansion,and ! = (cid:0) 1correspondsto acosmological
r constant.A speci(cid:12)cexoticform ofdarkenergydenoted phantom energy,with ! < (cid:0) 1,hasalsobeen proposed [4],
a
and possessespeculiarproperties,such astheviolation ofthenullenergy condition (NEC)and theenergy density
increasestoin(cid:12)nity in a (cid:12)nitetime[4],atwhich pointthesizeoftheUniverseblowsup in a (cid:12)nitetime,which is
knownastheBigRip.Inthiscontext,theviolationoftheNEC presentsuswithanaturalscenariofortheexistence
oftraversablewormholes,and indeed ithasbeen shown thattheseexoticgeometriescan besupported byphantom
energy[5,6].Itisalsointerestingtonotethatrecent(cid:12)tstosupernovae,CM B and weakgravitationallensingdata
probablyfavoranevolvingequationofstate,withtheparametercrossingthephantom divide! = (cid:0) 1[7].
Despitethefactthatthedark energy equation ofstaterepresentsa spatially homogeneouscosmic(cid:13)uid and is
assumednottocluster,itispossiblethatinhomogeneitiesmayariseduetogravitationalinstabilities.M oreprecisely,
although the equation ofstate leading to the acceleration ofthe Universeon largescalesisan averageequation
ofstate corresponding to a background (cid:13)uid,itispossible thatdark energy condensatesmay possibly originate
from density(cid:13)uctuationsin thecosmologicalbackground,resultingin thenucleation through therespectivedensity
(cid:3)Electronicaddress:[email protected]
yElectronicaddress:[email protected]
zElectronicaddress:[email protected]
2
perturbations. Despitethefactthatoncein thedark energy regimethematerialsystem becomesgravitationally
repulsive,wemay considerthepossibility oftheformation ofamattersystem thatoriginally obeysalltheenergy
conditions.Cosmologicalobservationsdonotruleout,and in somestudiesfavor,an evolvingequation ofstatefor
thedark energy. Itisthereforequitepossiblethatwhatweknow asdark energy today hasevolved from a more
benign(cid:13)uid.An over-densityofthis(cid:13)uidcouldinprinciplecommenceacollapseintoastarwhich.Suchamodelis
presentedinsectionIIIC.W ealsopointoutthateveninthecaseofadark-energy(cid:13)uid,thereisnode(cid:12)niteresolution
tothedebateofclusteringscales.Thisismainlyduetonon-linearity,especiallyintheveinofdarkenergyinteracting
withordinary(cid:13)uids.Itmayalsobepossibletogleansomeinformationonthecosmologicaldarkmatterbystudying
certainpropertiesofsuchgravitationalcondensates.(See[8]andreferencesthereinforcommentsontheseissues.)In
thiscontext,anumberofinhomogeneoussolutionshavebeentheobjectofanalysis,suchasthephantom wormholes
[5,6]mentionedabove,darkenergystars[9],andotherstructuressuchascondensatessupportedbythegeneralized
Chaplygin gas[10]which possiblyarisefrom density(cid:13)uctuationsin thegeneralizedChaplygin gasbackground,and
condensedstructuressupportedbythevanderW aalsequationofstate[11].Inarecentpaper[12],itwasalsoshown
thatthe4D Einstein-Klein-Gordonequationswithaphantom scalar(cid:12)eldpossessnon-singular,sphericallysymmetry
solutions,althoughastabilityanalysisonthesesolutionsindicatestheyareunstable.
Thedark energy starmodelsarealsoageneralization ofanew emergingpictureforan alternative(cid:12)nalstateof
gravitationalcollapse,namely,thegravastar(gravitationalvacuum star)models.ThelatterproposedbyM azurand
M ottola[13],hasane(cid:11)ectivephasetransitionat/nearwheretheeventhorizonisexpected toform,and theinterior
isreplacedbyadeSittercondensate.Thelatteristhenmatchedtoathicklayer,withanequationofstategivenby
p= (cid:26),which isin turn matched toan exteriorSchwarzschildsolution.Theissueofgravastarshasbeen extensively
analyzed in theliterature,and wereferthereaderto Refs.[14,15].Thegeneralization ofthegravastarpictureis
considered bymatchingan interiorsolution governed bythedarkenergyequation ofstate,! (cid:17) p=(cid:26) < (cid:0) 1=3,toan
exteriorSchwarzschild vacuum solution ata junction interface[9]. Thedynamicalstability ofthetransition layer
wasalso explored,and itwasfound thatlargestability regionsexistthataresu(cid:14)ciently closeto wheretheevent
horizon isexpected toform,sothatitwasargued thatitwould bedi(cid:14)culttodistinguish theexteriorgeometryof
thedarkenergystarsfrom anastrophysicalblackhole.Thus,thesealternativemodelsdonotpossessasingularityat
theoriginandhavenoeventhorizon,asitsrigidsurfaceislocatedataradiusslightlygreaterthantheSchwarzschild
radius.Thisrestrictionarisesfrom theobservedlackofenergyemissionduetosurfacecollisionsofinfallingmaterial
insuspectedblackholesystems.Infact,althoughevidencefortheexistenceofblackholesisveryconvincing,acertain
amountofscepticism regardingthephysicalrealityofeventhorizonsisstillencountered,andithasbeenarguedthat
despitethefactthatobservationaldatadoindeed providestrongargumentsin favorofeventhorizons,theycannot
fundamentallyprovetheirexistence[16].
Asmentioned above,recent(cid:12)tstoobservationaldataprobablyfavoran evolvingequationofstate,with thedark
energyparametercrossingthephantom divide! = (cid:0) 1[7].M otivated by thisfact,in aratherspeculativescenario
onemay theoretically considertheexistenceofadark energy star,with an evolvingparameterstartingoutin the
range(cid:0) 1 < ! < (cid:0) 1=3,and crossingthephantom divide,! = (cid:0) 1. Oncein thephantom regime,thenullenergy
conditionisviolated,whichphysicallyimpliesthatthenegativeradialpressureexceedstheenergydensity.Therefore,
an enormousnegativepressurein thecentermay,in principle,imply a topology change,consequently opening up
atunnel,and convertingthedark energy starintoawormhole[9,17].Onemay assumethatthetopology change
may occuratapproximately the Planck length scales,and once created may be self-sustained asshown in Ref.
[18]. In fact,thechangein topology isan extremely subtleissue,asin generalrelativity thesechangesprobably
entailspacetimesingularities.However,atthePlancklengthscalesquantum gravitye(cid:11)ectsdominateandspacetime
undergoesadeepandrapidtransformationinitsstructure,probablyproducingamultiply-connectedquantum foam
structure[19,20].ItwassuggestedinRef.[17]thatonecouldimagineanabsurdlyadvancedcivilization[21]pulling
awormholefrom thissubmicroscopicspacetimequantum foam andenlargingittomacroscopicdimensions.However,
inamoreplausiblescenario,thepossibilitythatin(cid:13)ationmightprovideanaturalmechanism fortheenlargementof
suchwormholestomacroscopicsizewasexplored[22].In thiswork,weoutlinethetheoreticaldi(cid:14)cultiesassociated
tothechangein topologyand presentamethod based on theCasimirenergyapproach.Although itisstillunsure
ifthismethodproducesatopologychange,itisextremelyusefulasthequasi-localenergydi(cid:11)erencecalculationmay
re(cid:13)ectormeasuretheoccurrenceofachangeintopology.Otherconceptsoftopologychangingspacetimeshavebeen
studied,forinstance:Usingsemi-classicaland M orse-indexmethodsin Refs.[23];higherorderback-reactionterms
dueto(cid:13)uctuationsofgauge(cid:12)eldsinthevicinityofablackholemayresultintheformationofawormhole-likeobject
[24];andmorerecentlyanapproachbasedonaRicci(cid:13)ow mayresultinquantum wormholes[25].
Oncethetopology changehasoccurred,with therespectiveopening ofa tunnel,then thedark energy starhas
been converted intowormholesupported byphantom energy.Asmentioned above,ithasrecentlybeen shown that
traversablewormholesmay,in principle,besupported by phantom energy [5,6],which apartfrom being used as
interstellarshortcuts,mayinduceclosedtimelikecurveswiththeassociatedcausalityviolations[26,27].Particularly
interestingsolutionswerefound[6],andbyusingthe\volumeintegralquanti(cid:12)er",itwasfoundthatthesewormhole
3
geometriesare,inprinciple,sustainedbyarbitrarilysmallamountsofaveragednullenergycondition(ANEC)violating
phantom energy.A complementaryapproachwastracedoutin[5],byconsideringspeci(cid:12)cchoicesforthedistribution
oftheenergydensitythreadingthewormhole.Recently,4D staticwormholesolutionssupported bytwointeracting
phantom (cid:12)eldswerefound aswell[28]. Despite the factthattraversablewormholesviolatethe NEC in general
relativity (seeRef. [29]fora recentreview),ithasbeen shown thatthe stressenergy tensorpro(cid:12)lemay satisfy
theenergy conditionsin thethroatneighborhood in dynamicwormholes(seeRef.[30]and referencestherein)and
in certain alternativetheoriesto generalrelativity [31]. Perhapsnotso appealing,onecould denotetheseexotic
geometriesconsistingofdarkenergystars(in thephantom regime)and phantom wormholesasphantom stars.W e
wouldliketostateouragnosticposition relativelytotheexistenceofdarkenergystarsandphantom wormholes,or
forthatmatterofphantom stars.However,itisimportanttounderstandtheirgeneralpropertiesandcharacteristics,
and weemphasizethatthepresenceofadarkenergy(cid:13)uid permeatingtheuniversemakesthestudyofdarkenergy
condensatesaphysicallyrelevantendeavor.
Thispaperisorganizedin thefollowingmanner:In sectionII,webrie(cid:13)yreview staticdarkenergystars,followed
byadeductionofgeneralsolutionsoftime-dependentspacetimes.InSectionIII,speci(cid:12)ctime-dependentdarkenergy
solutionsareoutlined,in particular,wepresentthespeci(cid:12)ccasesofaconstantenergydensity,theTolman-M atese-
W hitmanmassfunctionsolution,andaclassofmodelswithanon-zeroenergy(cid:13)uxterm,whichform from gravitational
collapse. In Section IV,wedescribethetheoreticaldi(cid:14)cultiesassociated with changesin topology and presentin
somedetailspeci(cid:12)cmethodsused in theliterature,namely,aCasimirenergyapproach involvingquasi-localenergy
di(cid:11)erencecalculationsthatmay re(cid:13)ectormeasuretheoccurrenceofatopologychange.W ealsobrie(cid:13)y review the
M orseindexanalysis.InsectionV,weconclude.
II. TIM E-DEPENDENT DARK ENERGY STARS
A. Staticspacetim e
In thissection,weprovidea briefoutlineofthemathematicalmodelsofstaticand spherically symmetricdark
energystarsconsidered in Ref.[9].Considerthefollowingtime-independentlineelement,in curvaturecoordinates,
representingadarkenergystar
dr2
ds2 = (cid:0) e2(cid:11)(r)dt2+ + r2(d(cid:18)2+ sin2(cid:18)d(cid:30)2); (1)
1(cid:0) 2m(r)=r
where(cid:11)(r)andm(r)arearbitraryfunctionsoftheradialcoordinate,r.Thefunctionm(r)canbeinterpretedasthe
quasi-localmass,and isdenoted asthemassfunction [9].Thefactor(cid:11)(r)isthe\gravitypro(cid:12)le"and isrelated to
p
thelocallymeasuredaccelerationduetogravity,throughthefollowingrelationship:A = 1(cid:0) 2m(r)=r(cid:11)0(r)[9,11],
wheretheprimedenotesaderivativewith respecttotheradialcoordinater. Theconvention used isthat(cid:11)0(r)is
positiveforaninwardlygravitationalattraction,andnegativeforanoutwardgravitationalrepulsion.
TheEinstein(cid:12)eldequationsaregivenby[9]
m0= 4(cid:25)r2(cid:26); (2)
m + 4(cid:25)r3p
(cid:11)0= r ; (3)
r(r(cid:0) 2m)
((cid:26)+ p)(m + 4(cid:25)r3p ) 2
p0 = (cid:0) r r + (p (cid:0) p); (4)
r r(r(cid:0) 2m) r t r
where(cid:26)(r)istheenergy density,p(r)istheradialpressure,and p(r)isthetangentialpressureorthogonaltop .
r t r
NotethatEq.(4)correspondstotheanisotropicpressureTolman-Oppenheimer-Volko(cid:11)(TOV)equation.
An additionalconstraintisplaced on thesystem ofequationsby considering thedark energy equation ofstate,
p (r)= !(cid:26)(r),andtakingintoaccountEqs.(2)and(3),wehavethefollowingrelationship
r
m + !rm0
(cid:11)0(r)= : (5)
r(r(cid:0) 2m)
Thereis,however,asubtlepointthatneedstobeemphasized[5,6].Thenotionofdarkenergyisthatofaspatially
homogeneouscosmic(cid:13)uid.Nevertheless,itcanbeextendedtoinhomogeneoussphericallysymmetricspacetimes,by
regarding thatthepressurein theequation ofstatep = !(cid:26) isa radialpressure,and thatthetransversepressure
may beobtained from Eq. (4). In addition to this,and asmentioned in theIntroduction,despitethefactthat
4
thedark energy equation ofstaterepresentsa spatially homogeneouscosmic(cid:13)uid and isassumed notto cluster,
inhomogeneitiesmay ariseduetogravitationalinstabilities.Thus,thedark energystargeometriesconsidered here
maypossiblyoriginatefrom density(cid:13)uctuationsinthecosmologicalbackground,resultinginthenucleationthrough
therespectivedensityperturbations[11].
InRef.[9],speci(cid:12)csolutionswerefoundbyconsideringthattheenergydensityispositiveand(cid:12)niteatallpointsin
theinteriorofthedarkenergystar.Inparticular,severalrelativisticdarkenergystellarcon(cid:12)gurationswereanalyzed
by imposing speci(cid:12)cchoicesforthemassfunction m(r),and through Eq. (5),(cid:11)(r)wasdetermined,consequently
providingexplicitexpressionsforthestress-energytensorcomponents.Thisinteriorsolutionwasfurthermatchedto
anexteriorSchwarzschildvacuum solutiongivenby
(cid:18) (cid:19) (cid:18) (cid:19)
2M 2M (cid:0)1
ds2 = (cid:0) 1(cid:0) dt2+ 1(cid:0) dr2+ r2(d(cid:18)2+ sin2(cid:18)d(cid:30)2); (6)
r r
atajunction interfacea.TheSchwarzschildspacetimepossessesan eventhorizonatr = 2M ,sothattoavoidthe
b
latter,thejunctionradiusliesoutside2M ,i.e.,a> 2M .
Thesurfacestressesonthethinshellaregivenby
r r !
1 2M 2m
(cid:27) = (cid:0) 1(cid:0) + a_2(cid:0) 1(cid:0) + a_2 ; (7)
4(cid:25)a a a
!
1 1(cid:0) M + a_2+ aa(cid:127) 1+ !m0(cid:0) m + a_2+ aa(cid:127)+ a_2m0(1+!)
P = q a (cid:0) qa 1(cid:0)2m=a ; (8)
8(cid:25)a 1(cid:0) 2M + a_2 1(cid:0) 2m + a_2
a a
where(cid:27) andP arethesurfaceenergydensityandthetangentialsurfacepressure[9,32,33],respectively.Theoverdot
denotesaderivativewithrespectto(cid:28),whichisthepropertimeonthejunctioninterface,andtheprimeheredenotes
aderivativewithrespecttothejunctionsurfaceradiusa.
Thedynamicalstabilityofthetransitionlayeraofthesedarkenergystarstolinearizedsphericallysymmetricradial
perturbationsaboutstaticequilibrium solutionswasalsoexplored.Itwasfoundthatlargestabilityregionsexistthat
aresu(cid:14)ciently closetowheretheeventhorizon isexpected toform,sothatitwould bedi(cid:14)culttodistinguish the
exteriorgeometryofthedarkenergystars,analyzedin[9],from anastrophysicalblackhole.
B. Tim e-dependentspacetim e
Inthissection,wegeneralizetheabovestaticdarkenergystarmodelstotime-dependentgeometries.Thisismainly
motivated by thefactthatrecent(cid:12)tsto supernovae,CM B and weak gravitationallensing data probably favoran
evolvingequationofstate,withthedarkenergyparametercrossingthephantom divide! = (cid:0) 1[7].
Inthefollowing,weconsideratime-dependentandsphericallysymmetricmetricgivenby
(cid:0) (cid:1)
ds2 = (cid:0) e2(cid:11)(r;t)dt2+ e2(cid:12)(r;t)dr2+ r2 d(cid:18)2+ sin2(cid:18)d(cid:30)2 : (9)
Notethatonemayalsode(cid:12)nethefunction(cid:12)(r;t)as
(cid:20) (cid:21)
1 2m(r;t)
(cid:12)(r;t)= ln 1(cid:0) ; (10)
2 r
wherethe‘massfunction’m(r;t)isnow time-dependent.
TheEinstein(cid:12)eldequationprovidesthefollowingnonzerocomponents:
e(cid:0)2(cid:12)
(cid:26)(r;t)= (2(cid:12)0r+ e2(cid:12) (cid:0) 1); (11)
8(cid:25)r2
e(cid:0)2(cid:12)
p (r;t)= (2(cid:11)0r(cid:0) e2(cid:12) + 1); (12)
r 8(cid:25)r2
(cid:12)_e(cid:0)((cid:11)+(cid:12))
f(r;t)= ; (13)
4(cid:25)r
( (cid:20) (cid:21) (cid:16) (cid:17))
1 1
p(r;t)= e(cid:0)2(cid:12) (cid:11)00+ (cid:11)02(cid:0) (cid:11)0(cid:12)0+ ((cid:11)0(cid:0) (cid:12)0) + e(cid:0)2(cid:11) (cid:11)_(cid:12)_(cid:0) (cid:12)(cid:127)(cid:0) (cid:12)_2 ; (14)
t 8(cid:25) r
5
wheretheprimedenotesapartialderivativewithrespecttotheradialcoordinater,andtheoverdotapartialderivative
withrespecttothetimecoordinatet.Notethepresenceofanenergy(cid:13)uxterm intheradialdirection,Tt = (cid:6) f(r;t),
r
whichdependson(cid:12)_.
An importantissuein thetime-dependentdarkenergystarswith theparametercrossingthephantom divideare
theenergy conditions,in particular,thenullenergy condition (NEC).TheNEC isde(cid:12)ned asT k(cid:22)k(cid:23) (cid:21) 0,where
(cid:22)(cid:23)
k(cid:22) isanynullvector,andconsequentlyprovides(cid:26)+ p (cid:6) 2f(cid:21) 0.Thelatterde(cid:12)nition,takingintoaccountthe(cid:12)eld
r
equations(11)-(13),isgivenby
h i
1
(cid:26)+ p (cid:6) 2f= e(cid:0)2(cid:12) ((cid:11)0+ (cid:12)0)(cid:6) 2(cid:12)_e(cid:0)((cid:11)+(cid:12)) (cid:21) 0: (15)
r 4(cid:25)r
Inthiscontext,onemayconsiderageneralizationoftheequationofstatep = !(cid:26),givenby
r
p (r;t)= !(r;t)[(cid:26)(r;t)(cid:6) 2f(r;t)]; (16)
r
or
p (r;t)(cid:7) 2f(r;t)= !(r;t)(cid:26)(r;t); (17)
r
wheref(r;t)= (cid:7) Tt istheenergy(cid:13)uxterm,asnotedabove.However,onecancomeupwithaninterestingclassof
r
solutionsconsideringthefollowingequationofstate:
p (r;t)= !(r;t)(cid:26)(r;t): (18)
r
Throughoutthiswork,weessentiallyusetheequationofstategivenbyEq.(18).
Takingintoaccountthe(cid:12)eldequations(11)and(12),then Eq.(18)providesthefollowingrelationship
h i
1
!(r;t)(cid:12)0(r;t)= (cid:11)0(r;t)+ [1+ !(r;t)] 1(cid:0) e2(cid:12)(r;t) : (19)
2r
A similaranalysiswascarriedoutinRef.[34],inthecontextoftime-dependentwormholes.
Equation(19)maybeformallysolvedintermsof(cid:11)(r;t),andprovidesthefollowinggeneralsolution
Z (cid:26) h i(cid:27)
1 1
(cid:11)(r;t)= !(r(cid:22);t)(cid:12)0(r(cid:22);t)r(cid:22)(cid:0) [1+ !(r(cid:22);t)] 1(cid:0) e2(cid:12)(r(cid:22);t) dr(cid:22): (20)
r(cid:22) 2
Thus,inprinciple,if!(r;t)and(cid:12)(r;t)areknown,then (cid:11)(r;t)maybeobtainedfrom Eq.(20).
Asanalternative,Eq.(19)canbeformallyintegratedfor(cid:12)(r;t)toyieldthegeneralsolution
(cid:20) Z (cid:21)
1 e(cid:0)(r(cid:22);t)(1+ !(r(cid:22);t)) 1
(cid:12)(r;t)= (cid:0) ln (cid:0) 2F(t)+ dr(cid:22) + (cid:0)(r;t); (21)
2 !(r(cid:22);t)r(cid:22) 2
whereF(t)isanintegrationfunction,andthefactor(cid:0)(r;t)isde(cid:12)nedas
Z (cid:20) (cid:21)
2r(cid:22)(cid:11)0(r(cid:22);t)+ r(cid:22)+ 1
(cid:0)(r;t)= dr(cid:22): (22)
r(cid:22)!(r(cid:22);t)
A particularlysimpleandinterestingtoymodelisthespeci(cid:12)ccaseofapurelytimedependentparameter! = !(t),
sothatthegeneralsolution(21)takestheform
( )
h (cid:16) Z (cid:17)i
(cid:12)(r;t)= (cid:0)1 2(cid:11)(r;t)(cid:0) ln r(cid:0) 1+!!(t()t) (cid:0) 2F(t)+ 1+ !(t) e2(cid:11)!((r(cid:22)t;)t)r(cid:22)!(1t) dr(cid:22) : (23)
2 !(t) !(t)
In thenextsection,weanalyzespeci(cid:12)csolutions,namely,thatofaconstantenergydensity,theTolman-M atese-
W hitman massfunction,which wereextensivelyexplored in Ref.[9],and acollapsingmodelwith anonzeroenergy
(cid:13)uxterm.
6
III. SPECIFIC TIM E-DEPENDENT DARK ENERGY STAR SOLUTIONS
A. Constantenergy density
Considerthespeci(cid:12)ccaseofaconstantenergydensity,(cid:26)(r;t)= (cid:26) ,sothatEq.(11)providesthesolution
0
(cid:20) (cid:21)
1 F (t)
(cid:12)(r;t)= (cid:0) ln 1(cid:0) Ar2+ 1 ; (24)
2 r
withA = 8(cid:25)(cid:26) =3andF (t)isafunctionofintegration.
0 1
SubstitutingintoEq.(20),onearrivesat
Z
3A!(r;t)r3+ Ar3(cid:0) 3F(t)
1
(cid:11)(r;t)= + F (t); (25)
2r[3r(cid:0) Ar3+ 3F (t)] 2
1
whereF (t)isanotherfunction ofintegration.OnemayfurthersimplifytheanalysisbyconsideringthatF (t)= 0,
2 1
which isphysicallyjusti(cid:12)ed bytheimposition ofa(cid:12)nitemassfunction attheoriginr= 0forallvaluesofthetime
coordinatet.NotethatconsideringF (t)= 0impliesthatthemassfunctionisnottime-dependent,andthe(cid:13)uxterm
1
f(r;t)iszero,as(cid:12)_= 0.
Forinstance,considerthespeci(cid:12)cexampleofaseparationofvariablesoftheparameter!(r;t)givenby
!(r;t)= ! (t)! (r): (26)
1 2
Choosingthefollowingfunctions:
! (t)= ! + !(cid:22) tanh[(cid:27)(t(cid:0) t)]; (27)
1 0 0 0
(cid:21)2
! (r)= (cid:0) ; (28)
2 1+ (r=R)2
where! ,!(cid:22) ,(cid:27),t,(cid:21)2,andR areconstants.Thefactor1=(cid:27) maybeinterpretedasthe\relaxationtime",describing
0 0 0
thewidthofthetime-dependence.
SeeFig.1foraqualitativedescription of!(r;t)given byEqs.(26)-(28).Theleftplotrepresentsthebehaviorof
! = !(0;t)atthecenterr= 0.W ehaveconsidered thefollowingvalues:R = 1,(cid:14) = 1,! = 3=4,!(cid:22) = 2=3and
0 0
t = 8.
0
10 20 t 30 40 50
0
–0.2
–0.4
–0.4
0,t)–0.6 w(r,t)–0.8
w(–0.8 –1.2
–1 0 0
10 0.2
20 0.4
–1.2 t . 30 0.6 r
0.8
40
–1.4 50 1
FIG.1:Leftplot:Forthiscaseweconsidered thebehaviorof! = !(0;t)atthecenterr= 0;forthedashed curvewehave
(cid:27)= 0:3andforthesolid curve(cid:27)= 0:1.Rightplot:Forthiscaseweconsidered the! = !(t;r)dependencewith(cid:27)= 0:3.For
bothcases,wehaveassumedthefollowingnumericalvalues:R = 1,(cid:14)= 1,! = 3=4,!(cid:22) = 2=3andt = 8.
0 0 0
Substitutingthefunctions(26)-(28)intoEq.(25),yieldsthefollowingsolution
( )
1 (1(cid:0) Ar2)A(cid:22)!1(t)(cid:0)1
(cid:11)(r;t)= ln + F (t); (29)
4 [1+ (r=R)2]A(cid:22)!1(t) 2
7
where,fornotationalsimplicity,theconstantA(cid:22)isde(cid:12)nedas
3AR2(cid:21)2
A(cid:22)= : (30)
R2A + 1
ThefunctionF (t)canbeabsorbedthrougharede(cid:12)nitionofthetimecoordinateasbefore,sothatwithoutasigni(cid:12)cant
2
lossofgeneralityonemayimposetheconditionF (t)= 0.
2
Thepressurepro(cid:12)leisgivenby
3(cid:21)2R2A! (t)
1
p = (cid:0) ; (31)
r 8(cid:25)(R2+ r2)
3A R2Ar2(R2+ 2r2)+ Ar6+ ! (t)(cid:21)2R2[3Ar2! (t)(cid:21)2R2(cid:0) 4(Ar4+ R2)]
1 1
p = ; (32)
t 32(cid:25) (1(cid:0) Ar2)(R2+ r2)2
withp = p atthecenter,r= 0.
r t
Theanalysissimpli(cid:12)esbyconsideringapurelytime-dependentparameter,i.e.,! = !(t).Thus,Eq.(25)takesthe
form
1
(cid:11)(r;t)= (cid:0) [1+ 3!(t)]ln(1(cid:0) Ar2)+ F (t): (33)
2
4
ThefactorF (t)can beabsorbed into a rede(cid:12)nition ofthe timecoordinate,so thatwithouta signi(cid:12)cantlossof
2
generality,onecanassumeF (t)= 0.
2
Thepressurepro(cid:12)leisgivenbythefollowingrelationships:
3A!(t)
p = ;
r 8(cid:25)
(cid:2) (cid:3)
3A 4!(t)+ Ar2+ 3Ar2!2(t)
p = : (34)
t 32(cid:25)(1(cid:0) Ar2)
Notethatp = p atthecenter,r= 0,asexpected.
r t
B. Tolm an-M atese-W hitm an m assfunction
An interestingexampleistheTolman-M atese-W hitman massfunction considered in Ref.[9].Asin theexample
outlinedabove,weimposethat(cid:12)_= 0,sothatthe(cid:13)uxterm f(r;t)iszero.Thus,considerthefollowingchoiceforthe
time-independentmassfunction,givenby
(cid:18) (cid:19)
1 1+ 2br2
0
(cid:12)(r;t)= (cid:12)(r)= ln ; (35)
2 1+ br2
0
whereb isanon-negativeconstant[9].Thelattermaybedetermined from theregularityconditionsand the(cid:12)nite
0
characteroftheenergydensity attheorigin r= 0,and isgiven by b = 8(cid:25)(cid:26) =3,where(cid:26) istheenergydensity at
0 c c
r= 0.
Now,considertheradialandtemporaldependentcaseof! = !(r;t)givenbythefunctions(26)-(28).Substituting
thesefunctionsandEq.(35)intoEq.(20),yieldsthefollowingsolution
(cid:11)(r;t)= 1ln((cid:0)1+ br2(cid:1)(cid:6)(t)(cid:0)1+ 2br2(cid:1)(cid:7)(t) (cid:20)1+ (cid:16)r(cid:17)2(cid:21)(cid:4)(t)) + F (t); (36)
0 0 2
2 R
wheretheF (t)isafunctionofintegrationwhichmaybereabsorbedinarede(cid:12)nitionofthetimecoordinate,sothat
2
withoutalossofgeneralityweimposeF (t)= 0,asbefore.Fornotationalsimplicity,wehaveconsideredthefollowing
2
de(cid:12)nitions
1+ bR2(2bR2(cid:0) 3)(cid:0) b(cid:21)2R2! (t)(1(cid:0) 2bR2)
0 0 0 1 0
(cid:6)(t)= ; (37)
2(1(cid:0) 2R2b)(1(cid:0) R2b)
0 0
2b(cid:21)2R2
0
(cid:7)(t)= ; (38)
1(cid:0) 2R2b
0
b(cid:21)2R2! (t)(2bR2(cid:0) 3)
0 1 0
(cid:4)(t)= ; (39)
2(1(cid:0) 2R2b)(1(cid:0) R2b)
0 0
8
respectively.
Thestress-energytensorcomponentsaregivenby
b(3+ 2br2)
0 0
(cid:26)(r)= ; (40)
8(cid:25)(1+ 2br2)2
0
bR2(cid:21)2(3+ 2br2)! (t)
0 0 1
p (r;t)= (cid:0) ; (41)
r 8(cid:25)(R2+ r2)(1+ 2br2)2
0
h
p(r;t)= b 4R4b3r6(1(cid:0) !(t)(cid:21)2)2+ 8R2b3r8(1+ ! (t)(cid:21)2)+ 4R4b2r4(2(cid:0) 3!(t)(cid:21)2+ 3! (t)(cid:21)4)
t 0 0 1 0 1 0 1 1
+ 4R2b2r6(4+ 9! (t)(cid:21)2)+ R4br2(3(cid:0) 16!(t)(cid:21)2+ 9!2(t)(cid:21)4)+ 2R2br4(3+ 14! (t)(cid:21)2)
0 1 0 1 1 0 1
i.h i
+ br6(3+ 8br2+ 4b2r4)(cid:0) 12!(t)R4(cid:21)2 32(cid:25)(1+ br2)(1+ 2br2)3(R2+ r2)2 : (42)
0 0 0 1 0 0
Notethatp = p atthecenter,r= 0.
r t
Forsimplicity,considering a purely time-dependentparameter! = !(t),and substituting (35)into Eq. (20),
providesthefollowingsolution
(cid:20) (cid:21)
(cid:11)(r;t)= 1ln (cid:0)1+ br2(cid:1)1(cid:0)!2(t) (cid:0)1+ 2br2(cid:1)!(t) + F(t); (43)
0 0
2
wheretheF(t)isafunctionofintegrationwhich,asbefore,maybeabsorbedintoarede(cid:12)nitionofthetimecoordinate,
sothatonemayconsiderF(t)= 0withoutasigni(cid:12)cantlossofgenerality.
Thestress-energytensorcomponentsaregivenby
b(3+ 2br2)!(t)
0 0
p = !(t)(cid:26)= ; (44)
r 8(cid:25)(1+ 2br2)2
0
bf4b2r4(1+ !)[3+ br2(1+ !)]+ br2[3+ !(9!+ 16)]+ 8b2r4+ 12!g
p = 0 0 0 0 0 ; (45)
t 32(cid:25)(1+ br2)(1+ 2br2)3
0 0
withp = p atthecenter,r= 0.
r t
C. A classofm odelswith a non-zero (cid:13)ux term
Inthissection,weconstructasetofmodelswithanon-zeroenergy(cid:13)uxterm,whereatearlytimespossessesasmall
inhomogeneityintheregionnearr= 0whichgrowsduetogravitationalcollapse.Forthespeci(cid:12)ccaseconsideredin
thissection,weassumeforsimplicitythattheparameter,whicheventuallycrossesthephantom divideinthecentral
region,ispurelytime-dependent,i.e.,! = !(t),andisgovernedbyanequationofstateoftheform
p (r;t)= !(t)(cid:26)(r;t); (46)
r
andthatthesystem tendstoisotropyforlarger.
Fortheenergydensity,wegeneralizetheM bonye-Kazanasdensitypro(cid:12)le[35](alsoutilizedbyDymnikova[36])to
areasonabletime-dependentmodelgivenby
(cid:26)(r;t)= (cid:26)0a(t)e(cid:0)(r=r0)n ; (47)
where(cid:26) ,r and n areappropriately chosen constants. Here,thetime-dependentfunction a(t)ischosen so that
0 0
thecollapsewillasymptoteatlatetimes,formingastaticstar.Noteforthesakeofclaritythatthetime-dependent
function a(t)should notbeconfused with thejunction interfaceradiusintroduced in Eqs.(7)-(8).Thispro(cid:12)lehas
beenextremelyusefulintheinvestigationsofnon-singularblackholes(i.e.,horizonsnotshieldingasingularity)[35],
including deSittercoreblack holes[36]and,morerecently,asa modelforgravastars[15](supplemented with an
appropriateequationofstate).
Theequationofstate(46)thenyields
pr(r;t)= !(t)(cid:26)0a(t)e(cid:0)(r=r0)n ; (48)
9
Atthisstage,itisusefultowritethesolutiontothe(cid:12)eldequationsasfollows[37]:
" #
Z
8(cid:25) r
e(cid:0)2(cid:12)(r;t) = 1(cid:0) b2(t)+ (cid:26)(r(cid:22);t)r(cid:22)2dr(cid:22) ; (49a)
r
0+
( " #)
Z
r
e2(cid:11)(r;t) = e(cid:0)2(cid:12)(r;t) exp h(t)+ 8(cid:25) [p (r(cid:22);t)+ (cid:26)(r(cid:22);t)]e2(cid:12)(r(cid:22);t)r(cid:22)dr(cid:22) ; (49b)
r
0+
" #
Z
1 r
f(r;t)= (cid:0) 2b(t)b_(t)+ (cid:26)_(r(cid:22);t)r(cid:22)2dr(cid:22) e2[(cid:12)(r;t)(cid:0)(cid:11)(r;t)] ; (49c)
r2
(cid:16) (cid:17) (cid:16) 0+ (cid:17) (cid:16) (cid:17)
r r r r
p(r;t)= p0+ f_ + 1+ (cid:11)0 p + (cid:11)_+ (cid:12)_ f+ (cid:11)0(cid:26); (49d)
t 2 r 2 r 2 2
respectively,wherethe explicitcoordinatedependence hasbeen dropped in Eq. (49d),and asbefore,theprime
denotesapartialderivativewithrespecttotheradialcoordinater,andtheoverdotapartialderivativewithrespect
tothetimecoordinatet.Thefunctionsb(t)andh(t)aretwoarbitraryfunctionsofintegration.Forastar,b(t)isset
tozerotoavoidasingularityatthecenter.However,atlatetimethisfunctionneednotvanishasitisusefulforthe
wormholecon(cid:12)guration.Notethatwiththeprescriptionoftheenergydensityandradialpressure,theentiresystem
ofequationsmay,inprinciple,besolvedfortheunknowns.W eexploitthisfacthere.
To ensure thatthe starcrossesthe phantom divide and yetdoesnotcollapse forin(cid:12)nite time,the following
prescriptionsaremade:
(cid:2) (cid:3) (cid:0) (cid:1)
a(t)= a0 (1+ (cid:15))(cid:0) e(cid:0)k0t ; !(t)= !0(cid:0) !1 1(cid:0) e(cid:0)k1t ; (50)
where0and 1subscriptsdenoteconstantquantities. W enow havean in(cid:12)nitefamily ofsolutionswith thedesired
physicalproperties. Atthisstageweshould notethatthegenerated space-timestend toM inkowskispace-timeas
r ! 1 . One would need to therefore cut-o(cid:11) the solution atsomer = r and patch itto an appropriatedark
(cid:3)
energy exterior. However,given how smalltheenergy density and pressuresofthecurrently acceleratinguniverse
are(whencomparedtothoseofanaveragestar),theasymptoticM inkowskiapproximationisprobablyareasonable
approximation.
Surprisingly,forcertainvaluesofn,theequations(49a)-(49d)mayactuallybeintegratedtoyieldananalyticresult.
Theexpressionsareratherunwieldlyhoweversoinstead weplotthevariousrelevantparametersin (cid:12)gures2and 3.
From the(cid:12)guresitcan benoted thattheinitialinhomogeneityisverysmall,mostpronouncednearthecenter,and
thespace-timealmost(cid:13)ateverywhere. Thereisan inward (cid:13)ow ofenergy,dueto thegravitationalcollapseascan
beseen bythenegativevaluesoftheenergy(cid:13)uxplot.Atlatetime,themagnitudeoftheenergy(cid:13)uxdecreasesand
asymptotesto zero,indicating thehaltofthecollapse. Atsomepointduring thecollapse,thephantom divideis
crossedandtheconditionsforpossiblewormholeformationareestablished.W eshow thisin(cid:12)gure4whereitmaybe
seenthattheNEC violationismostsevereinthecenter.
IV. TOPOLOGY CHANGE
A. The Casim irenergy approach
Although(50)alsoallowsforanordinary(positivepressure)(cid:13)uidatearlytimes,allofthegeometriesconsideredin
theprevioussectionsweremodelledsothattheevolvingparameter!(r;t)j startsoutintherange(cid:0) 1< ! < (cid:0) 1=3,
r=0
then crossesthephantom divide,and (cid:12)nallyendsup in thephantom regime,! = p =(cid:26)< (cid:0) 1.Oncein thephantom
r
regime,thenegativeradialpressureexceedstheenergydensity,whichinprinciplemayimplyatopologychange.Itis
stilluncertainhow toobtainthistopologychange,andifpossible,isriddledwhichdi(cid:14)culties,suchasthetheoretical
appearanceofclosedtimelikecurves.Itislikelythatcrossingthephantom divideisaccompaniedbyalargequantum
(cid:13)uctuationofthemetric.Thenacrucialquestionis[27]:whathappenswhenthemetric(cid:13)uctuationsbecomelarge?
Concerning the geometry ofspacetime undergoing quantum (cid:13)uctuations,thisdoesnotseem to be a sourceof
disagreement,butwhen weturn to thequestion ofwhetherornotthetopology ofspacetimeundergoesquantum
(cid:13)uctuations,theproblem becomesmoresubtle.ItwasJ.A.W heeler[19,20]who(cid:12)rstconjectured thatspacetime
could be subjected to a topology (cid:13)uctuation atthe Planck scale. Thismeansthatspacetime undergoesa deep
and rapid transformation in itsstructure.Thechangingspacetimeisbestknown asspacetimefoam,which can be
takenasamodelforthequantum gravitationalvacuum.Someauthorshaveinvestigatedthee(cid:11)ectsofsuchafoamy
spaceonthecosmologicalconstant,forinstance,oneexampleisthecelebratedColemanmechanism,wherewormhole
10
10-3 10-7
5
energy density1234 0 energy flux-1----94272.....55000 0
0 1,000 -14.5 500
15 t 15 1,000t
10 2,000 10 1,500
r 5 0 r 5 0 2,000
10-3
10-30.0 ure 1.5
s
ure-2.5 pres-1.0
al press--75..50 ansverse --63..05 0
radi-101.50 0 tr-81.55 10 1,000t
10 1,000 r 5 2,000
r 5 2,000 t 0
0 3,000
1.125
1.1 -0.75
-0.8
1.075
g_rr1.05 g_tt-0.85
-0.9
1.025 0
0 -0.95
110.00 75 r50 25 0 2,000 1,500 1,000t500 -1.0100 75 5r0 25 0 2,0001,5001,00050t0
FIG.2: Parametersfora samplestellarmodelpresented in section III-C.Plotted on theverticalaxes(from leftto right,
startingfrom thetop)are(cid:26)(r;t),Tt(r;t)p (r;t),p(r;t),and(cid:12)nallye2(cid:12)(r;t),and(cid:0)e2(cid:11)(r;t).
r r t
contributionssuppressthecosmologicalconstant,explainingitssmallobservedvalue[38].Nevertheless,how torealize
suchafoam-likespaceandalsowhetherthisrepresentstherealquantum gravitationalvacuum arestillunknown.W e
can mention someresultsabouttopologicalconstraintson theclassicalevolution ofgeneralrelativisticspacetimes.
Theyaresummarizedintwopoints[27]:
1.Incausallywell-behavedclassicalspacetimesthetopologyofspacedoesnotchangeasafunctionoftime.
2.Incausallyill-behavedclassicalspacetimesthetopologyofspacecansometimeschange.
From thequantum pointofview wecanseparatetheproblem oftopologychangegeneratedbyacanonicalquan-
tization approach and a functionalintegralquantization approach. TheHawkingtopology changetheorem isthus
enough to show thatthetopology ofspacecannotchangein canonically quantized gravity [39]. In theFeynman