Final fate of spherically symmetric gravitational collapse of a dust cloud in Einstein-Gaus

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

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aFinalfateofsphericallysymmetricgravitationalcollapseofadustcloudinEinstein-Gauss-BonnetgravityHidekiMaeda AdvancedResearchInstituteforScienceandEngineering,WasedaUniversity,Okubo3-4-1,Shinjuku,Tokyo169-8555,Japan(Dated:February7,2008)Wegiveamodelofthehigher-dimensionalsphericallysymmetricgravitationalcollapseofadustcloudincludingtheperturbativee ectsofquantumgravity.Then(≥5)-dimensionalactionwiththeGauss-BonnettermforgravityisconsideredandasimpleformulationofthebasicequationsisgivenforthespacetimeM≈M2×Kn 2withaperfect uidandacosmologicalconstant.ThisisageneralizationoftheMisner-Sharpformalismofthefour-dimensionalsphericallysymmetricspacetimewithaperfect uidingeneralrelativity.Thewholepictureandthe nalfateofthegravitationalcollapseofadustclouddi ergreatlybetweenthecaseswithn=5andn≥6.Therearetwofamiliesofsolutions,whichwecallplus-branchandtheminus-branchsolutions.Aplus-branchsolutioncanbeattachedtotheoutsidevacuumregionwhichisasymptoticallyanti-deSitterinspiteoftheabsenceofacosmologicalconstant.Bounceinevitablyoccursintheplus-branchsolutionforn≥6,andconsequentlysingularitiescannotbeformed.Sincethereisnotrappedsurfaceintheplus-branchsolution,thesingularityformedinthecaseofn=5mustbenaked.Ontheotherhand,aminus-branchsolutioncanbeattachedtotheoutsideasymptotically atvacuumregion.Weshowthatnakedsingularitiesaremasslessforn≥6,whilemassivenakedsingularitiesarepossibleforn=5.Inthehomogeneouscollapserepresentedbythe atFriedmann-Robertson-Walkersolution,thesingularityformedisspacelikeforn≥6,whileitisingoing-nullforn=5.Intheinhomogeneouscollapsewithsmoothinitialdata,thestrongcosmiccensorshiphypothesisholdsforn≥10andforn=9dependingontheparametersintheinitialdata,whileanakedsingularityisalwaysformedfor5≤n≤8.Thesenakedsingularitiescanbegloballynakedwhentheinitialsurfaceradiusofthedustcloudis ne-tuned,andthentheweakcosmiccensorshiphypothesisisviolated.PACSnumbers:04.20.Dw,04.40.Nr,04.50.+hI.INTRODUCTIONEinstein’sgeneraltheoryofrelativityhassuccessfullypassedmanyobservationaltestsandisnowacentralparadigmingravitationphysics.GeneralrelativityexplainssuchgravitationalphenomenaastheperihelionshiftofMercury’sorbit,gravitationallensing,redshiftinthelightspectrumfromextragalacticobjects,andsoon.Oneofthemostintriguingpredictionsofthetheoryistheexistenceofaspacetimeregionfromwhichnothingcanescape,i.e.,ablackhole.Ithasbeenconsideredthatblackholesareformedfromthegravitationalcollapseinthelaststageofheavystars’lifeorinhigh-densityregionsofthedensityperturbationsintheearlyuniverse.The rstanalyticmodelofblack-hole

formationingeneralrelativitywasobtainedbyOppenheimerandSnyderin1939,whichrepresentsthesphericallysymmetricgravitationalcollapseofahomogeneousdustcloudinasymptotically atvacuumspacetime[1].Inthisspacetime,thesingularityformedisspacelikeandhiddeninsidetheblack-holeeventhorizon,sothatitisnotvisibletoanyobserver.However,itwasshownlaterthatthisisnotatypicalmodelandthesingularitiesformedingenericcollapsearenaked,i.e.,observable[2,3,4,5].

Ingeneralrelativity,itwasproventhatspacetimesingularitiesinevitablyappearingeneralsituationsandunderphysicalenergyconditions[6].Gravitationalcollapseisoneofthepresumablescenariosinwhichsingularitiesareformed.Whereanakedsingularityexists,thespacetimeisnotgloballyhyperbolic,sothatthefuturepredictabilityofthespacetimebreaksdown.Inthiscontext,Penroseproposedthecosmiccensorshiphypothesis(CCH),whichprohibitstheformationofnakedsingularitiesingravitationalcollapseofphysicallyreasonablematterswithgenericregularinitialdata[7,8].TheweakversionofCCHprohibitsonlytheformationofgloballynakedsingularities,i.e.,thosewhichcanbeseenbyanobserveratin nity.IftheweakCCHiscorrect,singularitiesformedingenericgravitationalcollapsearehiddeninsideblackholes,andthefuturepredictabilityofthespacetimeoutsidetheblack-holeeventhorizonisguaranteed.Ontheotherhand,thestrongversionofCCHprohibitstheformationoflocally

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

nakedsingularitiesalso,whichcanbeseenbysomelocalobserver.ThestrongCCHassertsthefuturepredictabilityofthewholespacetime,i.e.,globalhyperbolicityofthespacetime.

TheCCHisoneofthemostattractiveandimportantunsolvedproblemsingravitationphysics.ValidityoftheCCHisassumedinthemanystrongtheoremssuchastheblack-holeuniquenesstheoremorthepositiveenergytheoreminasymptotically atspacetime.Atpresent,however,thegeneralproofoftheCCHisfarfromcomplete.Onthecontrary,therearemanycounterexample“candidates”ingeneralrelativity.(See[9]forareview.)

Theformationofasingularitymeansthataspacetimeregionwithin nitelyhighcurvaturecanberealizedinthevery nalstageofgravitationalcollapse.Itisnaturallyconsideredthatquantume ectsofgravitycannotbeneglectedinsuchregions,sothattheclassicaltheoryofgravitycannotbeappliedthere.Therefore,nakedsingularitiesgiveusachancetoobservethequantume ectsofgravity.Fromthispointofview,HaradaandNakaoproposedaconceptnamedthespacetimeborder,whichisthespacetimeregionwhereclassicaltheoriesofgravitycannotbeapplied[10].Thespacetimeborderisane ective“singularity”inclassicaltheory,andthentheCCHcanbenaturallymodi edtomoreapracticalversion,whichprohibitstheformationofnakedspacetimeborders.Ifthemodi edCCHistrue,spacetimeregionswherequantume ectsofgravitydominateareneverobserved.Ontheotherhand,ifitisviolated,thereisapossibilityinprincipleforustoobservesuchregionsandobtaininformationusefultotheconstructionofthequantumtheoryofgravity,whichisstillincomplete.Fromthispointofview,studiesofthe nalfateofgravitationalcollapsearequiteimportant.

Uptonow,manyquantumtheoriesofgravityhavebeenproposed.Amongthem,superstring/M-theoryisthemostpromisingcandidateandhasbeenintensivelyinvestigated,whichpredictshigher-dimensionalspacetime(morethanfourdimensions).Inthistheory,whenthecurvatureradiusofthecentralhigh-densityregioningravitationalcollapseiscomparablewiththecompacti cationradiusofextradimensions,thee ectsofextradimensionswillbeimportant.Suchregionscanbemodelede ectivelybyhigher-dimensionalgravitationalcollapse.

Arecentattractiveproposalforanewpictureofouruniverse,whichiscalledthebraneworlduniverse[11,12,13],isbasedonsuperstring/M-theory[14].Inthebraneworlduniverse,weliveonafour-dimensionaltimelikehypersurfaceembeddedinthehigher-dimensionalbulkspacetime.BecausethefundamentalscalecouldbearoundtheTeVscaleinthisscenario,thebraneworldsuggeststhatthecreationoftinyblackholesintheupcominghigh-energycolliderispossible[15].Fromthispointofview,thee ectsofsuperstring/M-theoryonblackholesorgravitationalcollapseshouldbeinvestigated.

However,thenon-perturbativeaspectsofsuperstring/M-theoryarenotunderstoodcompletelysofar,althoughtheprogressinrecentyearshasbeenremarkable.Giventhepresentcircumstances,takingtheire ectsperturbativelyintoclassicalgravityisonepossibleapproachtostudyingthequantume ectsofgravity.TheGauss-BonnettermintheLagrangianisthehighercurvaturecorrectiontogeneralrelativityandnaturallyarisesasthenextleadingorderoftheα′-expansionofheteroticsuperstringtheory,whereα′istheinversestringtension[16].SuchatheoryiscalledtheEinstein-Gauss-Bonnetgravity.

Inapreviouspaper,theauthorpresentedamodelofthen(≥5)-dimensionalsphericallysymmetricgravitationalcollapseofanulldust uidinEinstein-Gauss-Bonnetgravity[17].Itwasshownthatthespacetimestructureofthegravitationalcollapsedi ersgreatlybetweenn=5andn≥6.In vedimensions,massivetimelikenakedsingularitiescanbeformed,whichneverappearinthegeneralrelativisticcase,whilemasslessingoing-nullnakedsingularitiesareformedinthen(≥6)-dimensionalcase.

Inthispaper,weconsiderthen(≥5)-dimensionalsphericallysymmetricgravitationalcollapseofadust uidwithsmoothinitialdatainEinstein-Gauss-Bonnetgravity.Ingeneralrelativity,thesamesystemhasbeenanalyzedbymanyresearchersbothforn=4[2,3,4,5]andforn≥5[18].Theyshowedthatthesingularityformediscensoredforn(≥6),whileitisnakedforn=4.Forn=5,thesingularitycanbecensoreddependingontheparametersintheinitialdata.

Thispaperisorganizedasfollows.InSec.II,forthen(≥5)-dimensionalspacetimeM≈M2×Kn 2withaperfect uidandacosmologicalconstant,whereKn 2isthe(n 2)-dimensionalEinsteinspace,wede neascalaronM2,ofwhichdimensionismass,andgiveasimpleformulationofthebasicequationsinEinstein-Gauss-Bonnetgravity.InSec.III,usingthisformalism,weinvestigatethe nalfateofthen(≥5)-dimensionalsphericallysymmetricgravitationalcollapseofadustcloudwithoutacosmologicalconstant.SectionVisdevotedtodiscussionandconclusions.InAppendixA,wereviewthestudyofthegeneralrelativisticcaseforcomparisonandgivesomecomplements.Throughoutthispaperweuseunitssuchthatc=1.Asfornotationwefollow[19].TheGreekindicesrunµ=0,1,···,n 1.

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

II.MODELANDBASICEQUATIONS

Webeginwiththefollowingn-dimensional(n≥5)action:

n√(R 2Λ+αLGB)+Smatter,S=dx2κ2n

whereRandΛarethen-dimensionalRicciscalarandthecosmologicalconstant,respectively.κn≡(2.1)√

2gµνR,

Hµν≡2 RRµν 2RµαRαν 2RαβRµανβ+RµαβγRναβγ 1

xµ=e Φ (2.4)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

˙2 e 2ΨS′2)cannotbezero.Lemma2Ifp= ρ,thenS2+2(n 3)(n 4)α(k+e 2ΦS

Proof.Iftherelation

˙2 e 2ΨS′2)=0S2+2(n 3)(n 4)α(k+e 2ΦS(2.11)

issatis edatamoment,thenthe(t,t)and(r,r)componentsofthe eldequation(2.3)give(

n

1)(

n

2)

8

α

(

n

3)(

n 4)+Λ=κ2np,

respectively.Eqs.(2.12)and(2.13)giveacontradictionp= ρ.2

Herewegiveade nitionofascalaronM2withthedimensionofmasssuchthat

m≡(n 2)Vnk 2

S ,

m′=Vnk′

2ρSSn 2,

m˙= Vnk 2pSS˙n 2,

0= S˙′+Φ′S˙+Ψ˙S′,

m=(n 2)Vnk 2

rdr(3.1)(2.13)(2.16)(2.17)(2.18)(2.19)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

onthehypersurfacewithaconstantt.FromEq.(2.19),weobtain

e2Ψ=S′2

1S′Sn 2Vn 2.(3.3)

Fromthisequation,we ndthattheremayexistbothshell-crossingsingularities,whereS′=0,andshell-focusingsingularities,whereS=0.HereafterweassumeS′>0,orequivalentlym′>0,inorderthatshell-crossingsingularitiesmayberemovedfromourconsideration.

FromEq.(2.20),weobtainamasterequationofthesystem:

S˙2=f S2

1+8 ακ2nm

2

F(S)+Sd 2n 2,

whered 2n 2isthelineelementofthe(n 2)-dimensionalunitsphere.F(S)isde nedby

2

F(S)≡1+S1+8 ακ2nM

F(Sdt2+SΣ) 2Σd 2n 2.

BecausetheinducedmetricmustbethesameonbothsidesofthehypersurfaceΣ,wehave

1=F(S˙2 S˙2Σ)TΣΣ(3.5)(3.8)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

Herewede neβby

β≡

xµ=

xµ=T˙ Σ r

and

nµdxµ= S˙Σdt+T˙Σdr

withaconsistentchoiceforthesign.

AsseenfrominsideofΣ,thenon-zerocomponentsoftheextrinsiccurvatureKabofΣarecalculatedas

Ktt=0

and

Kii= S.Σ

AsseenfromoutsideofΣ,weobtain

1

Ktβ˙

t=F(SΣ)

and

Kii=β

1+f(r0),

whichisconsistentwithEqs.(3.15)and(3.17).

FromEq.(3.10),we nallyobtaintheequationsofmotionforthehypersurfaceΣas

dTΣ1+f(r0)

dt 2=1 F(SΣ)+f(r0),

=f(r0) S2Σ1+8 ακ2nM6(3.17)(3.16)(3.15)(3.14)(3.13)(3.19)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

Lemma3TheGB-LTBsolutioncanbeattachedatr=r0totheoutsideGB-SchwarzschildsolutiononlyinthesamebranchandthenM=m(r0).

˙2asProof.FromEq.(3.4),wealsoobtainSΣ

dSΣ

2 α 1 1Sn 1(n 2)Vn 2Σ .(3.22)

FromEqs.(3.21)and(3.22),itisfoundthatonlythesolutioninthesamebranchcanbeattachedandthen

M=m(r0)

issatis ed.2

IV.FINALFATEOFGRAVITATIONALCOLLAPSE(3.23)

Inthissection,weconsiderthe nalfateofgravitationalcollapse.Firstweconsiderthepossibilityofbounce.Wede nev(t,r)as

v≡S

(n m(r)12)Vn 2=rn 1M(r),(4.5)

(4.6)f(r)=r2b(r),

whereM0≡M(0)<∞andb0≡b(0)<∞aresatis edfortheregularityatthecenter.BythefollowingLemma,shell-focusingsingularitiesareneverformedintheplus-branchsolutionforn≥6.

Lemma4Bounceinevitablyoccursintheplus-branchsolutionforn≥6.

Proof.Eq.(3.4)gives

v˙2=b v2

1+4 αM

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

issatis edinthatcase.2

Weobtain

dG(v)1+4 αM/vn 1

1+4 αM/vn 1 (n 5) αM,(4.10)

whichreadsthatG(v)isadecreasingfunctionfortheminus-branchsolutionandtheplus-branchsolutionwithn=5.Since 11+4 αM(4.11)G(1)=b

isobtained,weassume

G(1)=b(r)

holdsfor0≤r≤r0.

Nowwecanshowthefollowingtheorem:11+4 αM(r)≥0 (4.12)

Theorem1Letusconsiderthen(≥5)-dimensionalgravitationalcollapseofasphericaldustcloudwithpositivemassinEinstein-Gauss-Bonnetgravitywithoutacosmologicalconstant.Then,thesingularityformedintheoutsidevacuumregionis(i)timelikeinthe ve-dimensionalplus-branchandinthe ve-dimensionalminus-branchwith0<m(r0)<3 αV31/(2κ2αV31/(2κ25)(ii)outgoing-nullinthe ve-dimensionalminus-branchwithm(r0)=3 5)(iii)1spacelikeinthen(≥6)-dimensionalminus-branchandinthe ve-dimensionalminus-branchwithm(r0)>3 αV3/(2κ25).Proof.TrivialfromLemma3,Lemma4andtheresultin[24].2

FromTheorem1,thefollowingcorollaryholds:

Corollary1Letusconsiderthe ve-dimensionalgravitationalcollapseofasphericaldustcloudwithpositivemassinEinstein-Gauss-Bonnetgravitywithoutacosmologicalconstant.Then,theweakcosmiccensorshiphypothesisisviolatedintheplus-branchandintheminus-branchwith0<m(r0)<3 αV31/(2κ25).

Nextletusconsiderthetrappedsurfacesinthecollapsingsolution.Inordertode nethetrappedsurface,weconsiderthetimeevolutionofthearealradiusalongafuture-directedradialnullgeodesic,whichwedenotedS/dt|+anddS/dt| foroutgoingandingoingnullgeodesics,respectively.Future-directedradialnullgeodesicsobey

dr1+f

dt

dt +˙+S′dr≡S2 α

˙+S′dr≡S

2 α 1 1Sn 1(n 2)Vn 2 1 1Sn 1(n 2)Vn 2 1/2+ 1/2

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

De nition1AtrappedsurfaceisametricspherewithdS/dt|+<0anddS/dt| <0.

De nition2Atrappedregionisaregionwheretrappedsurfacesexist.

De nition3Anapparenthorizonisaboundaryofatrappedregion.

Lemma5dS/dt| <0holdsinthecollapsingGB-LTBsolution.

Proof.TrivialfromEq.(4.15).2Lemma6Thereisnotrappedsurfaceinthecollapsingplus-branchGB-LTBsolution.

Proof.FromEq.(4.14),dS/dt|+<0isequivalentto

1+S2

2 α 1Sn 1(n 2)Vn 2(4.16)

intheplus-branchsolution,whichcannotbesatis edbecausetheleft-hand-sideispositivede nitewhiletheright-hand-sideisnegativede nite.2

ByLemma4,singularitiesareneverformedintheplus-branchsolutionforn≥6.Intheplus-branchsolutionwhenn=5,thesingularityformedshouldbenakedbyLemma6.Hereafterweconcentrateontheminus-branchsolution.Lemma7Inthecollapsingminus-branchGB-LTBsolution,atrappedregionisobtainedby

m>1(n 2)Vn 2

2κ2n

Proof.FromEq.(4.14),dS/dt|+≤0isequivalentto

F(t,r)≡1+S2(Sn 3+αS n 5).(4.18)

1+8 ακ2nm

2κ2n

2[Sn 3+αS n 5].(4.20)

Werepresentthetimewheneachshellreachesthesingularityast=ts(r).Theregularregionofspacetimeist<ts(r).ThefollowingLemmawillbeusedlater.

Lemma8t=ts(r)isanon-decreasingfunction.

Proof.Ift=ts(r)hasadecreasingportion,thenwecan ndconstantst1,t2,r1,andr2suchthatt1<t2and0<r1<r2with

S(t1,r2)=S(t2,r1)=0.(4.21)

˙<0,weobtainS(t1,r2)>S(t1,r1)andS(t2,r1)<S(t1,r1),respectively.TheseFromtheassumptionsS′>0andS

twoinequalitiesgive

S(t1,r2)>S(t2,r1),(4.22)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

whichcontradictsEq.(4.21).2Weconsiderwhetherthesingularityisnakedorcensored.Future-directedingoinggeodesics,whichsatisfydt/dr<0,cannotemanatefromthesingularitybyLemma8,whilefuture-directedoutgoinggeodesics,whichsatisfydt/dr>0,maydo.Lemma5andthecontrapositionofthefollowingLemmaimplythatitissu cienttoconsideronlythefuture-directedoutgoingradialnullgeodesicsto ndwhetherthesingularityisnakedorcensored.Theproofissimilartothefour-dimensionalcasein[27].

Lemma9Ifafuture-directedoutgoingcausal(excludingradialnull)geodesicemanatesfromthesingularity,thenafuture-directedoutgoingradialnullgeodesicemanatesfromthesingularity.

Proof.Inthespacetime(2.9),thetangenttoacausalgeodesicsatis es

e2Φ dt

dλ 2+L2

dλ 2≥e2Ψ dr

dr≥eΨ Φ,(4.25)

wherewetakethepositiverootforthefuture-directedgeodesics.Thisgives

dtCG

dr,(4.26)

wherethesubscriptsrepresentcausal(excludingradialnull)geodesicsandoutgoingradialnullgeodesics,respectively.Nowsupposethatt=tCG(r)extendsbacktoasingularitylocatedat(r,t)=(rs,ts(rs)).Letpbeanypointont=tCG(r)tothefutureofthesingularity.Applyinginequality(4.26)atp,weseethatthet=tRNG(r)throughpcrossest=tCG(r)fromaboveandhencepointstRNG(r)onthisradialnullgeodesicpriortopmustlietothefutureofpointsont=tCG(r)priortop,inthesenseoftRNG(r)>tCG(r)forr∈(rs,r ),wherer correspondstop.Thus,theradialnullgeodesics,whichnecessarilylieatt<ts(r),mustextendbacktothesingularityatr∈[rs,r ),andsomustemergefromthesingularity.2

Theorem2Inthecollapsingminus-branchGB-LTBsolution,massivesingularitiesforn≥6andsingularitieswithm>3 αV31/(2κ25)forn=5arecensored.

Proof.Eq.(4.17)givesthatthecentralsingularitieswithm>0forn≥6andm>3 αV31/(2κ25)forn=5areinthetrappedregion.ByLemma5andthecontrapositionofLemma9,itissu cienttoshowthatthefuture-directedoutgoingradialnullgeodesicscannotemergefromthesingularity.Past-directedingoingradialnullgeodesicsemanatingfromaspacetimeeventinthetrappedregioncannotreachthesingularityatS=0becausedS/dt|+<0holdsthere.Therefore,thereisnofuture-directedradialnullgeodesicemanatingfromthesingularityinthetrappedregion.2

ByTheorem2,ifnakedsingularitiesareformedintheminus-branchGB-LTBsolution,theyaremasslessforn≥6,whiletheyhavemasswith0≤m≤3 αV31/(2κ25)forn=5.Inthenexttwosubsections,weinvestigatethe nalfateofthegravitationalcollapsefurther.First,weconsiderthehomogeneouscollapsewithf=0asthesimplestcase;subsequentlywemovetotheinhomogeneouscase.

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

A.Homogeneouscollapse

Hereweconsiderhomogeneouscollapsewithf=0,whichisrepresentedbythe atFriedmann-Robertson-Walker(FRW)solution.Then,wehaveS=ra(t)and

ds2= dt2+a(t)2(dr2+r2d 2n 2),(4.27)

wherea(t)isthescalefactor.FromEq.(4.7),we ndthatonlytheminus-branchispossibleandM=M0,whereM0isapositiveconstant.Eq.(4.7)isthenreducedto

M0=an 3a˙2+α an 5a˙4.(4.28)

Theorem3Inthecollapsing atFRWsolutionwithadust uidinEinstein-Gauss-Bonnetgravity,thebig-crunchsingularityisspacelikeforn≥6,whileitisingoing-nullforn=5.

Proof.FromEq.(4.28),we ndthatthescalefactorabehavesas

a ( t)4/(n 1)(4.29)

neart=0,wherewesettheoriginoftcorrespondingtoa=0.Thebig-crunchsingularityisformedatt=0,wheretheKretschmanninvariantK≡RµνρσRµνρσdivergesas

K=O(1/( t)4).(4.30)

ThestructureofthesingularityisdeterminedbyEq.(4.29).Wetakethelineelementofthe atFRWsolutionwiththescalefactora=( t/t0)ptotheconformally atformas

ds2=a(t(η))2( dη2+dr2+r2d 2n 2),(4.31)

wheredη≡dt/a(t)andt0isaconstant.Therangeofηis ∞<η<∞and ∞<η<η0forp=1and0<p<1,respectively,whereη0isaconstant.Thus,thesingularityisspacelikefor0<p<1,whileitisingoing-nullforp=1[28].2

ByTheorems1and3,globalstructuresofthehomogenouscollapseofadustcloudwithf=0areshowninFig.1.Ingeneralrelativity,thisisthen-dimensionalOppenheimer-Snydersolution,whichrepresentsblack-holeformationforn≥4.InEinstein-Gauss-Bonnetgravity,thesolutionalsorepresentsblack-holeformationforn≥6andforn=5withm(r0)≥3 αV31/(2κ2αV31/(2κ25).Forn=5withm(r0)<3 5),ontheotherhand,thesolutionrepresentsgloballynakedsingularityformation.

B.Inhomogeneouscollapse

Nextweconsiderinhomogeneouscollapse.Firstweconsiderthedistributionoftheapparenthorizonaroundr=0.FromEq.(4.17),thetrappedregionisgivenby

M>r 2vn 3+α r 4vn 5.

FromEq.(4.18),theapparenthorizonisgivenby

M=r 2vn 3+α r 4vn 5.(4.33)(4.32)

ByTheorem2,onlythesingularityatr=0maybenakedforn≥6.Ontheotherhand,thesingularityatr=rswith0≤rs≤rahmaybenakedforn=5,whererahisde nedby

4rahM(rah)=α .(4.34)

FromEqs.(4.32)-(4.34),weobtain

4rsM(rs)≤α (4.35)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

(A)

(B)

(C)

(D)FIG.1:Globalstructuresofthen(≥5)-dimensionalhomogeneouscollapseofasphericallysymmetricdustcloudwithf=0.Zigzaglinesrepresentthecentralsingularities. +( )correspondstothefuture(past)nullin nity.BEHandCHstandfortheblack-holeeventhorizonandtheCauchyhorizon.Theglobalstructureisrepresentedby(A)forn≥6.Forn=5,theglobalstructuresare(B),(C),and(D)whichcorrespondtom(r0)>3 αV31/(2κ2αV31/(2κ2αV31/(2κ25),m(r0)=3 5),andm(r0)<3 5),respectively.(A),(B),and(C)representblack-holeformation,while(D)representsagloballynakedsingularityformation.Ingeneralrelativity,theglobalstructureisrepresentedby(A)forn≥4.

forn=5.

Nextwederivetherelationbetweenthetimewhenthesingularityoccursandthetimewhentheapparenthorizonappearsforeachr.Herewewritethemasterequation(4.7)again:

v˙= b v2

1+4 αMv1 n 1/2.(4.36)

Integratingthisequationwithrespecttov,weobtain

1

t(v,r)=

vdv1+4 αMv1 n

isanarbitraryfunctionofr.We ndt (r)=t(1,r),whereristobetreatedasaconstantintheaboveequationandt (r)meansthetimeforeachrwhenthearealradiusScoincideswiththecomovingradiusr.Wecansetsothatt (r)=ti,wheretiisaconstant,byusingthefreedomoftheradialcoordinatesuchasrt¯≡r¯(r).Thismeansthat

S=rissatis edatamomentt=ti,whichwehavede nedastheinitialtimeinEq.(4.2).Moreover,wecanset (r)≡0hereafter.ti=0byusingthefreedomofthetimetranslation.Thus,wesett

ThetimewheneachshellreachesthesingularityisgivenfromEq.(4.37)by

ts(r)= 1 (r), 1/2+t/(2 α)(4.37)dv

1+4 αMv1 n0 1/2./(2 α)(4.38)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

FromEq.(4.37),wealsoobtainthetimet=tah(r)whentheapparenthorizonappearsforeachrby

1dvtah(r)≡ 1/2,1 nvah(r)/(2 α)1+4 αMv 1 0dvdv= 1/2+ 1/2,1 n1 n0v(r)ah/(2 α)/(2 α)1+4 αMv1+4 αMv vah(r)dv(4.39)=ts(r) 1/2,1 n0/(2 α)1+4 αMv

wherev=vah(r)isgivenbysolvingEq.(4.33)algebraically.

Inthenexttwosubsections,weconsiderwhetherthesingularityappearingisnakedorcensored.Wetreatthesingularitiesatr=0inthecaseofn≥5andatr=rsinthecaseofn=5,separately.

1.Singularitiesatr=0forn≥5

Weconsidernotonlyregularbutalsosmoothinitialdataatthesymmetriccentersuchas

M(r)=M0+M2r2+M4r4+···,

b(r)=b0+b2r2+b4r4+···,

whereM2,M4,···andb2,b4,···areconstants.ExpandingEq.(4.37)aroundr=0,weobtain

t(v,r)=t(v,0)+r2(4.40)(4.41)

2 α 1 √

√b0 v21

χ2(0)+···.(4.45)2

ByLemma8,χ2(0)isnon-negativeandweassumeχ2(0)>0inthispaper.FromEqs.(4.37)and(4.42)andtherelation t t+0= rt

2 α1

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

Proof.ByTheorems1and

2,

only

the

singularity

at

r

=

0hasthepossibilityofbeingnakedforn≥6.Adecreasingapparenthorizoninthe(r,t)-planeisasu cientconditionforthesingularityformedtobecensoredbecauseitshowstheentrapmentoftheneighborhoodofthecenterbeforethesingularity.ExpandingtheintegrandinEq.(4.39)inapowerseriesinvandkeepingonlytheleadingorderterm,weobtain

tah(r)=ts0+r2

M0

n 1

(r rs)q=rv

dr+qx

1+(r rs)q rv˙.1+r2b(4.51)

Problemsappearinthenullgeodesicequationwhenweconsiderr=rs.

Weconsidertheneighborhoodofr=rs.NowletusconsiderthecaseinwhichΞisexpandedaroundr=rsas

Ng0(q)x+i=1gi(q)xαiΞ(x,r)

dr+(q g0(q))x

r rs+h(q,x)(r rs)a,(4.54)

Wede nethenewcoordinatesasu≡(r rs)βandx≡x¯γ,whereβandγarepositiveconstants.ThenEq.(4.53)becomes

dx¯

γβux¯=Ξ2(¯x,u),(4.55)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

whereΞ2isde nedby

Ξ2(¯x,u)≡ Nx(αi 1)γ+1

i=1gi(q)¯

γβx¯γ 1u(a β+1)/β.(4.56)

Thus,ifN=1,wechooseγsuchthatγ=1/(1 α1)andthenEq.(4.55)iswrittenas

dx¯

βu(¯x η)=ηΞ3(¯x,u),(4.57)

wherewehaveintroducedaparameter0<η<∞andΞ3isde nedby

Ξ3(¯x,u)≡ (1 α1)[(q g0(q))η g1(q)]

ηβx¯α1/(1 α1)u(a β+1)/β.(4.58)

Then,ifwechooseβandηsuchthatβ≤(a+1)/2andη=g1(q)/(q g0(q))≡η0,respectively,Ξ3isatleastC1inu≥0andx¯>0.ThenwecanapplythecontractionmappingprincipletoEq.(4.57)to ndthatthereexistsasolutionsatisfyingx¯(0)=η0,andmoreoverthatitistheuniquesolutionofEq.(4.57)whichiscontinuousatu=0.Theproofisavailablein[2,3,29].2

Thus,thereisnoothersolutionwith0<x(0)<∞underthesituationsstatedinLemma10,sothatotherpossiblesolutionsmustcorrespondtox(0)=0or∞.

Theorem5Letusconsidertheminus-branchGB-LTBsolutionwithpositivemassandsmoothinitialpro lessatis-fyingχ2(0)>0.Then,thesingularityformedatr=0islocallynakedfor5≤n≤8andconsequentlythestrongcosmiccensorshiphypothesisisviolated.

Proof.Inorderto ndwhetherthesingularitiesarenakedornot,weinvestigatethegeodesicequationforafuture-directedoutgoingradialnullgeodesicwhichemanatesfromthesingularity.We rstadopttheso-calledroot-equationmethodtoobtainthepossiblebehaviorofanullgeodesicaroundr=0ifitexists[4,30];subsequentlyweshowtheexistenceofsuchageodesicbyLemma10.

TheconstantqinEq.(4.49)mustsatisfyq>1inthiscase,inwhichrs=0,andisdeterminedbyrequiringthatxhasapositive nitelimitx0forr→0alongthefuture-directedoutgoingradialnullgeodesics.Itisnotedthattheregularcentercorrespondstoq=1.Fromthel’Hospitalrule,weobtain

x0=limS

dS

qrq 1

qrq 1

v+rv ′r→0

=limr→0 ˙ S′+eΨ ΦS √,rq(4.61)S=x0

qrq 1

for6≤n≤9,while v+ M0α 1/4 v(5 n)/4 (4.63)v=x0rq 1

x0=lim1M0

r→0

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

Forn=5,thepositive niterootofEq.(4.64)isobtainedwithq=3as

x0=1M0

8 4/(n 1) M0

5

α

bysolvingthealgebraicequation

x30+ M0 1/4 1/4 (4.67)α x0+χ2(0)M0

r M0x+α 1/4x(5 n)/4r2(5 n)/(n 1)x+ M0

r x+ b0+ α 1/2 χ2(0)+O(r0)(4.71)

forn=5.Wenotethat2(5 n)/(n 1)≥ 1withequalityholdingforn=9.Then,byLemma10,wecanshowtheexistenceofanullgeodesicemanatingfromthesingularitywhichbehavesasEq.(4.69)aroundr=0,wherex0isgivenbyEq.(4.65)and(4.66)forn=5,6≤n≤8,respectively.However,wecannotapplyLemma10tothecaseofn=9,soothermethodsarerequiredtoshowtheexistenceofthenullgeodesicinthatcase.

Finallyweshowthatthecurvaturescalarsactuallydivergealongthenullgeodesic,i.e.,itisactuallyasingularnullgeodesic.Theenergydensityofadust(3.3)iswrittenas

ρ=[(n 1)M+rM′]

2κ2n[(n 2)R+α(n 4)LGB].(4.73)

Theshell-focusingsingularityisherecharacterizedbyv=0,r=0,M=M0,andM′=0forn≥5,thusρdivergesifrv′<∞issatis edforr→0alongthenullgeodesic(4.69).FromEq.(4.47),weobtain

v ′ M0

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

for6≤n≤9,while

v′ b0+ 1/2rχ2(0)(4.75)α

forn=5alongthenullgeodesic(4.69),sothatweobtainrv′→0forr→0.Therefore,ρandthecurvaturescalarsdivergealongthenullgeodesics(4.69).2

Unfortunately,the xed-pointmethodcannotbeappliedtoshowtheexistenceofthesingularnullgeodesicsforn=9.Iftheyexist,theybehaveasEq.(4.69)aroundr=0,wherex0isgivenbyEq.(4.68).

Thegeneralrelativisticcasehasbeeninvestigatedbyseveralauthors[18].WereviewtheiranalysesinAppendixAforcomparisonwiththecaseinEinstein-Gauss-Bonnetgravityandgivesomecomplements.

2.Singularitiesatr=rs>0forn=5

Nextweconsidersingularitiesatr=rswith0<rs≤rahinthe ve-dimensionalcase.WeexpandMandbaroundr=rs>0as

¯0+M¯1(r rs)+M¯2(r rs)2+···,M(r)=Mb(r)=¯b0+¯b1(r rs)+¯b2(r rs)2+···,

¯0,M¯1,M¯2,···and¯whereMb0,¯b1,¯b2,···areconstants.

Expandingt(v,r)inEq.(4.37)aroundr=rs,weobtain

t(v,r)=t(v,rs)+(r rs)¯χ1(v)+···,

wherethefunctionχ¯1(v)isde nedby

χ¯1(v)≡ 1(4.76)(4.77)(4.78)

v¯¯1/b1+M ¯2b0 v2

¯b0 v2

,q(r rs) S=x0(r rs)q 1 =lim,r→rsdr + S=x0(r rs)q v+rv′√=limr→rs(4.82)(4.83)

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

FromEqs.(4.37),(4.46),and(4.78),weobtain

1

b

v′=

¯

¯0v

4+4 αM 1/2

(¯χ1(v)+···)(4.85)

nearr=rs.

FromEq.(4.84)usingEqs.(4.36)and(4.85),weobtainthedesiredrootequation

¯¯rs(b0+(b0+x0=lim1 . 21/2r→rsq(r rs)q 1(¯b0+rs)(4.86)

2¯0/αTheterminthelargebracketinEq.(4.86)isnon-negativebecauseEq.(4.35)gives¯b0+rs≥¯b0+(M )1/2with¯0/αequalityholdingforrs=rah=(M )1/2.FromEq.(4.86),weobtain

¯x0=rsχ¯1(0)b0+¯0/αM )1/2

r rs b0+χ¯1(0)¯¯0/αM )1/2

1/2¯0/αM χ¯1(0),(4.90)

sothatrv′<∞forr→rsissatis edalongthenullgeodesic(4.88).Asaresult,we ndfromEqs.(4.72)and(4.73)thatthecurvaturescalarsdivergealongthisnullgeodesic(4.88)andthereforeitisasingularnullgeodesic.2

V.DISCUSSIONANDCONCLUSIONS

Inthispaper,wehaveinvestigatedthe nalfateofthegravitationalcollapseofadustcloudinEinstein-Gauss-Bonnetgravity.First,wehaveadoptedthecomovingcoordinatesandde nedascalaronM2,ofwhichdimensionismass,andgivenasimpleformulationofthebasicequationsforthen(≥5)-dimensionalspacetimeM≈M2×Kn 2includingaperfect uidandacosmologicalconstantinEinstein-Gauss-Bonnetgravity.Then,havingusedthisformalisminthesphericallysymmetriccasewithoutacosmologicalconstant,wehaveinvestigatedthe nalfateofthen(≥5)-dimensionalgravitationalcollapseofadustcloud.Wehaveassumedthat(i)thecouplingconstantoftheGauss-Bonnettermαispositive,whichisrequiredbyheteroticsuperstringtheory,(ii)positivemassofadust uid,(iii)smoothinitialdataaroundthesymmetriccenter,and(iv)S′>0,i.e.,noshell-crossingsingularities.

Therearetwofamiliesofsolutions,whichwecalltheplus-branchandtheminus-branchGB-LTBsolutions,respec-tively,althoughwehavenotobtainedanexplicitformofthesolutions.Inthegeneralrelativisticlimitα →0,theminus-branchsolutionisreducedtothen-dimensionalLema itre-Tolman-Bondisolution[18,21,31].Onthecontrarythereisnogeneralrelativisticlimitfortheplus-branchsolution.TheGB-LTBsolutionscanbeattachedatthe niteconstantcomovingradiusr=r0>0totheoutsidevacuumspacetimerepresentedbytheGB-SchwarzschildsolutionsinthesamebranchwithamassparameterM=m(r0).

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

Intheplus-branchsolutionwithn≥6,bounceinevitablyoccursandconsequentlysingularitiesareneverformed.Forn=5,atimelikenakedsingularityappearsintheoutsidevacuumregion,whichisasymptoticallyanti-deSitterinspiteoftheabsenceofacosmologicalconstant.Sincethereisnotrappedsurfaceintheplus-branchsolution,thesingularityformedinthedustregionmustbenaked,too.

Intheminus-branchsolution,theoutsidevacuumregionisasymptotically at.Asinthecaseofanulldust uid[17],the nalfateofgravitationalcollapseisquitedi erentdependingonwhethern=5orn≥6.Amassivenakedsingularityisformedforn=5,whichisprohibitedforn≥6andingeneralrelativity.

Intheoutsidevacuumregion,amassivetimelikenakedsingularityappearsifthemassofadustcloudsatis es0<m(r0)<3 αV31/(2κ2αV31/(2κ25)forn=5.Ifeithern≥6orn=5withm(r0)>3 5)issatis ed,thesingularityis12spacelike.Inthespecialcaseofn=5withm(r0)=3 αV3/(2κ5),itisoutgoing-null.

Wehavetreatedthehomogeneousandinhomogeneouscasesintheminus-branchsolution,separately.Inthehomogeneouscollapserepresentedbythen-dimensional atFriedmann-Robertson-Walkersolution,thesingularityformedinthisFriedmannregioniscensored,whichisspacelikeforn≥6andingoingnullforn=5.Asaresult,thestrongCCHholdsforn≥6orn=5withm(r0)≥3 αV31/(2κ25),whiletheweakCCHisviolatedforn=5with120<m(r0)<3 αV3/(2κ5).Theglobalstructureforn≥6isthesameasthatofthegeneralrelativisticcase,whichisthen-dimensionalOppenheimer-Snydersolutionrepresentingblack-holeformation.

Intheinhomogeneouscollapsewithsmoothinitialdata,thestrongCCHholdsforn≥10.Forn=9,thestrongCCHholdsdependingontheparametersintheinitialdata.Ontheotherhand,anakedsingularityisformedinthedustregionfor5≤n≤8.Thenakedsingularityismasslessfor6≤n≤8,whileitismassiveforn=5.Thus,atleastthestrongCCHisviolatedfor6≤n≤8orn=5withm(r0)≥3 αV31/(2κ25),whiletheweakCCHisviolatedfor21n=5with0<m(r0)<3 αV3/(2κ5).Althoughwehavenotsolvedthenullgeodesicequationinthewholespacetime,thesingularitycanbegloballynakedifwetakethelimitr0→0+εfor6≤n≤8andr0→rah+εforn=5,whereεisasu cientlysmallpositiveconstantsuchthatalightrayemanatingfromthesingularityreachesthesurfacer=r0intheuntrappedregion.Thenthelightraycanescapetoin nityandconsequentlytheweakCCHisviolated.

Unfortunately,wecouldnotshowtheexistenceofsingularnullgeodesicsforn=9althoughwecouldshowtheirpossiblebehaviornearthesingularity.Thesituationisthesameforn=5ingeneralrelativity,asshowninAppendix

A.Otherapproachessuchasthecomparisonmethodarerequiredtoshowthenakednessofthesingularityinthesecases[5].

Ingeneralrelativity[18,21]withsmoothinitialdata,theweakCCHholdsforn≥6[18,21].Thepresentresultandtheresultin[17]implythatthee ectsoftheGauss-BonnettermworsenthesituationfromtheviewpointofCCHratherthanpreventnakedsingularityformation.

TheformationofmassivenakedsingularitiesinEinstein-Gauss-Bonnetgravityforn=urdimensionswithsphericalsymmetry,ithasbeenshownbyLakeunderverygenericsituationswithoutusingtheEinsteinequationsthatmassivesingularitiesformedfromregularinitialdataarecensoredbyadoptingtheMisner-Sharpmassasaquasi-localmass[32].Aswasalreadypointedoutin[17],thehigher-dimensionalcounterpartoftheMisner-Sharpmassisnotanappropriatequasi-localmassinEinstein-Gauss-Bonnetgravity,andthereforehisresultdoesnotcon ictwithours.Lemma3impliesthat,inEinstein-Gauss-Bonnetgravity,ourde ke’sresultmustbeextendedinEinstein-Gauss-Bonnetgravityforn≥6byadoptingourquasi-localmass.

Theformationofmassivesingularitiesinodddimensionsisconsideredtobeacharacteristice ectofLovelocktermsintheaction,whichincludetheGauss-Bonnettermasthequadraticterm[33].TheGauss-Bonnettermbecomes rstnontrivialin vedimensions,sothatmassivesingularitiescanbeformedonlyin vedimensions.Ifweaddthehigher-orderLovelocktermsintheaction,massivenakedsingularitiesmustbeformedinallodddimensions.

Inthispaper,wehaveassumedthattheequationofstateofmatterisdustinthe nalstagesofgravitationalcollapse.Althoughitisaverystrongassumption,itisnotcompletelyruledout[34].Actually,wehavelittleknowledgeoftheequationofstateintheveryadvancedstageofgravitationalcollapse.However,ofcourse,thee ectsofpressureshouldbeinvestigated.Ingeneralrelativity,therehavebeenmanystudiesincludingpressurebothinfourdimensions[35,36]andinhigherdimensions[37,38,39,40].

Amongthem,ascalar eldmustbeespeciallyaddressedinthehigher-dimensionalcontextbecauseitarisesnaturallyinsupergravity[41]andplaysacentralroleinmoderncosmology[42].Inthebrane-worldscenario,ascalar urdimensions,theCCHhasbeenproveninsphericallysymmetriccollapseofamasslessscalar eldingeneralrelativity[36].Thisresultwillholdinhigherdimensionsbecausehigher-dimensionale ectsmakesingularitiescensored.However,it’sjustconceivablethatthee ectsoftheGauss-Bonnettermwillchangedrasticallythewholepictureandthe nalfateofgravitationalcollapse.

Amongscalar elds,adilaton eldisparticularlyofinterestbecauseitnaturallyarisesinthelow-energye ectivetheoryofsuperstringtheory[16,43].Indeed,itiscoupledtotheGauss-Bonnetterm,andconsequentlytheGauss-Bonnettermdoescontributetothe eldequationseveninfourdimensions[16,44].Inanycase,thequasi-localmassde nedinthispapershouldbeusefulinthestudyofgravitationalcollapseinEinstein-Gauss-Bonnetgravity.

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

Lastly,severalnotesshouldbemade.Inthispaper,wehaveassumedsmoothinitialdataaroundthesymmetriccenter.Analyseswithmoregeneralinitialdataareneededtodeterminewhetherthenatureofthesingularitiesandthepictureofgravitationalcollapseobtainedinthispaperaregenericornot.Also,thestructureandthestrengthofthenakedsingularitiesarestillopen.Inthecaseofanulldust,theGauss-Bonnettermweakensthestrengthofthenakedsingularityandthenakedsingularityistimelikeandingoingnullforn=5andn≥6,respectively.Thesestudieswillbereportedelsewhere.

Acknowledgements:TheauthorwouldliketothankM.Narita,T.Harada,andM.Nozawafordiscussionsandusefulcomments.

APPENDIXA:GENERALRELATIVISTICCASE

Inthisappendix,wereviewthegeneralrelativisticcaseforn≥4andgivesomecomplementstothepreviousstudiesforn≥5[18].TheequationsingeneralrelativitycanbeobtainedfromthoseinEinstein-Gauss-Bonnetgravitybythelimitofα →0.

Firstweconsiderthedistributionoftheapparenthorizonaroundr=0.FromEq.(4.17),thetrappedregionisgivenby

M>r 2vn 3.

FromEq.(4.18),theapparenthorizonisgivenby

M=r 2vn 3,(A2)(A1)

sothatonlythesingularityatr=0maybenaked,inotherwords,nomassivenakedsingularityexists.

Wederivetherelationbetweenthetimewhenthesingularityoccursandthetimewhentheapparenthorizonappearsforeachr.FromEq.(4.36),weobtain

1/2.v˙= b+Mv3 n

1(A3)Integratingthisequationwithrespecttov,weobtain

t(v,r)=dv

v

(b+1/2Mv3 n).(A5)

FromEq.(A4),wealsoobtainthetimet=tah(r)whentheapparenthorizonappearsforeachrby

tah(r)≡ 1dv

0vah(r)

(b+Mv3 n)1/2

(b+

wherev=vah(r)isgivenbysolvingEq.(A2).

ExpandingEq.(A4)aroundr=0,weobtain

t(v,r)=t(v,0)++dvvah(r)1/2Mv3 n),(A6)r2

We give a model of the higher-dimensional spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation of the basic equations is given for the spacetime $M \approx M^2 \times K^{n-2}$ with a perfect fl

wherethefunctionχ2(v)isnowobtainedby

χ2(v)≡ 1

v

where

the

integrandis nitefor0≤v≤1.The

timewhenthecentralshellreachesthesingularityisgivenfromEq.(A4)by

ts0= 1 3/2 b2+M2v3 ndv,b0+M0v3 n(A8)dv

2χ2(0)+···.(A10)

ByLemma8,χ2(0)isnon-negativeandweassumeχ2(0)>0here.

FromEqs.(A4),(A7)and(4.46),weobtain

1/2 (rχ2(v)+···)v′=b0+M0v3 n(A11)

nearr=0.

Theorem7Letusconsiderthegravitationalcollapseofasphericaldustcloudwithpositivemassandsmoothinitialpro lessatisfyingχ2(0)>0ingeneralrelativitywithoutacosmologicalconstant.Then,thestrongcosmiccensorship

1/2hypothesisholdsforn≥6andn=5if0<χ2(0)<M0.

Proof.FromEq.(A1),onlythesingularityatr=0hasthepossibilityofbeingnaked.Adecreasingapparenthorizoninthe(r,t)-planeisasu cientconditionfortheformedsingularitytobecensoredbecauseitshowstheentrapmentoftheneighborhoodofthecenterbeforethesingularity.ExpandingtheintegrandinEq.(A6)inapowerseriesinvandkeepingonlytheleadingorderterm,weobtain

tah(r)=ts0+r2

M01/(n 3)(n 1)/(n 3)

n 1r.(A12)

Forn≥6,thelasttermdominatesthesecondterm,andconsequentlytheapparenthorizonisdecreasingnearr=01/2inthe(r,t)-plane.Forn=5withχ2(0)<M0,theapparenthorizonisalsodecreasingnearr=0inthe(r,t)-plane.2

Nextweconsiderthenakednessofsingularitiesforn=4and5.WeshowthenakednessofsingularitiesinthesimilarmannertothecaseinEinstein-Gauss-Bonnetgravity.

Theorem8Letusconsiderthen(≥4)-dimensionalgravitationalcollapseofasphericaldustcloudwithpositivemassandsmoothinitialpro lessatisfyingχ2(0)>0ingeneralrelativitywithoutacosmologicalconstant.Then,thestrongcosmiccensorshiphypothesisisviolatedforn=4.

ingEqs.(A3)and(A11),weobtainfromEq.(4.62)thedesiredrootequation

x0=lim1

r→0

4 2/3.(A14)

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