A Connection of Apparent Horizon and Naked Singularities in Spherical Dust Collapse

SukratuBarve1

TheInstituteofMathematicalSciences

Chennai

Abstract

Weshowthatthebehaviouroftheoutgoingradialnullgeodesiccongruenceontheapparenthorizonisrelatedtothepropertyofnaked-nessinsphericaldustcollapsejustifyingthedifferenceinthePenrosediagramsinthenakedandcovereddustcollapsescenarios.Weprovideargumentssuggestingthattherelationshipcouldbegenerallyvalid.

1Introduction

Consideracloudofmatter(regularinitialCauchydata)collapsingin-definitelyunderitsowngravity.AsingularityeventuallydevelopsinthespacetimeanditisindicatedbythedivergenceoftheKretschmannscalar.Intheadvancedstagesofcollapsetrappedregionsareformed[1],[2]andthereexistsanullraywhichmarginallyescapestoinfin-ity(eventhorizon).Itisnotclearwhetherthesingularboundaryisentirelysurroundedbythetrappedregion.Inotherwords,itisnotknownifaportionoftheboundaryisexposedintheuntrappedregionandnon-spacelikegeodesicscanemanatefromit(nakedsingularities).ExactsolutionstoEinstein’sfieldequationswithcertainkindofsourcetermsareknowntoexhibitbothnakedandcoveredsingular-itiesdependinguponthesortofregularinitialdatachosen.Littleprogresshasbeenmaderegardingageneralclassificationofinitialdataaccordingtothecoveredornakedconsequenceoftheevolution.Itisnotknownifthedataleadingtonakedsingularitieshaszeromeasureinthesetofallpossible(orallpossiblephysicallyrelevant)Cauchydata.

ThecomplexityoftheproblemliesinthefactthattheCauchyinitialvalueproblemfortheEinsteinfieldequationswithsourcesislesstractable.Thesystematicsavailableaboutthegeneralproblemisfartoolessforanyimplicationforquestionslikeformationofnakedsingularities.Forinstance,eventhewellposednessoftheproblemisnotselfevidentandhastobeprovedindependentlyfordifferenttypesofsources[3],[4],[5].ItwouldbeindeeddifficulttofindorevenexpectaconservedormonotonicallybehavingfunctionoftheCauchysurfacewithrespecttoitsevolution,whichcouldbeexpectedtoprovideinsightintotheprocessofcreationofnakedsingularities.Asaresultofthisdifficulty,alargenumberofinvestigationsthathavebeencarriedouthavebeenconcerningcertainexactsolutionsornumericalsimulations.Issueslikestrengthofthesingularities,gener-icity,behaviourwithrespecttochangeofsourceetc.havebeenstud-iedinexampleslikedust,nulldust,perfectandimperfectfluidsandscalarfields.However,theydonotsuggestanytypicalgeometricalfeaturewhichcouldbeexpectedtoarisebeforeanakedsingularityforms(Incaseofasingularitythesingularitytheoremsmakeuseofatypicalgeometricalfeatureviz.trappedregionstoproveitsexis-tence).Suchafeatureindicatingthepresenceofanakedsingularity

2

wouldbeinterestinginthelightoftheHoopconjectureortheissueofisoperimetricinequalitiesandcouldleadtosomegeometricalinsightintotheprocess.Thelackofindicationofexistenceofsuchafeatureisevidentinthefactthatoneisforcedtocheckfortheexistenceofnakedsingularitiesinadirectmannereveninparticularexamples.Tobeprecise,onechecksifnonspace-likegeodesicsemergefromthesin-gularboundaryusingdifferentialgeometryandtheformofthemetricintheexample.

Thispaperisafirststeptowardsanindirectcriterion.Preferably,thecriterionshouldbeapplicableawayfromthesingularboundary.Thatisanon-localproblemandgiventhedifficultiesofCauchyevolu-tion,thereisnoindicationavailableforwhatthecriterioncouldbe.Itwouldbeperhapsappropriateinsuchasituationtoexamineregionsnearthesingularity.Therearealsopartsofthesingularityfromwhichgeodesicscannotescape.Somecriterionapplicabletosuchaportionwouldalsobesignificant.Itwouldindicatethattheinformationabouttheexposureofapartoftheboundaryiscontainedelsewhereontheboundary.

Weillustratethemainfeaturesofsphericaldustcollapseinradialco-ordinates(figures1and2)andincausallycorrectPenrosediagrams(figures3and4).

Infigure3,thereisnoportionoftheboundarybeyondOthatisexposed.Infigure4,however,thereisanullportion.Inthispaper,weexamineifOwouldyieldtheinformationabouttheexistenceofanexposedportion.Oisacoveredpointofthesingularboundary(withorwithoutanakedportion).Thispointiswheretheapparenthorizonmeetsthesingularity.Mathematically,oneworkswithpointsontheapparenthorizonintheapproachtotheconcernedpoint.Thiscouldbelookeduponasapropertyofthemarginallytrappedregionswhichconstitutetheapparenthorizon.Oneisthereforeworkingwithoutgoingnullcongruenceswithzeroexpansion.Ifasingularitymeetssucharegion,thepointinthestrictsensewillnothaveanynon-spacelikegeodesicsemergingfromitselfandwillthereforebecovered.Weprovethecriterionforthedustcase.Weofferasimplejusti-ficationforthedifferenceinthePenrosediagramsinthenakedandcovereddustcases.However,thecriterionisnotdustspecificandshouldbefurtherexaminedingeneralcases.

Theplanofthepaperisasfollows.Wefirstmotivatetheviewthatthetangentvectortoageodesicinacongruencerepresentsa’fluxden-

3

sity’analogoustothesituationinordinaryelectrostaticswhereonehaslinesofforce.Wewishtoinvestigatethetangentvectorontheapparenthorizon.Thenextsectiondescribestheselfsimilardustmodelwherewediscusstheaboveanddemonstratetheconnectionwithnakedness.Inthenextsectionweshowthattheresultscanbeextendedtothegeneraldustcase.FinallyweturntotheconformaltransformationleadingtothePenrosediagramforthenakedcasefromsphericalco-ordinatespointingoutthatitshoulddivergeatthesingu-larboundary.Wearguethatinageneralcase,thisdivergencewouldpreventingoinggeodesicsfromreachingthepointofinterestwhichwouldthensuggestthatanexposedingoingnullboundaryexists.

2FluxofCongruence

Thescenarioofsphericaldustcollapseisshownintheradialco-rdinatesinfigures1and2.Observethatinthenakedcase(figure2)severalgeodesicsemergefromthecentrebetweentheeventandap-parenthorizonsandfallintothetrappedregioncrossingtheapparenthorizon(Typicallythisisacaseofalocallynakedsingularity2).Allalongtheapparenthorizonintheapproachtothecentre,onewouldfindnullgeodesicswhichwouldhaveoriginatedatthenakedcentre.Asdrawninthefigure,theyappeartointersecttheapparenthorizoninasequenceofpointswhichagglomerateintheapproachtothecen-tre(intheEuclideansenseofthefigure).Ifthisideaofagglomerationcouldbemadeprecisethenitwouldbeofinteresttostudyifthey(orthegeodesics)tendtoclusternearapointontheapparenthorizon.Suchapropertywouldperhapscontaininformationaboutthesource,ifexistent,forthelinesofthecongruenceinamanneranalogoustoelectrostatics.Onecouldexpectanenhancementofclusteringofthenullcurvesinthenakedcaseasthereisasourceoflinesinthevicinity(nakedportionofthesingularboundary).Nosuchsourceatthecentreexistsinthecoveredcaseandthecongruencecouldperhapsexhibitadifferentcharacteristicbehaviour.Motivatedbythis,wedefinethequantityoffluxdensity.GivenamanifoldM,onedefinesacongruencetobeasetofcurvessuchthattheirunionisMandeachpointofMhasasinglecurvecontainingit.Thisisanalogoustotheconceptof

linesofforceinelectrostatics.Sowedefineaquantityanalogoustoelectrostaticfluxdensityforageodesicscongruence.Wefirstaffinelyparametrizethecongruenceandtreatthetangentvectorξliketheelectricfieldoftheanalogy.Forany3dimensionalhypersurfacewedefineξµdsµtobetheinfinitesimalfluxacrosstheinfinitesimalthreeareadsµofthehypersurface.Thefluxdensityisobviouslythetangentvectoritself.Itisthistangentvectorthatweexamineinthelimitofapproachtothesingularity(alongtheapparenthorizon).

Itmightappearfromthemotivationinthepreviousparagraphthatonewouldattempttocheckforsourcesoftheflux(liketaking∇horizon.Einelectrostatics).inanattempttoInsteadmoveawayofthis,fromwetheexaminesourceit(nakedonthepartapparentofthesingularboundaryorsetof‘points’wherecausalcurvesoriginateonthesingularityormorepreciselysetofidealpointstoT.I.F.s[2])forsearchingforanon-localcriterion.Wefindthatthoughweareunabletoformulateanycriterionawayfromthesingularity,weareabletoexaminethepointwheretheapparenthorizonmeetstheboundarywhichisalways(marginally)trappedandcanneverbeasource.

Thegeneralexpressionfortangentvectorisnotavailableinaclosedanalyticform.However,onecancalculateitinaspecialcasewhichwedescribebelowandsubsequentlyshowhowtogeneralizetheresults.

3SelfSimilarDustModel

Thecollapseofasphericalcloudofpressurelessfluidisgivenbythefollowingmetric[6])

ds2

=dt2

−

R′2

whichresemblesarelationbetweenkineticandgravitationalpotentialenergiesofashell.

R

˙2=F√

F

󰀋

(3)

whereasingularityboundaryisformedatt=t0(r).Thecentral

shellfocussingsingularity,whichisthelimitasr→0alongthislocusisofinterestandturnsouttobenakedforsomeinitialdata.

Theself-similarmodelistheoneinwhichF(r)=λrwhereλisaconstant(whichdecidesifthecentralsingularitywillbenakedornot)andf(r)=0.3

Wechoosethescalingt0(r)=r.Aselfsimilarco-ordinatez=t/risintroduced.WenotetheexpressionsforRandR′whichwillbeusefulinthesubsequentanalysis

R=rλ−2/3(3/2(z−1))2/3(4)

R′=

󰀈

2λ/3

2

󰀋

(5)

Wecastthemetricintodoublenullco-ordinates.Itisnotdifficulttoshowthatds2=r2󰀆z2−R′2󰀇

dudv(6)where

du=dr

z−R′(z)(7)dv=

dr

z+R′(z)

(8)

Thedoublenullform(ds2=C2(u,v)dudv)turnsouttobeuseful

whenaffineparametersalongnullgeodesicsaretobecalculated.Forinstance,alonganoutgoingradialnullgeodesic(du=0),theaffineparameteris󰀊

u=constantC2dvuptoamultiplicativeandanadditiveconstant.

∂t

+r

∂

Nowletusturntocalculatingthetangentvectortotheoutgoingnullradialgeodesiccongruence,whichisourprimaryinterest.Assumethevectortobeoftheform

ξ=(Q(t,r),Q(t,r)

󰀄

˙′/2R′=01+f/R′+Q2R

(10)

Wehaveprovidedtheexpressionsforthemostgeneraldustcase

here.Onemayreadofftheexpressionsfortheselfsimilarcasebysettingftozeroandusingequation5forR′.Theequationabovetakestheform

1/Q=

󰀅

˙′R

u=constant

(z−3)23

z−1

󰀃1/3

dz(13)

orusingthefactthatdu=0fromequation(du)

󰀅

1/Q=

1

u=constant

z2−1

󰀂

2λ/3

b−R′(b)

󰀁

z

dz(14)

Theintegraloverzistobeevaluatedfromr=0totheapparent

horizon,whereweshallbeinterestedinevaluatingthetangentvector.ThelattercanbeshowntobethecurveR=Fandturnsouttobethelocusz=1−2λ/3.

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Theintegralbeingoverz,itisimportanttoknowwhichvalueofzalongtheoutgoingnullcurveyieldsr=0,thelowerlimitoftheintegral.ThisissueasitisshownfurtherleadstothedifferenceinthebehaviourofQinthenakedandcoveredcases.Considerthentheequation

r=e

󰀉󰀊

−

db

b−R′(b)=0hasnorealroot.

Theintegrandthereforedoesnotdivergeanywhereandalsoremainspositive(orentirelynegative)allovertherealline.Itcanbecheckedthatb−R′(b)>0foranyonerealbwhichwouldbesufficienttoclaimthattheintegrandispositive.Also,b−R′(b)isboundedsincebistobelimitedtothenonsingularregionz<1(z=1isthesingularitycurveitself).So,inorderthattheintegraldiverge,therangeofintegrationshouldbeinfinite.Wehavechosentolimitthefinalpointtotheapparenthorizonz=1−2λ/3andhencetheinitialpointmustbez=−∞.

Infactashorterintuitiveargumentispossible.ItisknownfromtheTolmanBondidustmodelthatcentralsingularityformsat(t0(0),0).IfoneassumesthePenrosediagramforthecoveredcase(whichisindeedwhatthiscaseturnsouttobe),thenullrayscrossingtheap-parenthorizonbeginatthecentreattCase(ii)

r=(z−z−)

1

db

󰀋

(17)

b=z−

(Itcanbeeasilycheckedthatα<0)Atz=z−,therefore,rvanishes.

Thusinconclusionofthisanalysis,wenotethatthelowerlimitofintegralfor1/Qdiffers.Itis−∞whenR′(b)−b=0canneverhavearealsolutionandistheroot(closesttoapparenthorizon)whenasolutionexists.

Thisobservationplaysthekeyroleinfurtheranalysis.Makingnoteofthisconsiderequation(1/Qsecondone).Analyzingthevariousfactorsintheintegrandonefindsthattheintegrandwoulddivergeifz=−1(z=1sincewearenotonthesingularboundary).

zwilltakethevalue−1incasei.Incaseii,thefollowingtakesplace.Considerb−R′usingequation(R’).Itiseasytoseethatb−R′<0forallb<0.So,therootb−cannotbenegative.Henceitiscertainlygreaterthan−1.Thuszcannottakethevalue−1incaseiiintheintegralfor1/Q.

Thustheintegranddivergesas1/(z+1)incaseiandisfiniteincaseii.

ExpandingtherestoftheintegrandfactorinaTaylorseriesaboutz=−1,onecaneasilycheckthatintegraldivergeslogarithmicallyincaseiwhilestayingfiniteincaseii.

Thus,Qvanishesincaseiiandstaysnonzero(andfinite)incaseii.

Returningnowtoequation9,wecannowseethatξbehavesindifferentwaysincaseiandcaseiiontheapparenthorizon,inparticularasoneapproachesthepointOonthePenrosediagrams√shown(figures3and44).Itcanbecheckedthatthefactor

Figure4isadiagramofalocallynakedsingularity.Theselfsimilarcloudwhichweexaminehereturnsouttobegloballynaked.However,thestructurenearOisthesameasanylocallynakedcaseandfigure4canbeused.

4

9

Frompreviousanalysisofnakedsingularities(selfsimilarcases)usinganalysisforemergenceofgeodesics(rootsanalysis),itcanbecheckedthatcaseicorrespondstothecoveredcaseandcaseiicorre-spondstothenakedsingularmetric.

4Extensiontothegeneraldustcase

Inthegeneraldustcase,theequation10yieldsnoclosedanalyticsolutionwhichwouldhaveclearlybeenuseful.However,wenotethatweareinterestedonlyinthebehaviourofQinthelimitofapproachtopointOontheapparenthorizon.

Tothisendthefollowingobservationplaysanimportantrole.Itisshownthatgivenadustsolution,onecanconstructamodifieddustsolution(modifieddistribution)whichinasuitablelimitapproachesthegivendustsolution[7].Thekeyresultthatmakesthisconstructionusefulisthatitisprovedthatnakedmodifieddistributionsreproducenakeddustsolutionsgivenandcoveredmodifieddistributionsrepro-ducecoveredones.Onecanthenworkwiththemodifieddistributionforthegivendustsolutionandtakethelimitwhichpreservesnakedorcoverednature.Weoutlinetheconstructionin[7]belowa)Marginallyboundcase(f=0)

ImagineashellofradiusrcinthegivenTolmanBondidustmodel.Replacetheinterioroftheshellbyaselfsimilardustmetric,matchingthefirstandsecondfundamentalformsattheinterfacer=rc.Itcanbeshownthatthisrestrictstheselfsimilarityparameterλwhichappearsinthemassfunction.Thisspecifiestheselfsimilarsolutioncompletely.Nowtakingthelimitasrctendstozero,onecanshow[7]thatthematchingconstraintdoesimplythattheinteriorselfsimilarsolutionstaysnakedinthelimitiftheoriginaldustsolutionwasnakedandlikewiseinthecoveredcase.

b)NonMarginallyboundcase(f=0)

Theconstructionissimilarinthiscaseexceptforanadditionalinterface.Twoshells,rc1andrc2(sayrc110

secondfundamentalformsateachoftheinterfaces.Asbefore,thiscanbeshowntoconstraintheinteriorselfsimilarsolutionuniquelygivenrc1andtheoriginaldustsolution.Again,thepropertyofbeingnakedorcoveredispreservedinthelimit(rc2→0)likethepreviouscase[7].

WenowconsiderQinthemodifieddistributionforanygivendustsolution.Intheselfsimilarpartofthelatter,resultsoftheprevioussectionapply.Sincethecongruenceofoutgoinggeodesicsissmooth,soisQ.ThismakesQcontinuousacrosstheinterface/sinthemodifieddistribution.Nowimaginethegivendustsolutionasthelimitingcaseofthemodifieddistribution.InthelimitofapproachtopointOontheapparenthorizon,onehastoevaluateQintheselfsimilarpart.BecauseofcontinuityofQ,thesamebehaviourwillcontinuetoholdinthelimitoftheinterface/stendingtozerowhentheoriginaldustsolutionisreproduced.Makinguseofthefactthatthepropertyofbeingnakedorcoveredispreservedinthislimit,oneconcludesthatthebehaviourofQintheselfsimilarnakedandcoveredcasescontinuestoholdinthegeneraldustscenarioaswell.

5ConformaltransformationandPen-rosediagram

Thetendencyofthenullgeodesicsofthecongruencetoclusterintheapproachtothesingularboundaryisbasicallyduetotheinappropri-atenatureoftheco-ordinatesystemattheboundary.Ifonewishestodepicttheboundaryasacurveinaparticularco-ordinatesystem,thenullcongruencehastobewelldefined(inthesensethattheprop-ertythatoneandonlyonecurvepassesthrougheverypointshouldholdevenwhenthecongruenceisextendedtotheboundary).Forinstance,inthenakeddustcase,whenoneusessphericalco-ordinatesitcanbeseenthatseveralradialnullgeodesicsappeartoemergefromthecentralsingularitywiththesametangentvector[8].

Theissueabouttheco-ordinatesystembeingappropriateforsuchanextensioncouldthusrelatedtothebehaviourofξ.

Fromatechnicalpointofview,thecalculationsusingtheradialco-ordinatescouldbeperformedinaconformallyrelatedmetricwhichavoidstheproblemofclusteringifitoccurs.Theconformaltransfor-mationwouldbetheoneleadingtothestructureofthesingularityas

11

depictedinthePenrosediagram.

WenowarguethattheunderaconformaltransformationwhichdivergesinthelimitofO,ξwhichtendtoanon-zerolimittransformtovectorfieldswhichvanishinthelimit.

Recallthatwedefinedξforanygeodesiccongruenceusinganaffineparametrization.Underconformaltransformations,affineparametersalongnullgeodesicschange(unliketimelikegeodesicswhichdonotremaingeodesiccurves,nullgeodesicsdostaysoprovidedtheaffineparameterchangesappropriately).Infinitesimalparameterdstrans-formstoΩ2(xµ)ds[3],whereΩ2isaconformaltransformation.Thusitisobviousthatξµ=dxµ/dsiffiniteandnon-vanishinginthelimitwillvanishunderΩ2transformationprovidedthelatterdivergesthere.Thuswefindthatatleastinthedustcase,onerequiresacon-formaltransformationwhichdivergesontheapparenthorizoninthelimitofapproachtothesingularityinthenakedcaseasagainstthecoveredcasewheretheradialco-ordinatesareappropriatetodescribethesingularitystructure.5ThisjustifiesthedifferenceinthestructureofthesingularboundarynearOinfigures3and4.

6Apossiblegeneralscenario

Considerthecasesofcollapseinwhichthesingularityformedmeetstheboundaryofthetrappedregion(orevencrossingitasinnakedcases)Nowitwouldbeofinteresttoexamineξingeneralontheapparenthorizonandcheckifitvanishesornotintheapproachtothesingularboundary.Ifitdoesnot,thenoneinvokesthedivergingconformaltransformationtoobtainthecorrectcausaldepiction.Theimmediatequestionwouldbethenakedorcoverednatureofsuchasingularity.Wecertainlyknowthatitisnakedinthedustcasewhentheconformaltransformationdiverges.Wepresentanargumentsug-gestingitsvalidityinageneralscenariorelaxingtheassumptionofdustandsphericalsymmetry.

LetpointObetheintersectionofthesingularityandtheapparenthorizonasbefore.

Theorem:NoingoingnullgeodesiccanreachpointOafteracon-formaltransformationiftheconformaltransformationdivergesatO.Proof:

Consideraspace-likehypersurfacefromwhichaningoingnullgeodesicreachesOifpossible.FromtheRaychaudhariequations,itcanbeshownthatonceanullgeodesichasnegativeexpansion,itwillreachaconjugatepointafterafiniteamountofaffineparameterhaselapsed.Iftheconformaltransformationdivergesinthelimit,thenanullgeodesicreachingOwouldimplyanelapseofinfiniteamountofaffineparameter6.Thisisacontradiction.HencetheconjugatepointmustoccurbeforeOonthenullcurve,beyondwhichthegeodesiccannotbeextended.SothegeodesiccannotreachO.2

Lemma:Thereexistsaningoingnullboundary(includingO)tothepastofOifnoingoinggeodesicreachesO.

Proof:

Considerasequenceofingoingnullgeodesicsegments{Λntureendpointsontheapparenthorizon,theendpointsapproaching}withfu-Oasn→∞.LettherebenoingoingnullboundarytothepastofO,ifpossible.Thentherewillbealimitingnullgeodesicof{ΛnreachesO.Thiscontradictstheprevioustheorem.2

}whichInthePenrosediagram(figure4)onecanimagineaningoingnullgeodesicwhichreachesthenullsingularboundaryatpointP.Thisistheconjugatepointforthatgeodesic.WehavesimplyjustifiedthattherewillbeaboundarytothespacetimeinplaceofthegeodesiccurvebetweenPandO.

IftheapparenthorizonisspacelikeintheapproachtoO,theaboveportionofboundaryiscertainlyexposedintotheuntrappedregionandisthereforenaked.Onemayaskiftheapparenthorizonisalways

spacelikeinthenakedcase.(Itiscertainlytruefordust[9]).IfasingularportiontothepastofOexists,thenitcannotbetrapped.IfsuchaportionexistedthenitwouldsimplyappearfromaPenrosediagramwithsuchaportionthatcausalcurveswouldemanatefromthem.Theseargumentsaremadeprecisebelow(FordefinitionsofIFs,ProperIFs,andTIFs,see[2],[4]).

Theorem:ExistenceofaningoingnullboundarytothepastofOimpliestheexistenceofTIFs.(TheportionoftheboundarycontainstheidealpointofaTIF)

Proof:

Considerasequence{Λnoutthisproof.Alsonote}thatasbefore.J+(Λi)This⊂J+choice(Λj)forisallfixedj>through-i(Thenullcurvesaresuccessivelytothepast).SinceeachΛihasafutureendpointontheapparenthorizonEi,allpointsofΛiexceptEiareuntrapped.ChooseonesuchpointQi.I+(Qi)isnonempty.BychoosingQi+1tothecausalpastofQiforeveryi(wecanalwaysdothatsincethenullcurvesaresuccessivelytothepast),oneobtainsasequenceof(proper)indecomposablefuturesetsofQiwhicharenested.(Onemaybeginthesequenceatanyi)ThelimitingIFasi→∞isthereforenonempty.

ThisIFwillbeproperiffthereisalimitof{Qnspacetime.ThereisaboundarytothepastofO.}whichThereforeisaatpartleastofoneQsequence(constructedasdescribedabove)existswhichfailstohavealimitingQwithinspacetime.7ConsiderthisQsequence(thereareactuallyanuncountablyinfiniteofthem).Thecorrespond-inglimitingIFoftheQiswillbeaTIFsincethereisnopointinspacetimeofwhichitisthefuture.2

7SummaryandConclusion

Thetangentvectorfieldtoanullgeodesiccongruencebeingthoughtofas‘fluxdensity’ofacongruenceofgeodesicsisexaminedforbe-haviourontheapparenthorizonintheapproachtothesingularity(pointO)inthedustcollapsemodel.Thereisacorrelationwiththepropertyofnakednesswiththisbehaviour.DemandingthatthevectorvanishesatthecoveredpointOforcesthedivergenceoftheconformaltransformationatOwhichleadstothePenrosediagramforthenakedscenario.Sincethefluxvanishesinthecoveredcase,thereisnosuchdivergenceandhencethePenrosediagramsinthetwocasesdiffer.OnedemandsthatthefluxvanishesatOinageneralcollapsescenarioonthegroundsthatOiscoveredwhencalculatedusingametricexhibitingthecorrectcausalstructure,andincaseitdoesnot,oneusesasuitableconformaltransformation(i.eonewhichdivergesatO)inordertoobtainthecorrectcausalstructurenearO.WeshowthattherewillbenoingoingnullgeodesicreachingOifthelatteristhecaseandarguethatitindicatestheexistenceofaportionofsingularitywhichisuntrapped.

Inconclusion,wehaveshownthattheinformationaboutwhetherthesingularityformedincollapseisnakediscontainedattheintersec-tionoftheapparenthorizonandsingularboundaryinthesphericaldustcase.Wealsosuggestthatitholdsinthecaseofageneralcol-lapse.Itshouldalsobenotedthattheprocedureofcheckingifanappropriateconformaltransformationisnecessarydoesnotdirectlyinvolvecheckingforemergenceofcausalcurvesfromthesingularity.

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Singularity

Boundary of cloud

App.Hor.

Event Hor.

Figure1:Collapseofsphericaldustleadingtoacoveredsingularity

17

SingularityApp.Hor.NullGeo.EventHorizon

Figure2:Collapseofsphericaldustleadingtoanakedsingularity

18

Singularity

O

App.horizon

Boundary ofcloud

Figure3:PenroseCarterdiagramforcollapseofsphericaldustleadingtoacoveredsingularity

19

Singularity

O

App. hor.

Nakedsingularportion

Boundaryof cloud

Figure4:Penrose-Carterdiagramforcollapseofsphericaldustleadingtoa(locally)nakedsingularity

20

O

Ei+2

Ei+1

Ei

QiQi+1Qi+2Sequenceof null curves.App. Hor.

Boundary to the past of O.

P

Figure5:ConstructionofasequenceofingoingnullgeodesicsapproachingO.

21