Volume 59A, number 2 PHYSICS LE~~ERS 15 November1976

THE BEGINNING AND THE END OF A BLACK HOLE

FrankJ. TIPLER *

De~,artmentof PhysicsandAstronomy,UniversityofMaryland,CollegePark,Maryland20742,USA

Received14 June 1976Revisedmanuscript received26 July 1976

It is tho~~vnthat a black hole cannotexist forever: it must haveabeginning or anend. Furthermore, it is demon-strated that’ genericcasually symmetric black holesviolate cosmiccensorship.

Oneof then~ostinterestingproblemsin contem- in theeventhorizon.Usingthesedefinitions,we canporarygeneralr~lativitiyis thatof blackhole lifetimes. prove:This problemhaS mostrecentlybeendiscussedby Theorem1: A blackhole whosesurfaceis genesatedHawking[1] in ~onnectionwith thequantumevapor- by future-completenull geodesicsmusthavea begin-ation of blackh~les.It has,however,classicalaspects. ning if the followingconditionshold:For example,the centralhypothesisof classicalblack (a) the Einsteinequations;hole physics— ~osmiccensorship[2, 3] — implies (b) theweakenergycondition;thata blackhol~will existforeverafter its formation. (c) the genericcondition.But If all blackl~olesmustexistforeverin the future Theorem2: A blackholecannotexistforever;itdirection,it is ppssiblethatsomeblackholesexist musthaveabeginningor anend(orboth)if condi-foreverin the p~stdirection also?I shallanswerthe lions(a),(b), (c) of theorem1 hold.question,showi~igfirst thatanyphysicallyrealistic Proofoftheorem1: Supposethe blackhole didblackholewhic~obeyscosmiccensorshipmusthave not haveabeginning.This wouldmeanthatatleastabeginning,an~secondthatany physicallyrealistic onegeneratort~of J(91)hadnopastendpoint,andblackholemust~haveeitherabeginningor an end. furtherthatt~couldbecontinuedinto the pastonFinally, I shallshowthata specialclassof blackholes J(

9~)for infinite affme parameterlength.Since thedefinitely have~nend: genericcausallysymmetric blackholeis generatedby future-completegeodesics,blackholesvioldte cosmiccensorhip. n can also be continuedinto the future for infinite

Thereare severaldefmitionsof ablackholegiven affme parameterlength.However,by conditions(a),in the1lterature~The oneusedby Hawkingin refs. (b),(c) andproposition4.4.5 of ref. [4], i~musthave[4, 5J1stoo restrictive— it requiresasymptoticpre- apair of conjuptepoints.But if ~ hadapair of con-dictability, whi~hby definition rulesout thepossibili- jugatepoints,J(9~)couldnot be achronal,by prop.ty of blackhole terminations.We will follow osition4.5.12of [4]. This contradictspropositionCarter[6], Misi~er,Thorne,andWheeler [7] — and 6.3.1 of [4]. Theproofof theorem2 is similar.Hawkingin [1] — anddefineablackhole to bethe The word “first” usedin thedefmition of theendregion of spacet~meoutsider (9k). J (9k) is then of ablackholeis somewhatvague;it canbe madethe surfaceof theblackhole. We wili saythatablack morepreciseif we assumethe regionJ(9~)canbeholehas abegin~singif everygeneratorofJ(9’) hasa foliatedby asequenceof spacelikehypersurfacespastendpoint,or is pastincomplete.We will say that (i.e.,r(9~)is assumedto be stablycausal).Theablackholehasan endif atleastonegeneratorof spacelikehypersurfacesarelabeledby a“universalJ(9+) is not fut~irecomplete,andfurtherwe will say time parameter”t; theword “first” in theabove defi-thattheblackholeendswhensingularitiesfirst appear nition will mean“first accordingto thetime param-

etert”. This is areasonabledefinitionfor theendof* Presentaddress:Departmentof Mathematics, University of ablack hole, sincein the caseof approximatelyaxi-

California,Rerkóley, CA 94720,USA. symmetric blackholes,thehorizongeneratorswill ter-

107

Volume59A, number2 PHYSICSLETTERS 15 November1976

minate(for theappropriatefoliation) at aboutthe future-complete.Then all generatorsofJ(9~)couldsametime t [1], while deviationsfrom axial symme- be extendedinto the future inJ(9~)for infinitetry will tend to beradiatedaway [8, 9]. Thusin gen- valuesof the affine parameter.Since the blackhole iseral theendof a blackholecanbe defmedby the causallysymmetric,all generatorscanbeextendedintoterminationof onegenerator.It isnot necessaryto thepastin J—(9~)for infinite valuesof theaffinerequire thatall generatorsterminate, parameter.We cannow derivea contradictionby pro-

It shouldbe notedthat the Schwarzschild,Reissner- ceedingin the samewayasin the proofof theorem1.Nordstrom,Kerr, andKerr-Newmanblackholes doexist forever: thesesolutionsdo notsatisfy thegenericcondition.However,theseblack holesare causally Referencessymmetric— we say that acausallysymmetricblackhole is onefor whicht(S) flJ(9~)is isometricto [11S.W. Hawking, Comm.Math. Phys.43 (1975) 199.

J(S) flJ—(9~)for somepartialCauchysurfaceS [21R. Penrose,Riv. NuovoCim. 1, Num. spec.(1969)252.andi~fl S * 0 for any generatori~ofJ~(9~).That is, [31S.W. Hawking, Phys.Rev.Lett. 26 (1971)1344.

[41S.W. HawkingandG.F.R. Ellis, Thelargescalestructurecausallysymmetricblackhole solutionshavea space. of spacetime(CambridgeUniversityPress,Cambridge,like hypersurfaceSfor which theportion of the event 1973).horizonto the futureof S isidenticalin structureto [51S.W. Hawking, in: Blackholes,LesHouches1972, eds.the portion of the eventhorizonto the pastof S. We C. DeWitt andB.S. DeWitt (GordonandBreach,N.Y.,now showthat a genericcausallysymmetricblack 1973)i. 31.[61B. Carter,in: Blackholes, LesHouches1972,eds. C.hole violatescosmiccensorship: Dewitt andB.S. Dewitt (GordonandBreach,N.Y., 1973),

Theorem3: The surfaceof a causallysymmetric ~ 134.blackholehasatleastonefuture-incompletegenerator [7j C.W. Misner,K.S. ThorneandJ.A. Wheeler,Gravitationprovidedconditions(a), (b), and(c)of theorem 1 (Freeman,San Francisco,1973)p. 875.hold. [81 W.H. Pressand S.A. Teukoisky, Ap. J. 185 (1973)649.

Proof: Supposethat all generatorsofJ(9~)were [91 R.H. Price,Phys.Rev. D5 (1972)2419.

108