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1、VOLUME17,NUMBER8PHYSICAL.REVIEWI,ZTTKRS22AUGUsT1966SINGULARITIESINTHEUNIVERSES.W.HawkingDepartmentofAppliedMathematicsandTheoreticalPhysics,UniversityofCambridge,Cambridge,England(Received30June1966)Itisshownthatspace-timecannotbesingularityfreeiftheEinsteinequationshold,matterhasnormalproperties,an

2、dsomereasonableglobalconditionshold.ThiswouldsuggestthattheEinsteintheoryprobablybreaksdowninverystrongfields.Thepredictionofasingularitybyaphysicaltheoryisusuallytakenasanindicationthatthetheoryhasbrokendown.ThusthequestionofwhetherEinsteinstheoryofgeneralrela-tivitypredictssingularitiesofspace-tim

3、einallreasonablecosmologicalmodelsisclear-lyofconsiderableimportance.Ithas beenknownforsometimethatcertainexactsolutionsoftheEinsteinequationhavesingularitiesbutitwasclaimedthatthismightsimplybeaconsequenceofthehighsym-metryofthesesolutionsandthatthesingular-itiesmightbeavoidedbymakingsmallpertur-ba

4、tions.ThiswasshownnottobethecasebyPenrosewhoprovedthatanapproximate-lysphericalstarcollapsinginauniversewithanoncompactCauchysurfacemustgotoasin-gularity.PenrosesmethodcanalsobeappliedtoprovethatanapproximatelyhomogeneousandisotropicopencosmologicalmodeLmusthaveasingularityinthepast.Howeverthemethod

5、canonlybeappliedtomodelswithnoncompactCauchysurfacesanditmightbethattheuniversehasacompactCauchysurface(a,closedmodel)orevennoCauchysurfaceatall.Theaimofthispaperistoindicatethat,oncertainreason-ableassumptions,itisstillpossibletoprovethatsingularitiesmustoccur.Onlytheresultsandabriefindicationofthe

6、methodwillbegiv-enhere.Thefulldetailsoftheproofsarebe-ingpublishedelsewhere.WeassumetheEinsteinequations:Ra-axR=TabSPace-timelssaidtobesinglarltyfreeifitistime-complete(alltimelikege-odesicscan beextendedtoarbitarylength)andifthemetricisaCtensorfield.Theorem1.Space-timecannotbesingular-ityfreeif (1)

7、theenergy-momentumtensorobeystheinequalityTagM-2uaMTforanytimelikeornullvectorw(forafluidwithdensitypandisotropicpressurepthisissat-isfiedifp+p0andp,+3p0;thisisaveryreasonablerequirement);(2)thereexistsaCauchysurfaceHwedefineaCauchysurfaceasaclosed(notnecessarilycompact)space-likesurfacewithoutbound

8、ary,whichintersectseverytimelikeandnulllineonceandonceonly;and(3)theunitnormalsvtoHhaveapositivelowerboundofdivergenceonH;i.e.,va.aC&O.Ifcondition(2)weresatisfiedonewouldex-pectcondition(3)tobesatisfiedinauniverseexpandingeverywhere.Indeeditmightalmostserveasadefinitionofwhatwemeanbyanexpandingunive

9、rse.Nevertheless,itwouldbeimpossibletotestitbyobservation.Wethereforepresenttwotheoremswhichreplacethisconditionbyotherconditions.AfurthertheoremisgiveninthenotebyGeroehaceom-paningthispaper.Theorem2(a).Spacetimeisnotsingularityfreeifconditions(1)and(2)holdandif(4)thereisapointPonHsuchthatallthepast

10、-direct-edtimelikegeodesicsthroughpstartconverg-ingagain.Thatis,8=p,o.&becomesnegativeoneachpast-directedtimelikegeodesicthroughp,wherepistheunittangentvectortothegeodesic.Condition(4)isfairlyseverebutitwillbesatisfiedinanapproximatelyhomogenousandisotropicexpandinguniverseobeying(1)and(2).Itwouldbe

11、testablebyobservation.Theorem2(b).Space-timeisnotsingular-ityfreeifcondition(1)holds,thereisacom-pactCauchysurfaceH,and(5)onHtheener-gy-momentumtensorsatisfiesthestrictinequal-itya5jaTSVSe&MISeTabaforanytimelikeornullvectorsea.Condition(5)requiresthatthedensityisnon-zeroonH.Itisassumedforsimplicityb

12、utisreallystrongerthanisnecessary.Allthatisneededisthateverytimelikegeodesicshouldencountersomematterorevensomerandom-lyorientedcurvature.PenrosehaspointedoutthattheuniversemightnotpossessaCauchysurfaceatall.HeinstancestheRiesner-Nordstromsolution444VOLUME17,NUMBER8PHYSICALREVIEWLETTERS22AUGUST1966a

13、sanexampleofaspace-timewhereitisim-possibletofindaconnectedspacelikesurfacecuttingeverytimelikeline.Wewillcallacon-nectedclosedspacelikesurfacewithoutbound-aryasliceandaslicewhichdoesnotintersectanytimelikeornulllinemorethanonceapar-tialCauchysurface.Itiseasytoshowthatifspace-timeadmitsaslicetheneit

14、herthatsliceisapartialCauchysurfaceorspace-timehasacoveringspaceinwhicheachconnectedcomponentoftheimageofthesliceisapar-tialCauchysurface.Wemayapplythefollow-ingtheoremtospace-timeortothecoveringspacesinceasingularityinthecoveringspaceimpliesoneinthespacecovered.Theorem3.Spacetimeisnotsingularityfre

15、eifcondition(1)holdsand(6)thereexistsacompactpartialCauchysurfaceHwhoseunitnormalsv&arediverging,(7)everypointqhasaneighborhoodWsuchthateverytimelikeandnulllinefromqleavesWanddoesnotre-en-ter.Condition(7)isreallyastatementaboutcau-sality.Itwouldseemaveryreasonablere-quirement.Finally,wemakeabriefmen

16、tionofthetech-niqueusedtoprovetheseresults.ApointPissaidtobeconjugatetoapointqalongatimelikegeodesicyifPliesonthecausticoftimelikegeodesicsthroughq.ThepointPissaidtobeconjugatetoaspacelikesurfaceHifitliesonthecausticofthegeodesicsnor-maltoH.Bytheformulaforvariationofarclengthitcanbeshownthata,geodes

17、icynor-maltoHfromHtoapointxcannotbethelong-esttimelikelinefromHtoxifthereisapointconjugatetoHonybetweenHandx.BytheconditionsofthetheoremsitispossibletoshowthatthereisapointconjugatetoHwith-inaboundeddistanceboneverygeodesicnor-maltoH.Ontheotherhand,itispossibletoshowthattherearepointsfromwhichtherei

18、salongesttimelikelinetoHof finitelengthgreaterthanb.Thisestablishesacontradic-tionwhichshowsthattheassumedconditionsareincompatiblewithspace-timebeingsingu-larityfree.TheauthorisverygratefultoDr.R.PenroseandMr.B.Carterforhelpandadvice.H.P.Robertson,Rev.Mod.Phys.5,62(1933).A.Raychaudhuri,Phys.Rev.98,

19、1123(1955).3A.Komar,Phys.Rev.104,544(1956).4L.Shepley,Proc.Natl.Acad.Sci.U.S.52,1403(1964).5S.W.HawkingandG.F.R.Ellis,Phys.Letters17,246(1965).6E.M.Lifshitzandl.M.Khalatniko,Advan.Phys.12,185(1963).R.Penrose,Phys.Rev.Letters1457(1965).S.W.Hawking,Phys.Rev.Letters15,689(1965).C.W.Misner,J.Math.Phys.4

20、,924(1963).R.P.Geroch,followingLetterPhys.Rev.Letters17,445(1966).R.Penrose,privatecommunication.B.Carter,Phys.Letters21,423(1966).E.H.KronheimerandR.Penrose,TheStructureofCausalSpaces(tobepublished).SINGULARITIESINCLOSEDUNIVERSESR.P.Geroch*PalmerPhysicalLaboratory,PrincetonUniversity,Pririceton,New

21、Jersey(Received6July1966)Inrecentyears,severaltheoremshaveappearedwhichshowthatthesingularitiesob-servedincertainexactsolutionsoftheEinsteinequationscannotbeavoidedbysmallpertur-bationsinthesesolutions.Thesetheoremsarenotsufficientlystrong,however,tosettlecompletelythequestionwhetheranynonsingu-lars

22、olutionsofEinsteinsequationscanpro-videareasonablemodelofourownuniverse.Inparticular,shouldthetrappedsurfaceofPenroseortheexpandingCauchysurfaceofHawkingoccurinourownimmediatevicinity,wecould,bysuitablydistributingmassesandthusaffectingthelocalgeometry,destroythesesymptonsofprospectivecollapse.Wouldwethussavetheentireuniversefromevolvingintoasingularstate?Thisseemsunlikely,andonewouldliketoexpressthisfeelingasatheorem.Weshouldlikeheretopresentaresultwhichtakesafurthersteptowardthemorelimitedgoalofestablishingthat,withsomesuitable445