arXiv:gr-qc/0412012v2 17 Mar 2005Geons with spin and chargeJormaLouko1, RobertB.Mann2,3andDonaldMarolf41School ofMathematical Sciences,Universityof Nottingham, NottinghamNG72RD,UK2PerimeterInstituteforTheoretical Physics, OntarioN2J2W9, Canada3Departmentof Physics, Universityof WaterlooWaterloo, OntarioN2L3G1, Canada4PhysicsDepartment, UCSB,SantaBarbara, CA93106, USA(RevisedFebruary2005)arXiv:gr-qc/0412012)AbstractWeconstructnewgeon-typeblackholesinD 4dimensionsforEinsteinstheorycoupledtogaugeelds. Astaticnondegeneratevacuumblackholehasageonquotientprovidedthespatial sectionadmitsasuitablediscreteisometry,andanantisymmetric tensor eldof rank2or D 2withapure F2actioncanbeincludedbyanappropriate(andinmostcasesnontrivial) choiceof theeldstrengthbundle. Wendrotatinggeonsasquotientsof theMyers-Perry(-AdS) solutionwhenDis oddandnot equal to7. For other Dweshowthatsuchrotatinggeons,iftheyexistatall,cannotbecontinuouslydeformedtozeroangularmomentum. Withanegativecosmological constant, weconstructgeonswithangularmomentaonatorusattheinnity. Asanexampleofanonabeliangauge eld, we show that the D = 4 spherically symmetric SU(2) black hole admitsa geon version with a trivial gauge bundle. Various generalisations, including bothblack-branegeonsandYang-MillstheorieswithChern-Simonsterms, arebrieydiscussed.jorma.louko@nottingham.ac.ukmann@avatar.uwaterloo.camarolf@physics.ucsb.edu1Contents1 Introduction 22 StaticEinstein-U(1)geons 42.1 Gibbons-Wiltshiremetric . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Geonquotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Examples: Constantcurvaturetransversalspace . . . . . . . . . . . . . . 92.4 Generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Angularmomentum 113.1 Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Einstein-SU(2)geon 155 Conclusionsanddiscussion 161 IntroductionThetermgeon, short forgravitational-electromagneticentity, wasintroducedin1955 by John Archibald Wheeler to denote a classical gravitational conguration, possi-blycoupledtoelectromagnetismorotherzero-masselds,thatappearsasalong-livedmassiveobjectwhenobservedfromadistancebutisnotablackhole[1,2,3,4,5,6].While a precise denition of a geon seems not to have been sought, the examples studiedin [1, 2, 3, 4, 5, 6] had spatial topology R3and with one exception an asymptotically atinnity, wheremasscouldbedenedbywhatarenowknownasADMmethods. TheoneexceptionwasMelvinsmagneticuniverse[4, 7], whichwassubsequentlydeemednottoqualifyasageon,preciselybecauseofitscylindrical innity[8]. Unfortunately,whileMelvinsuniverseisstable[8,9, 10],thesegeonswerenot,owingtothetendencyof amasslesseldeithertodispersetoinnityortocollapseintoablackhole. Thistendencyhasmorerecentlybeenmuchstudiedinthecontextofcriticalphenomenaingravitationalcollapse[11].In1985, Sorkin[12] denotedby topological geons gravitational congurationswhose spatial geometry is asymptotically at and has the topology of a compact manifoldwithonepuncture,theomittedpointbeingattheasymptoticinnity. ThisgeneralisesWheelers geoninthesense of allowingnontrivial spatial topologyandablackholehorizon. An example of a topological geon that is a black hole is the space and time ori-entable Z2 quotient of the Kruskal manifold known as the RP3geon [13]. This spacetimehas an asymptotically at exterior region isometric to a standard Schwarzschildexteriorandis hencean eternalblackhole. However,whileinKruskalthe spatialhypersurfacesareRS2wormholeswithtwoasymptopias, thespatial topologyoftheRP3geonis2RP3 pointatinnity.1As an eternal black and white hole spacetime, the RP3geon is not an object one wouldexpect to be formed in astrophysical processes. Its interestis that as an unconventionalblack hole, it provides an arena for probing our understanding of black holes both in theclassical and quantum contexts. For example, in [13] the RP3geon was used to illustratea classical topological censorship theorem, and the Hamiltonian dynamics of sphericallysymmetricspacetimeswithgeon-typeboundaryconditionswasinvestigatedin[19,20].TheHawking(-Unruh)eectontheRP3geonwasanalysedforscalarandspinoreldsin[18, 21],addressingquestionsabout thegeonsentropyanditsstatisticalmechanicalinterpretation. The(2 + 1)dimensional asymptoticallyAdSanalogueoftheRP3geonwasusedin[22,23,24]toprobeAdS/CFTcorrespondence.The purpose of this paper is topresent newfamilies of geon-type blackholes inD 4spacetimedimensions forEinsteins theorycoupledtogaugeelds. Weshallnotattempttogiveaprecisedenitionof geon-typeblackhole, butwerequirethespacetimestobetimeorientableandfoliatedbyspacelikehypersurfaceswithasingleasymptoticregion. Wealsorequiretheasymptoticregionto bestationaryandsimplein a sense that allows conservedcharges to be dened by appropriate integrals. We alsoconsider geon variants of certain black branes that appear in string theory. We nd thatwhiletheRP3geongeneralisesreadilyintostaticvacuumgeonsinanydimension, theinclusion of U(1) gauge elds is more subtle and depends sensitively on the dimension, inmost cases requiringthe eld strength to be a section of a bundle that is twistedby thespatial fundamental grouporbysomequotientthereof. SuchtwistingisinparticularrequiredforelectricallychargedReissner-NordstrominanyspacetimedimensionandformagneticallychargedReissner-Nordstrominevenspacetimedimensions. Similarly,while the rotating (2+1)-dimensional black hole has geon-type variants for any values ofthe angular momentum [25, 26, 27], we nd that that the options for building geons fromthe rotating Kerr-Myers-Perry(-AdS) solutions [28, 29, 30, 31, 32] depend sensitivelyonthedimension. Withanegativecosmological constant, wealsopresentrotatinggeonswith a at horizon. As an example of a nonabelian gauge eld, we show that the D = 4spherically symmetricSU(2) black hole [33, 34, 35, 36, 37, 38, 39] admits a geon variantwithatrivialgaugebundle.Weusemetricsignature( + + ). StaticEinstein-U(1)geonsarediscussedinsection2andgeons withangular momentuminsection3. The D=4sphericallysymmetricSU(2)geonisconstructedinsection4. Section5presentsasummaryandconcludingremarks.1Thetime-symmetricinitial datafortheRP3geonwasdiscussedpriorto[13] in[5] and[14, 15].Anearlydiscussiononquotients of Kruskal canbe foundin[16] andamodernone in[17]. TheEuclidean-signaturesectionoftheRP3geonisdiscussedin[18].32 StaticEinstein-U(1)geonsInthis sectionwe discuss Einstein-U(1) geons that have astatic asymptotic region.Insubsection2.1werecallrelevantpropertiesofthenondegenerateGibbons-Wiltshireblackholeswith staticasymptoticregions [40], and in subsection2.2we constructgeonversionsof theseholes. Subsection2.3 presentsexamplesin the specialcase ofconstantcurvaturetransversalspace. GeneralisationsbeyondtheGibbons-Wiltshirespacetimes,includinggeneralisationsto multiple(i.e.,[U(1)]n) gauge elds,are discussedin subsec-tionin2.4.2.1 Gibbons-WiltshiremetricLet D 4, and let (/, ds2) be a (D2)-dimensional positive denite Einstein manifold:Writingds2=gIJ dyIdyJ, wehaveRIJ=(D 3)gIJ, where Risaconstant.Inlocal Schwarzschild-likecoordinates, theD-dimensional Lorentz-signatureGibbons-Wiltshiremetricreads[40]2ds2= dt2+dr2+ r2ds2, (2.1a) = rD3+Q2r2(D3) 2r2(D1)(D 2), (2.1b)where r > 0, and Q are real-valued constants and is the cosmological constant. ThespacetimehasaMaxwelleldwhoseFaradaytwo-formisF= DQrD2 dt dr , (2.2)whereDisaD-dependentconstantwhoseprecisevaluewill notbeneededinwhatfollows. Equations(2.1)and(2.2)solvetheEinstein-Maxwellequations,obtainedfromthe action whose gravitational part is proportional to_ g (R2) and Maxwell partto_ g FabFab. Whenisnonzero, itcanbenormalisedto [[=1withoutlossofgeneralitybyrescalingr,t,andQ.On par with the electric solution given by (2.1) and (2.2), we consider the magneticsolution[55]inwhich(2.2)isreplacedbyH= DQVds2, (2.3)whereDisa D-dependentconstantandVds2isthe volumeform (respectivelyvolumedensity)on(/, ds2)if /isorientable(nonorientable)[56]. Equations(2.1)and(2.3)solvetheEinstein-U(1)equationsobtainedfromtheactionwhosegaugeeldpartisproportionalto_ g Ha...fHa...f.Henceforthweassume(/, ds2)tobegeodesicallycomplete.2Forrelatedwork,see[41,42,43,44,45,46,47,48,49,50,51,52,53,54].4Weareinterestedinthesituationinwhichthemetric(2.1)extendsintoaneternalblack-and-whiteholespacetimeinwhichabifurcateeventhorizonat=0[57]sepa-ratestwoexteriorregions,bothofwhicharestaticwithrespecttothetimelikeKillingvector tand extendto a spacelikeinnity. This can only occurfor 0, as otherwisetis not timelikeat large r. When = 0, it is necessarythat > 0, and the parameterrange analysis and the conformal diagrams with suppressed(/, ds2) are as for Kruskaland nonextremal Reissner-Nordstrom [58]. When < 0, may take any value, and theparameter range analysis and the conformal diagrams are as in the four-dimensional case[51, 59], withthe correctionpointed out in [60, 61] to those diagrams that are shownin[51,59]assquares.3Let(/, ds2)denotethespacetimeconsistingofthefourconformalblocksadjacenttotheKillinghorizonintheconformal diagram. Weintroducein(/, ds2)standardglobal Kruskalnullcoordinates(U, V, y),inwhichydenotesapointin /. Themetricreadsds2= f dUdV+ r2ds2, (2.4)thehorizonisatUV= 0,andfandraresmoothpositivefunctionsofargumentUV .Thetwo-form(2.2)becomesF= DQf2 rD2 dU dV , (2.5)whiletheexpression(2.3)remainsvalidforH. Thestaticexteriorsare at UV< 0andtheinteriorsatUV >0. TherangeofUV dependsontheparameters; inparticular,UV isboundedbelowfor < 0butnotfor = 0. (/, ds2)maybeextendibletothefuture and past through further Killing horizons, but whether or not it is, the conformaldiagramsshowthat(/, ds2)containsallspacelikehypersurfacesthatconnectthetwostaticregions andaregiveninthelocal metric(2.1) byarelationbetweent andr.Workingwith(/, ds2)willthussuceforour purposes. If(/, ds2)isextendible,thequotientsintherestofthispapercanbeextendedinanobviousmanner.To summarise: The spacetime (/, ds2) is an eternal black hole, foliated bywormhole-likespacelikehypersurfacesof topology /R. Thestaticregionsarelo-callyasymptoticallyatfor=0andlocallyasymptoticallyanti-deSitterfor 4. Other examplesarisebysettingtheangularmomentaintherotatinggeonsofsubsection3.1tozero. = 1: HyperbolicspacesWhen = 1,_

/, ds2_is(D 2)-dimensionalhyperbolicspace. QuotientsofTypes(i),(ii)and(iii)existforallD. ForD>4,noncompactexamplesarisebywritingthemetricinPoincarecoordinates,

ds2=dz2+ dx2z2, (2.9)where z> 0 and dx2is the standard at metric on RD3, and letting G act on x RD3asin the = 0case. For D = 4, an exampleof Type(ii) ariseswhenG is generatedbyahyperbolicor ellipticMobiustransformation, andan exampleof Type(i)ariseswhenG is generated by a glide-reection, a hyperbolic Mobius transformation followed by thereection about the axis of this transformation. An example of Type (iii) arises when G isgeneratedbyaglide-reectionAand ahyperbolicorellipticMobiustransformation B,withtheparameterschosensothatthegroupactsfreely(cf. thepairsof hyperbolicelementsanalysedin [69]), and His generated byA and B2. Compact D = 4 examplesarediscussedin[70,71].2.4 GeneralisationsWenowconsidersomegeneralisationsbeyondtheGibbons-Wiltshirespacetimes.Thequotienttechniquegeneralisesimmediatelytobrane-likesolutionsinwhichthebrane dimensions are suciently inert. An example in vacuum with = 0 is the productofKruskalwithanypositivedeniteRicci-atspace. ThebranedimensionsmayoernewchoicesfortheisometryP, andthisfreedommayinparticularbeusedtomakespatiallyorientableornonorientablegeonsasdesired. Asanexample, inthestringyblack hole that is the product of the spinlessBTZ hole and S3T4, choosing Pto haveasuitablenontrivialactionontheT4factorgivesaspatiallyorientablegeon[22].Thetechniquealsogeneralisesimmediatelytoform eldsof more generalrankwithapure F2action. Ageneralisationto[U(1)]nformelds is possible andgives new10opportunities totwist the bundle bypermutingthe U(1) components. Terms otherthanF2in a form eld action mayhoweverbring about newphenomena. For example,aChern-Simonsterm, suchasthatin11-dimensional supergravity[72], mustbeatoprankformwhenthespacetimeisorientablebutatoprankdensitywhenthespacetimeisnonorientable. Finally, includingdilatonicscalarelds, suchasthosein[55, 73], isstraightforward.Thereexist generalisations withoutanasymptoticregionof thekindwehaveas-sumedabove. OneexampleistheBertotti-Robinson-typeextremal limitof (2.1), inwhichthespacetimeistheproductof (1 + 1)-dimensional anti-deSitterspaceandaconstant multiple of (/, ds2). The horizon is then a Killing horizon on the anti-de Sitterspace,and the quotientanalysisproceedsasin subsection2.2. In the limitofvanishingcurvature,thesespacetimesreducetogeon-likeversionsofRindlerspace[18].Another example is de Sitter space. The geon-like quotientyieldsa version in whichthespacelikehypersurfacesintheglobalfoliationarenotSD1butSD1/Z2 RPD1[74,75,76].Yetanotherexampleoccurswhen>0andtheconformal diagramisasforthenondegenerate Schwarzschild-de Sitter solution, with a pattern of black hole and cosmo-logical horizons repeating innitely in the horizontal direction [77]. It is possible to takeaZ2quotientusingoneofthehorizons, eitherablackholehorizonoracosmologicalone. Theconformaldiagrambecomesthenboundedfrom(say)theleftandinnitetotheright. However, itisalsopossibletotakeaquotientunderagroupgeneratedbytwoinvolutiveisometries,eachusingadierenthorizon,sothat theconformal diagrambecomes bounded both from the left and from the right [76]. In this case the xed time-likehypersurfacesofthetwoisometriescanbearbitrarilyboostedwithrespecttoeachother(cf.thediscussionintwo-dimensional dilatongravitycontextin[78],Figure14).ThesequotientsmightoeranarenaforprobingdS/CFTcorrespondence[79,80].Finally, the higher-dimensional BTZ hole [48, 81, 82, 83, 84] resembles a geon in thatthe exterior region is connected. It is however possible to take a Z2quotient of this hole,withtheeectthatthetopologyatinnitychangesandtheisometrygroupbecomessmaller. ThesepropertieswereinvestigatedintheAdS/CFTcontextin[85].3 AngularmomentumInthissectionweconstructrotatinggeons. Weconsiderrstangularmomentaonatorusandthenangularmomentaona(non-round)sphere.3.1 TorusLet(^, ds2)betheplanarvacuumGibbons-Wiltshireholewith0, the exterior is at rh 0. Transformingthegaugeeldcongurationfromthesingulargaugeusedin[37]toaregulargauge,bythematrix(9)in[37]with = /2,weobtainthegaugepotentialA(0)=12e_(n)u dt + (w 1)ijkinjdnk, (4.3)whereiarethePaulimatricesasin[37], n := (sin cos , sin sin , cos ),anduand15warefunctionsofrwiththenear-horizonexpansionsu =

k=1uk(r rh)k, (4.4a)w =

k=0wk(r rh)k. (4.4b)WedenetheKruskal coordinatesintheexteriorbyU= exp_12h1(t )andV= exp_12h1(t + ),whered/dr= p/Handwechoosetheadditiveconstantinsothat=h11ln_(r rh) + O_(r r0)_. ThehorizonisatUV0,risafunctionofUVwith the Taylor expansion r = rhUV+O_(UV )2_, and the metric takes the formds2= f dUdV+ r2d2, (4.5)wherefisafunctionof UV withtheTaylorexpansionf= 4h11+O(UV ). Thegaugepotential(4.3)takestheformA(0)=12e_(n)g (V dU UdV ) + (w 1)ijkinjdnk, (4.6)where g is afunctionof UV withthe Taylor expansiong =u1h11+ O(UV ). Thesolution given by (4.5) and (4.6) is clearly extendibleacross UV= 0 into a two-exteriorblackholespacetime(/, ds2)inthestandardfashion. AsA(0)(4.6)thenbecomesaglobally-denedsu(2)-valued1-formon /,thegaugebundleover /istrivial.Now, theonlyinvolutivefreely-actingisometryonthetwo-sphereistheantipodalmapP: (, ) ( , +),or n n. WiththisP,theisometryJ(2.6)plainlyleavesA(0)(4.6)invariant. Thequotientof(/, ds2)by_IdM, J_isthereforeageonwith a trivial gauge bundle. The ranges of the magnetic and electric charges in the geonareexactlyasinthetwo-exteriorhole.The dierence from the spherically symmetric U(1) geons, in which both the electricandmagnetic eldstrengths neededtobe twisted, arises fromthe nontrivial su(2)-valuednessofthegaugeeld. Itcanbeveriedthattheelectric andmagneticpartsofthe eldstrengthcomputedfromA(0)(4.6)are both in ahedgehog-likesu(2) cong-urationoverthetwo-sphere.5 ConclusionsanddiscussionInthis paper we have constructednewfamilies of geon-type blackholes inD4dimensions for Einsteins theory coupled to gauge elds, by taking Z2 quotients of eternalblackholesolutionsthathaveanondegenerateKillinghorizon. Inthestaticvacuumcase,theexistenceof thequotientwasequivalenttothe existenceofsuitableisometriesonthespatial sections, andantisymmetrictensorelds withapureF2actioncouldthenalwaysbeincludedbytakingtheeldstrengthtobeasectionofasuitable, and16inmost cases nontrivial, bundle. Withangular momentum, weconstructedrotatinggeons as quotients of the Myers-Perry(-AdS) solutionwhenDis oddandnot equalto7. Theserotating geonsare asymptoticallyat (anti-deSitter, respectively),but thespatial innity is a non-trivial quotient of the usual sphere. For other D we showed thatrotatinggeonquotientsoftheMyers-Perry(-AdS)solution, iftheyexistatall, cannotbe continuouslydeformed to the static case. With a negativecosmological constant, weconstructedgeonswithangularmomentaonatorusattheinnity. Finally,weshowedthattheD=4sphericallysymmetricSU(2)blackholeadmitsageonquotientwithatrivialgaugebundle.IntheGibbons-Wiltshiregeons of section2, we analysedtheabeliangaugeeldconguration in terms of the eld strength, rather than in terms of the gauge potential.In the electric geons and in the D = 4 magnetic geons, the gauge eld can be interpretedasaconnectioninaprincipalO(2) bundle. FortheelectricgeonsandforthemagneticD = 4 geons with 0, this follows by rst interpreting the gauge eld on the universalcovering space as a connection in the trivial O(2) bundle and then using the disconnectedcomponent of O(2) to twistthe bundleon taking the quotient. For the D = 4 magneticgeonwith = 1,asimilarargumentisnotavailablesincetheU(1) SO(2)bundleontheuniversalcoveringspaceisalreadynontrivial,buttheexistenceoftheO(2)bundlecan be veried directly.6For the D > 4 magnetic geons, the gauge potential issue wouldneedtobeaddressedintermsofgerbes[95].Alloursolutionshaveahighdegreeofsymmetry. FortheD= 4solutionswithanabelian gauge eld, this symmetry is dictated by the uniqueness theorems for stationaryblackholes [96]. Theuniqueness theorems areknownnot tohaveastraightforwardgeneralisation to D > 4 [97, 98], and there are suggestions that the uniqueness violationcouldbesevere[99]. Withanonabeliangaugeeld, blackholeswithlesssymmetryare knowntooccur eveninthe static D=4context [100, 101, 102]. One wouldliketounderstandtowhatextentourgeon quotientsgeneralisetoblackholeswithlesssymmetry. One would also like to understand at a more general level what kind of hair ageon may have. For example, despite the similarities between the spherically symmetricSU(2)blackholeandthesphericallysymmetricskyrmionblackholeof[103,104,105],theskyrmioneldcongurationisnotinvariant under theisometrythatyieldedtheSU(2) geon in section 4. What is the situation for, say, the spherically symmetric SU(n)blackholeswithn > 2[106]?From theviewpointofstringtheory,it wouldbeof interesttoexploregeon versionsof blackholesthataresolutionstosupergravitylimitsof M-theory. Stringtheoreticdualitieshaveprovidedamicroscopicstate-countingexplanation forthe entropyof cer-tainfamiliesof suchblackholes[107]: canD-branestatescorrespondingtogeonsbeidentiedanddistinguishedfromthosecorrespondingtootherblackholes? 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