Black hole Area-Angular momentum inequality in non-vacuum spacetimes

0.11 MB·English

by  José Luis Jaramillo

BlackholeArea-Angular momentum inequality innon-vacuum spacetimes
Jose´ Luis Jaramillo,1,2 Mart´ın Reiris,1 and Sergio Dain3,1
1 Max-Planck-Institutfu¨rGravitationsphysik,AlbertEinsteinInstitut,AmMu¨hlenberg1D-14476PotsdamGermany
2 Laboratoire Univers et The´ories (LUTH), Observatoire de Paris,
CNRS, Universite´ Paris Diderot, 92190 Meudon, France
3 Facultad deMatema´tica, Astronom´ıayF´ısica, FaMAF,Universidad Nacional deCo´rdoba
Instituto de F´ısica Enrique Gaviola, IFEG, CONICET,
Ciudad Universitaria (5000) Co´rdoba, Argentina
Weshowthatthearea-angularmomentuminequalityA ≥ 8π|J|holdsforaxiallysymmetricclosedouter-
moststablymarginallytrappedsurfaces.Thesearehorizonsections(inparticular,apparenthorizons)contained
inotherwisegenericnon-necessarilyaxisymmetricblackholespacetimes,withnon-negativecosmologicalcon-
stantandwhosemattercontentsatisfiesthedominantenergycondition.
PACSnumbers:04.70.-s,04.20.Dw,04.20.Cv
Introduction. Isolated and stationary black holes cannot (see[17]foradetaileddiscussionofthisconditioninthecon-
2 rotatearbitrarilyfast. ThetotalangularmomentumJ inKerr text of black hole extremality). The latter inequality acts as
1 solutionsthatareconsistentwithcosmiccensorship(i.e.with- a stability condition on S and, actually, is closely related to
0 outnakedsingularities)isboundedfromabovebythesquare thestablyoutermostconditionimposedonmarginallytrapped
2 ofthetotalmassM. Theheuristicstandardpictureofgravi- surfacescontainedinspatial3-slicesΣwhenprovingtheex-

n tationalcollapse[1]emphasizesthephysicalrelevanceofthis istenceofdynamicaltrappinghorizons[18]. Suchstablyout-
a boundandsuggestsitsgenericvaliditybeyondidealizedsitua- ermostconditionmeansthatthevariationofθ(ℓ) alongsome
J tions. Thetotalmass-angularmomentuminequalityJ ≤ M2 outwarddeformationofS inthesliceΣisnon-negative.That

0 has been indeed extended to the dynamical case of vacuum is,δ θ(ℓ) ≥0forsomespacelikeoutgoingvectorvtangentto
v
1 axisymmetricblackholespacetimes[2–7]. However,thisin- Σ(seethegeneralizationtospacetimenormalvectorsin[19];

equality involves global quantities. In order to gain further we refer to [18] for a discussion on operator δ ). Regard-
] v
c insightintothegravitationalcollapseprocessinthepresence ing now the vacuum dynamical case in [15], the inequality
q ofmatterand/ormultiplehorizons,itisalsodesirabletohave A ≥ 8π|J| is first proved for stable minimal surfaces S in
- a quasi-localversionofthe inequality. Thisattemptencoun- a spatial maximal slice Σ, i.e. S is a local minimum of the
r
g tersimmediately,though,theambiguitiesassociatedwiththe areawhenconsideringarbitrarydeformationsofS inΣ, and
[ quasi-local definition of gravitational mass and angular mo- then generalized for arbitrary surfaces, in particular horizon

2 mentum. In this context, an alternative (but related) bound sections.
v on the angular momentum can be formulated in terms of a ThepresentdiscussionoftheinequalityA≥8π|J|closely
3 horizonarea-angularmomentuminequalityA ≥ 8π|J|. This follows the strategy and steps in [15], adapting them to the
4 inequality was conjectured for the non-vacuum axisymmet- useofastabilityconditioninthespiritofthosein[9,10,17–
7
ricstationarycase(actually,includingthechargedcase)with 19], i.e. basedonmarginallytrappedsurfacesratherthanon
3
mattersurroundingthehorizonin[8]andthenprovedin[9– minimal surfaces. In the line of [18, 19] we will refer to a
.
6 11],whereasitsvalidityinthevacuumaxisymmetricdynami- marginallytrappedsurfaceS as(spacetime)stablyoutermost
0 calcasewasconjecturedanddiscussedin[12],partialresults (seeDefinition1below)ifforsomeoutgoingspace-likevector
1 were givenin [13, 14] and a complete proofin [15]. Equal- oroutgoingpastnullvectorXa itholdsδ θ(ℓ) ≥ 0. Then,it
1 X
ity holdsin the extremalcase. Here we reconciliate and ex- follows:
:
v tend both results by proving the validity of the inequality in
Theorem 1. Given an axisymmetric closed marginally
Xi fully general dynamical non-vacuum spacetimes (matter on trapped surface S satisfying the (axisymmetry-compatible)
thehorizonallowed),onlyrequiringaxisymmetryonthehori-
r spacetime stably outermost condition, in a spacetime with
a zon.Therestofthisletterisdevotedtoprovethisresult.
non-negative cosmological constant and fulfilling the domi-
The dynamical non-vacuum case. Proofs of A ≥ 8π|J|
nantenergycondition,itholdstheinequality
requiresomekindofgeometricstabilityconditioncharacter-
izingthesurfaceS forwhichtheinequalityisproved.Onthe A≥8π|J| , (1)
onehand,inthenon-vacuumstationarycasediscussedin[9– whereAandJ aretheareaandgravitational(Komar)angu-
11]surfacesS aretakentobesectionsofblackholehorizons larmomentumofS. Ifequalityholds,thenShasthegeometry
modeled as outer trapping horizons [16]. This entails, first, ofanextremeKerrthroatsphereand,inaddition,ifvectorXa
thevanishingoftheexpansionθ(ℓ) associatedwithlightrays inthestabilityconditioncanbefoundtobespacelikethenS
emittedfromS alongthe(outgoing)nullnormalℓa [i.e. S is isasectionofanon-expandinghorizon.
amarginallytrappedsurface]and,second,thatwhenmoving
towardsthe interiorofthe black holeone findsfullytrapped Notethataxisymmetryisonlyrequiredonthehorizonsur-
surfaces,sothatthevariationofθ(ℓ) alongsomefutureingo- face (this includes the intrinsic geometry of S and a cer-
ing nullnormalka is negative(outercondition): δ θ(ℓ) < 0 tain componentof its extrinsicgeometry,see below), so that
k
2
J accounts solely for the angular momentum of the black sense that L ℓa = L ka = 0, with ξa a unitvector tangent
η η
hole(horizon)in an otherwisegenericallynon-axisymmetric to S and orthogonalto ηa, i.e. ξaη = ξaℓ = ξak = 0,
a a a
spacetime. Actually, no other geometric requirement is im- ξaξ = 1. We can then write the induced metric on S as
a
posed outside S. Regarding the topology of the marginally q = 1η η +ξ ξ ,withη =ηaη ,sothat
ab η a b a b a
trapped surface S, this is always a topological sphere (for
J 6= 0)asaconsequenceofthestabilityconditioncombined Ω(ℓ) = Ω(η)+Ω(ξ)
a a a
with the dominant energy condition. Therefore, we can as- Ω(ℓ)Ω(ℓ)a = Ω(η)Ω(η)a+Ω(ξ)Ω(ξ)a , (6)
sume in the following that S is a sphere S2 without loss of a a a
generality. with Ω(η) = ηbΩ(ℓ)η /η and Ω(ξ) = ξbΩ(ℓ)ξ . In addition,
a b a a b a
Elements in the proof. The proof in [15] has two parts. wedemandΩ(ℓ) tobealso axisymmetric,L Ω(ℓ) = 0. Sec-
First, a geometric part providing a lower bound on the area a η a
ond, S is taken to be a marginal trapped surface: θ(ℓ) = 0.
A. Andsecond,apartmakinguseofvariationalprinciplesto
Wewillrefertoℓaastheoutgoingnullvector.Third,astabil-
relate thatlower area boundto an upperboundon the angu-
ity conditionmustbe imposedonS, namelywe demandthe
larmomentumJ and,inasubsequentstep, toproverigidity.
marginallytrappedsurfacetobeaspacetimestablyoutermost
Herewerecastthefirstgeometricpartinthenewsettingand
inthefollowingsense:
recoverexactlythefunctionalneededinthesecondvariational
part,sothatresultsin[13,15]canbedirectlyapplied. Definition1. GivenaclosedmarginallytrappedsurfaceS
Let us first introducesome notationand considera closed wewillrefertoitasspacetimestablyoutermostifthereexists
orientable2-surfaceS embeddedinaspacetimeM withmet- an outgoing (−ka-oriented) vector Xa = γℓa −ψka, with
ric g and Levi-Civita connection ∇ , satisfying the dom- γ ≥ 0andψ > 0,suchthatthevariationofθ(ℓ) withrespect
ab a
inant energy condition and with non-negative cosmological toXafulfillsthecondition
constantΛ ≥ 0. We denotethe inducedmetric on S as q ,
withLevi-CivitaconnectionD , Ricciscalar2Randvolumabe δXθ(ℓ) ≥0. (7)
a
element ǫab (we will denote by dS the area measure on S). If, in addition,Xa (inparticularγ, ψ) andΩ(ℓ) are axisym-
Let us consider null vectors ℓa and ka spanning the normal a
metric, we will refer to δ θ(ℓ) ≥ 0 as an (axisymmetry-
plane to S and normalized as ℓak = −1, leaving a (boost) X
a compatible)spacetimestablyoutermostcondition.
rescaling freedomℓ′a = fℓa, k′a = f−1ka. The expansion
θ(ℓ) andtheshearσ(ℓ) associatedwiththenullnormalℓa are Here δ denotes a variation operator associated with a de-
givenby ab formationof the surface S (c.f. for example [18, 20]). Two
remarks are in order. First, note that the characterization
1 of a marginally trapped surface as spacetime stably outer-
θ(ℓ) =qab∇ ℓ , σ(ℓ) =qc qd ∇ ℓ − θ(ℓ)q , (2)
a b ab a b c d 2 ab mostisindependentofthechoiceoffuture-orientednullnor-
mals ℓa and ka. Indeed, given f > 0, for ℓ′a = fℓa and
whereasthenormalfundamentalformΩ(ℓ)is k′a =f−1kawecanwriteXa =γℓa−ψka =γ′ℓ′a−ψ′k′a
a
(with γ′ = f−1γ ≥ 0 and ψ′ = fψ > 0), and it holds
Ω(ℓ) =−kcqd ∇ ℓ . (3) δ θ(ℓ′) = f ·δ θ(ℓ) > 0. Second, the proof of inequality
a a d c X X
(1)wouldonlyrequirethevectorXainthestabilitycondition
Transformationrulesunderanullnormalrescalingare to be outgoingpast null [16, 21]. We have, however, kept a
moregenericcharacterizationinDefinition1thatdirectlyex-
θ(ℓ′) =fθ(ℓ) , σ(ℓ′) =fσ(ℓ), Ω(ℓ′) =Ω(ℓ)+D (lnf). (4) tendsthe stably outermostconditionin [18] (in particular, S
ab ab a a a
isspacetimestablyoutermostifthereexistsa(−ka-oriented)
vectorforwhichS isstablyoutermostinthesenseof[19]).
WecharacterizenowthesurfacesS forwhichtheresultin
Wecannowestablishthelowerboundonthehorizonarea
Theorem1holds.First,weimposeStobeaxisymmetric,with
byfollowinganalogousstepstothosein[15].First,wederive
axialKillingvectorηa,i.e.L q =0.Moreprecisely,ηade-
η ab agenericinequalityonS,providedbythefollowinglemma.
finedonS hasclosedintegralcurvesandvanishesexactlyat
twopoints. Wenormalizevectorηa sothatitsintegralcurves Lemma 1. Given a closed marginallytrapped surface S
haveanaffinelengthof2π. Theassociatedgravitationalan- satisfyingthespacetimestablyoutermostconditionforanax-
gularmomentum(theKomarone,ifηa canbeextendedasa isymmetricXa,thenforallaxisymmetricαitholds
KillingvectortoaneighbourhoodofS)isexpressedinterms
1
ofΩa(ℓ)as DaαDaα+ 2α2 2R dS ≥ (8)
ZS(cid:20) (cid:21)
J = 81π ZSΩ(aℓ)ηadS , (5) ZShα2Ω(aη)Ω(η)a+αβσa(ℓb)σ(ℓ)ab+Gabαℓa(αkb+βℓb)idS ,
where the divergence-free character of ηa together with the whereβ =αγ/ψ.
transformationspropertiesofΩ(ℓ) in(4)guaranteetheinvari- To prove it we basically follow the discussion in section
a
anceofJ underrescalingofthenullnormals.Wealsoassume 3.3. of[22], allowingoneto essentially reducethenontime-
a tetrad(ξa,ηa,ℓa,ka) on S, adaptedto axisymmetryin the symmetric case to the time-symmetric one (cf. Th. 2.1 in
3
[23]forasimilarreasoning). First,weevaluateδ θ(ℓ)/ψfor In a second stage, under the assumption of axisymmetry
X
the vector Xa = γℓa −ψka providedby Definition 1, with we evaluateinequality(12) alongthelinesin [15]. First, we
axisymmetricγandψ(usee.g.Eqs.(2.23)and(2.24)in[20]) note that the sphericity of S follows from Lemma 1 under
andimposeθ(ℓ) =0. Wecanwrite theoutermoststablyanddominantenergyconditionstogether
withΛ ≥ 0since,uponthechoiceofaconstantαinEq. (8),
1δ θ(ℓ) = −γ σ(ℓ)σ(ℓ)ab+G ℓaℓb it implies (for non-vanishing angular momentum) a positive
ψ X ψ ab ab value for the Euler characteristic of S. Then, the following
h i
−2∆lnψ−D lnψDalnψ+2Ω(ℓ)Dalnψ (9) formfortheaxisymmetriclineelementonS isadopted
a a
− −DaΩ(aℓ)+Ω(cℓ)Ω(ℓ)c− 212R+Gabkaℓb . ds2 =qabdxadxb =eσ e2qdθ2+sin2θdϕ2 , (13)
(cid:20) (cid:21)
with σ and q functions on θ(cid:0)satisfying σ +q =(cid:1)c, where c
We multiply now the expression by α2 and integrate on S.
is a constant. This coordinate system can always be found
Using S αψ2δXθ(ℓ)dS ≥ 0, integrating by parts to remove in axisymmetry [30]. We can then write dS = ecdS0, with
boundaryterms,wecanwrite dS =sinθdθdϕ. Inaddition,thesquarednormηoftheaxial
R 0
Killingvectorηa =(∂ )a isgivenbyη =eσsin2θ.
ϕ
0≤ αβ −σ(ℓ)σ(ℓ)ab−G ℓaℓb dS Regardingthelefthandsidein(12),weproceedexactlyas
ab ab
ZS h i in[15]. Inparticular,choosingα= ec−σ/2,theevaluationof
+ α2 −Ω(ℓ)Ω(ℓ)a+ 12R−G kaℓb dS theleft-hand-sideininequality(12)resultsin(see[15])
a 2 ab
S
Z (cid:20) (cid:21) 1
+ S 2αDaαDalnψ−α2DalnψDalnψ dS ZS(cid:20)DaαDaα+ 2α2 2R(cid:21)dS (14)
Z
(cid:2) (cid:3) 1 dσ 2
+ 2α2Ω(ℓ)Dalnψ−2αΩ(ℓ)Daα dS . (10) =ec 4π(c+1)− σ+ dS .
a a 4 dθ 0
ZSh i " ZS (cid:18) (cid:19) ! #
From the axisymmetry of α and ψ, Ω(η)aDaα = Thesecondkeyingredientinthepresentdiscussionconcerns
Ω(η)aD ψ =0,andusing(6)wecanwrite theevaluationoftherighthandsidein(12),inparticularthe
a
possibility of making contact with the variational functional
0≤ αβ −σ(ℓ)σ(ℓ)ab−G ℓaℓb dS Memployedin[13,15].
ab ab
ZS h i DuetotheS2topologyofS,wecanalwaysexpressΩa(ℓ)in
+ α2 −Ω(η)Ω(η)a+ 12R−G kaℓb dS termsofadivergence-freeandanexactform.Writing
a 2 ab
ZS (cid:20) (cid:21) 1
+ 2(Daα)(αDalnψ−αΩa(ξ)) (11) Ω(aℓ) = 2ηǫabDbω¯+Daλ, (15)
S
Z h
−(αD lnψ−αΩ(ξ))(αDalnψ−αΩ(ξ)a) dS . withω¯ andλfixeduptoaconstant,fromtheaxisymmetryof
a a
q and Ω(ℓ) (functionsω¯ and λ are then axially symmetric)
i ab a
MakinguseoftheYoung’sinequalityinthelastintegral it followsthat Ω(η) = 1 ǫ Dbω¯ is the divergence-freepart
a 2η ab
DaαD α≥2Daα(αD lnψ−αΩ(ξ))−|αDlnψ−αΩ(ξ)|2 whereasΩa(ξ)istheexact(gauge)part.Inparticular,ηaΩa(ℓ) =
a a a 1 ǫ ηaDbω¯ andexpressingξa asξ =η−1/2ǫ ηa,wehave
2η ab b ab
inequality(8)followsforallaxisymmetricα.
Inequality (8) constitutes the first key ingredient in the 1
Ω(ℓ)ηa = ξaD ω¯ . (16)
present discussion and the counterpart of inequality (15) in a 2η1/2 a
[15] [inserting their Eqs. (30) and (31)]. In this spacetime
version,thegeometricmeaningofeachtermininequality(8) PluggingthisexpressionintoEq.(5)andusing(13)wefind
is apparent. For our present purposes, we first disregard the
1 π 1
positive-definite gravitational radiation shear squared term. J = ∂ ω¯dθ = (ω¯(π)−ω¯(0)) (17)
θ
Imposing Einstein equations, we also disregard the cosmo- 8Z0 8
logicalconstantandmatterterms[29], undertheassumption
whichisidenticaltotherelationbetweenJ andthetwistpo-
of non-negativecosmologicalconstantΛ ≥ 0 andthe domi-
tentialω inEq. (12)of[13]. Asaremark,wenotethatifthe
nantenergycondition(notethatαkb+βℓbisanon-spacelike
axialvectorηaonS extendstoaspacetimeneighbourhoodof
vector).Therefore
S (something not needed in the present discussion), we can
DaαDaα+ 12α2 2R dS ≥ α2Ω(aη)Ω(η)adS. (12) dreelfiantieonthξeaωtwais=t veξcatDoraωo¯fhηoaldass. ωInat=heǫvaabccuduηmb∇ccaηsde,aandtwthiset
ZS(cid:20) (cid:21) ZS potential ω satisfying ω = ∇ ω can be defined, so that ω¯
a a
Thisgeometricinequalitycompletesthefirststagetowardsthe andωcoincideonS uptoaconstant.Notehoweverthatω¯ on
lowerboundonA. S canbedefinedalways.
4
FromEqs. (15)and(13)andthechoiceofα,wehave assumptionon the spacetimegeometry. A criticalingredient
in the present derivation is the stability assumption (7), ba-
α2Ω(η)Ω(η)a = α2 D ω¯Daω¯ = 1 dω¯ 2 . (18) sically the stably outermost condition in [19] that naturally
a 4η2 a 4η2 dθ extendsthe stability conditionin [18] (in the contextof spa-
(cid:18) (cid:19) tial 3-slices) to generalspacetime embeddingsof marginally
Usingthisand(14)in(12)werecoverexactlythebound trappedsurfaces. Thisstabilitycondition,essentiallyequiva-
lenttotheouterhorizonconditionin[16],impliesthat(axially
M−8
A≥4πe 8 , (19) symmetric)[31]outertrappinghorizons[16]satisfyinequality
(1), independentlyofafuture/pastconditiononθ(k) (respec-
withtheactionfunctional tively,θ(k) < 0 orθ(k) > 0). Therefore,quasi-localmodels
of black hole horizons [16, 20, 24] satisfy the area-angular
1 dσ 2 1 dω¯ 2 momentuminequality(1),thatprovidesaquasi-localcharac-
M= +4σ+ dS , (20)
2π dθ η2 dθ 0 terizationofblackhole(sub)extremality.Inparticular,theva-
ZS"(cid:18) (cid:19) (cid:18) (cid:19) #
lidityof(1)is equivalentto thenon-negativityofthesurface
inRef. [15],sothattherestoftheproofreducestothatinthis gravityκofdynamicalandisolatedhorizons[24](withκ=0
reference.Namely,theupperboundin[13]forJ in the extremal case), the present result therefore endorsing
thephysicalconsistencyoftheirassociatedfirstlawofblack
e(M−8)/8 ≥2|J|, (21) holethermodynamics[25].Finally,inRef. [15]thefollowing
questionis posed: howsmall a black hole can be? Though,
togetherwithinequality(19)leadtothearea-angularmomen- accordingtoinequality(1)rotatingclassicalblackholescan-
tuminequality(1)and,inaddition,arigidityresultfollows:if notbe arbitrarilysmall, underthe lightof Eq. (8) one could
equalityin(1)holds,first, theintrinsicgeometryofS isthat haveviolationsofA ≥ 8π|J|innearextremalsemi-classical
ofanextremeKerrthroatsphere[12]and,second,thevanish- collapseduetocorrectionsviolatingthedominantenergycon-
ing of the positive-definiteterms in (8) implies in particular, dition,inparticularrelevantwhentheblackholeissmall.This
forspacelikeXain(7),thevanishingoftheshearσ(ℓ)sothat isalsoconsistentwiththeviolationsofinequality(1)foundin
ab
S isaninstantaneous (non-expanding)isolatedhorizon[24]. Ref. [26], in the contextof blackholesaccretingmatter that
Discussion. We have shown that axisymmetric stable violates the null (and therefore the dominant) energy condi-
marginallytrappedsurfaces(inparticular,apparenthorizons) tion. Equation (8) provides a tool to estimate such possible
satisfy the inequality A ≥ 8π|J| in generically dynamical, violations.
non-necessarilyaxisymmetric,spacetimeswithordinarymat-
terthatcanextendtothehorizon. Therearetwokeyingredi- Acknowledgments. It is a pleasure for J.L.J. to thank M.
entsenablingtheremarkableshiftfromtheinitialdatadiscus- Ansorg for highlighting the physical and geometrical rele-
sion of inequality(1) in [15] to a (purelyquasi-local)space- vanceofthisproblemandforhisencouragement.S.D.thanks
time result. First, the derivation of the geometric inequality J.M.Espinarfordiscussions. S.D.andJ.L.J.thankA.Acen˜a,
(8)wherethespacetimeinterpretationofeachtermintheright M.E. Gabach Cle´ment for enlightening discussions. J.L.J.
handsideistransparentand,moreimportantly,theglobalsign also thanks C. Barcelo´, R.P. Macedo, M. Mars, A. Nielsen
is controlledbystandardphysicalassumptionsonthe matter and J.M.M. Senovilla. J.L.J. acknowledges the Alexander
energycontent. Thisrelaxesthecounterpartmaximalslicing von Humboldt Foundation, the Spanish MICINN (FIS2008-
hypothesisin[15]. Second,usingthesphericaltopologyofS 06078-C03-01)andtheJuntadeAndaluc´ıa(FQM2288/219).
weexpressthequadratictermcontrollingtheangularmomen- S.D. issupportedby CONICET (Argentina). Thisworkwas
tumintheinequalityintermsofapotentialω¯ livingsolelyon supported by grant PIP 6354/05 of CONICET (Argentina),
the sphere and leading to an exact match with the key vari- grantSecyt-UNC(Argentina)andthePartnerGroupgrantof
ational functionalin [15]. This permits to avoid any further theMPIforGravitationalPhysics(Germany).
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