The area-angular momentum inequality for black holes in cosmological spacetimes

0.26 MB·English

by  Maria Eugenia Gabach Clement

The area-angular momentum inequality for
black holes in cosmological spacetimes
MAR´IA EUGENIA GABACH CLEMENT,
[email protected]
MART´IN REIRIS,
[email protected]
5
1
0 WALTER SIMON,
2 [email protected]

b
e For a stable marginally outer trapped surface (MOTS) in an axi-
F ally symmetric spacetime with cosmological constant L 0 and with

2 matter satisfying the dominant energy condition, we pr>ove that the
area A and the angular momentum J satisfy the inequality 8p J
]
c A√ 1 L A 4p 1 L A 12p which is saturated precisely for the∣ e∣x≤-
q
trem(e −Kerr-/deS)it(ter−fam/ily o)f metrics. This result entails a universal
-
r upper bound J J 0.17 L for such MOTS,which is saturated for
g max
[ oneparticular∣e∣x≤treme≈config/uration. Ourresultsharpenstheinequality
8p J A, [7,14], and we follow the overall strategy of its proof in the
2
v sen∣se∣≤that we first estimate the area from below in terms of the energy
3 correspondingtoa“massfunctional”,whichisbasicallyasuitablyregu-
4 larised harmonic mapS2 H2. However, inthecosmological case this
2
mass functional acquires→an additional potential term which itself de-
7
0 pendsonthearea. Toestimatethecorresponding energy intermsofthe
1. angularmomentumandthecosmological constant weuseasubtlescal-
0 ing argument, a generalised “Carter-identity”, and various techniques
5
fromvariational calculus, including themountainpasstheorem.
1
:
v PACS: 02.30.Xx,04.70.Bw,02.40.Hw,02.40.Ma,02.40.Vh
i
X Keywords: area inequality,apparent horizon,cosmologicalconstant
r
a
1 Introduction
Someremarkable area inequalitiesforstablemarginallyouter trapped surfaces (MOTS) have
been provenrecently [7], [9], [14], [16], [6], [10]. In particular,foraxially symmetricconfig-
urationswitharea A andangularmomentumJ, thereisthebound[7], [14]
A
J , (1.1)
8p
which is saturated for extreme Kerr black∣h∣o≤les. Although a cosmological constant L does
not explicitly enter into (1.1), this inequality holds in the presence of a non-negative L . On
theotherhand, when L 0,stableMOTSobeythelowerbound
> A 4p L −1, (1.2)
≤
1
1 INTRODUCTION
saturated for the extreme Schwarzschild-deSitter horizon [12]. This readily implies the uni-
versal upperbound
J 2L −1 (1.3)
which, however,can neverbe saturated even in theory (leaving practical considerations aside
∣ ∣≤( )
in viewofthefact that L −1 isoforder10122.
The situation bears some analogy to stable MOTS in (not necessarily axially symmetric)
spacetimes with electromagnetic fields and electric and magnetic charges Q and Q . In
E M
this case the inequalities A 4p Q2 [9] with Q2 Q2 Q2 (saturated for extreme Reissner-
E M
Nordstro¨m horizons) and A 4p L −1 imply the (unsaturated) bound Q2 L −1. There is how-
≥ = +
everthestrongerbound[16]
≤ L A2 4p A 16p 2Q2 0 ≤ (1.4)
which is saturated for extreme Reissner-Nordstro¨m-deSitter configurations and, moreover,
− + ≤
improvestheuniversalchargeboundto Q2 4L −1.
Returning to the present axially symmetric case, the main objective of this article is to
≤( )
incorporate explicitly the cosmological constant into inequality (1.1) and determine how it
controlstheallowedvaluesoftheangularmomentum. We provethefollowingtheorem.
Theorem 1.1. Let S be an axially symmetric, stable MOTS together with an axially sym-
metric 4-neighborhood of S called N,g . On N,g we require Einstein’s equations to
ij ij
hold, with L 0 and with matter satisfying the dominant energy condition. Then the angular
momentumJ andthearea A ofS sat(isfy ) ( )
>
A L A L A
J 1 1 , (1.5)
8p √ 4p 12p
∣ ∣ ≤ (3−2 )( −1 )0.17
J J 1 . (1.6)
max 8L √4 3 3 L
∣ ∣ ≤ = √ ( −√ )≈
Here (1.5) is saturated precisely for the 1-parameter family of extreme Kerr-deSitter (KdS)
horizonswhiletheuniversalbound(1.6)is saturatedfor oneparticularsuchconfiguration.
Theproofofthis theoremwillbesketched in Sect. 4, whiledetailsare postponedto Sect.
5. We discuss now its scope and the main differences, similarities and difficulties compared
to theones citedabove.
As L 0, the main inequality (1.5) is stronger than both (1.1) and (1.2); in particular it
forbids the black hole to rotate as fast as its non-cosmological counterpart. Concerning the
>
saturation of (1.5), we observe the same pattern as in the previous inequalities: the extreme
solutions set a bound to the maximum values of charges and/or angular momentum. The
non-vanishingcosmologicalconstantdoesnot changethisproperty ofextremeblack holes.
Inequality (1.6) is obtained in a straightforward manner from (1.5) and makes use of an
interesting feature of the extreme KdS family. Given L 0 there exists a maximum value
for the angular momentum which is attained at a certain value of the area A. This property
is not shared by extreme Kerr horizons (L 0), where the>value of A determines the angular
momentum as 8p J A. Note also that, as opposed to (1.3), (1.6) is sharp and improves the
=
numericalfactor from0.5 to0.17approximately.
∣ ∣=
2
1 INTRODUCTION
AsstatedinTheorem1.1,theinequality(1.5)holdsbetweentheareaandangularmomen-
tum of stable MOTS’s. Nevertheless, due to the analogy between stable MOTS and stable
minimal surfaces in maximal slices, one can prove an analogous result for this type of sur-
faces as well (see [6] for a discussion of the similarities of these surfaces within the context
ofgeometricinequalities).
Note that matter satisfying the dominant energy condition (DEC) is allowed. The energy
condition is required in order to dispose of the matter terms and to arrive at the ’clean’ in-
equality (1.5) where matter does not appear explicitly. However, for electromagnetic fields
we expectto obtainan inequalitybetween area, angularmomentum,electromagneticcharges
Q , Q and cosmological constant which should reduce to (1.5) for Q 0 and to (1.4) when
E M
J 0. We discussa correspondingconjecturein Sect. 6.
=
WenowcommentontheproofTheorem1.1whichisnotastraightforwardgeneralisation
=
of previous results. To explain this we recall briefly the basic strategy of [7], [14] that leads
to (1.1). Starting with the stability condition one obtains a lower bound for the area of the
MOTS in terms of a “mass functional” M. This M is the key quantity in the proof, and
dependsonlyonthetwistpotentialandthenormoftheaxialKillingvector. Thenon-negative
cosmological constant and the matter terms (satisfying the DEC) neither appear in M nor
laterinthediscussionin thiscase. Therefore, theproblemreduces tovacuumand withL 0.
Then, a variational principle is used to obtain a lower bound for M. The key point in this
step is the relation between M and the “harmonic energy” of maps between the two-sph=ere
and the hyperbolic plane. This allows to use and adapt a powerful theorem by Hildebrandt
et al. [13] on harmonic maps, which gives existenceand uniqueness of the minimiserfor M.
Thisminimiser,in turn,givestherighthandsideof(1.1).
In the present work where we strengthen (1.1) to (1.5), two important obstacles appear.
Firstly, the area A now appears not only as upper bound on the corresponding functional M
but also explicitly in M itself. This makes the variational principle hard to formulate. We
overcomethisprobleminessenceby“freezing”AaswellasJ tocertainvaluescorresponding
to an extreme KdS configuration, and by adapting the dynamical variables in M suitably.
Secondly, the relation of M to harmonic maps mentioned above no longer persists, whence
theproofofexistenceanduniquenessofaminimiserforMhastobedoneherefromscratch.
WeproceedbyprovingfirstthateverycriticalpointofMisalocalminimum. Finallyweuse
themountainpass theoreminorderto get thecorrespondingglobalstatement.
Ourpaper isorganisedas follows.
In Sect. 2 we recall and adapt some preliminary material, in particular the definition of
angularmomentumforgeneral2-surfaces,aswellasthedefinitionofastableMOTS.InSect.
3 we discuss relevant aspects of the KdS metric, focusing on the extreme case. In Sect. 4 we
sketchtheproofofTheorem1.1,postponingthecoreoftheargumenttothreekeypropositions
which areproveninSect. 5.
InSect. 6weconjectureageneralisationofourinequalitytothecasewithelectromagnetic
field along thelinesmentionedabovealready,and wealsodiscussbriefly thecaseL 0.
<
3
2 PRELIMINARIES
2 Preliminaries
2.1 The geometric setup
WeconsideramanifoldN whichistopologicallya4-neighborhoodofanembedded2-surface
S ofsphericaltopology. N carriesametricg andaLevi-Civitaconnection . (Latinindices
ij i
from ionwardsrun from0 to3,and themetrichassignature , , , ). Thefield equations
∇
are
G L g 8p T (− + + +) (2.1)
ij ij ij
whereL isthecosmologicalconstant,andtheenergymomentumtensorT satisfiesthedom-
=− + ij
inant energy condition. In Sections 2 and 3 we allow L to have either sign; this enables us
to compare with and to carry over useful formulas from work which focuses on Kerr-anti-
deSitter, inparticular[4]and [5].
We next introduce null vectors ℓi and ki spanning the normal plane to S and normalized
as ℓik 1. We denoteby q g 2l k theinduced metricon S,the correspondingLevi-
i ij ij (i j)
Civita connection by D and the Ricci scalar by 2R. e and dS are respectively the volume
i ij
elemen=t−and the area measure=on S+. The normalisation l ki 1 leaves a (boost) rescaling
i
freedomℓ′i fℓi,k′i f−1ki. Whilethisrescalingaffectssomequantitiesintroducedbelowin
=−
anobviousway,ourkeydefinitionssuchas theangularmomentum(2.4)andthedefinitionof
stability (2.1=2) are in=variant, and the same applies to all our results. The expansion q (ℓ), the
shears (ℓ) andthenormalfundamentalformW (ℓ) associatedwiththenullnormalℓi aregiven
ij i
by
1
q (ℓ) qij ℓ , s (ℓ) qk ql ℓ q (ℓ)q W (ℓ) kjqk ℓ . (2.2)
i j ij i j k l 2 ij i i k j
= ∇ = ∇ − =− ∇
2.2 Twist and angular momentum
We now assume that S as well as W (ℓ) are axially symmetric, i.e. there is a Killing vector h i
i
on S such that
Lh qij 0 Lh W (iℓ) 0. (2.3)
Thefield h i isnormalizedso thatitsintegralcurveshavelength2p .
= =
WedefinetheangularmomentumofS as
1
J W (ℓ)h idS , (2.4)
8p S i
which willberelated to theKomaran=gular∫momentumshortly.
By Hodge’stheorem, thereexistscalar fields w and l on S, defined up to constants,such
that W (ℓ) has thefollowingdecomposition
i
1
W (ℓ) e Djw D l . (2.5)
i 2h ij j
= +
4
2 PRELIMINARIES 2.3 Stablemarginallyoutertrappedsurfaces
From axial symmetryitfollowsthat
1 1
h iW (ℓ) e h iDjw h −1/2x iD w (2.6)
i 2h ij 2 i
= =
whereh h ih and x i isaunitvectortangent toS and orthogonaltoh i.
i
Wenowrecall from[7]thaton anyaxiallysymmetric2-surfaceonecanintroduceacoor-
=
dinatesystemsuch that
q dxidxj e2ce−s dq 2 es sin2q dj 2 (2.7)
ij
forsomefunctions anda constantcw=hich isrelat+ed totheareaA ofS viaA 4p ec. In such
acoordinatesystemwecan writeJ as
=
1 p 1
J w ′ dq w p w 0 , (2.8)
8 0 8
where here and henceforth a=p−rim∫e denotes =th−e de[riv(at)iv−e w(.r.)t.] q . From now onwards we
assumethat the Killingvectorh i on S extends to N as a Killing vectorof g . Of course this
ij
implies(2.3). Moreover,itfollowsthat Lh l Lh k 0. Usingthefirst equationweobtain
h iW (ℓ)= kjℓi= h . (2.9)
i i j
Inserting(2.9)in(2.4)weseethat itindeed=co−incid∇eswiththeKomarangularmomentum
1
J ih jdS . (2.10)
8p S ij
Wefinallyintroducethetwistvecto=r ∫ ∇
w e h j kh l. (2.11)
i ijkl
Iftheenergymomentumtensorvanishes=onN,w∇ehave w 0. Hencethereexistsatwist
[i j]
potentialw , definedup toaconstant,suchthatw w . Therestrictionofthisscalarfield to
i i
S is easilyseen to coincidewiththew introducedin (2.5∇), whic=h justifiesthenotation.
Inwhat followswewillrefer to thepair s ,w =o∇n S as thedata.
( )
2.3 Stable marginally outer trapped surfaces
We now take S to be a marginally trapped surface defined by q (ℓ) 0. We will refer to ℓi as
theoutgoingnullvector,which leadsto thenamemarginallyoutertrapped surface (MOTS).
=
Moreover, following [2] (Sect. 5) we now consider a family of two-surfaces in a neigh-
borhood of S together with respective null normals l and k and we impose the following
i i
additionalrequirementsonS and itsneighborhood.
Definition 2.1. A marginally trapped surface S is stable if there exists an outgoing ( ki-
oriented) vector Xi g ℓi y ki, with g 0 and y 0, such that the variation d of q (ℓ) with
X
−
= − ≥ >
5
3 KERR-DESITTER
respect to Xi fulfillsthecondition
d q (ℓ) 0. (2.12)
X
Two remarks are in order here. Firstly, it is easy to see (cf. Sect. 5 of [2]) that stability
≥
of S w.r.t. some direction Xi implies stability w.r.t all directions “tilted away from” ℓi. In
particular,sinced −y kq (ℓ) d Xq (ℓ) stabilityw.r.t. anyXi impliesstabilityinthepastoutgoing
nulldirection ki. Thislatterconditionsuffices as requirementforall ourresults.
≥
The other remark concerns the relation between stability and axial symmetry. We recall
−
thatin[7],[14],inequality(1.1)wasprovenunderthesymmetryrequirements(2.3)andunder
a stability condition similar to Definition 2.1 which, however, required y to be axially sym-
metricaswell. (Axialsymmetryofg wasalsoassumedbutnotusedintheproof). Incontrast,
in the present theorem (1.1) we impose the stronger symmetry requirement that S as well
as its neighborhood N are axially symmetric. In this case it suffices to impose the stability
condition (2.1) as above, namely without explicitly requiring axial symmetry of y , since the
existenceofanaxiallysymmetricfunctiony thenfollowsautomatically,cf. Thm. 8.2. of[2].
Moreover, for strictly stable MOTS (which satisfy d q (ℓ) 0 in addition to (2.12)) there fol-
X
̃
lows even axial symmetry of the surface itself if its neighborhood is axially symmetric (cf.
≡/
Thm. 8.1. of[2]).
3 Kerr-deSitter
In this section we review some relevant properties of the event horizons of the Kerr-deSitter
(KdS) solutions, making use of [4], [5], and references therein. Other aspects of the rich and
complexstructureofthesespacetimescan befoundin [11].
3.1 The metric, the horizon and the angular momentum
In “Boyer-Lindquist”coordinates,theKdSmetricis
z asin2q 2 r 2 r 2 c sin2q r2 a2 2
ds2 dt df dr2 dq 2 adt df (3.1)
r 2 k z c r 2 k
+
=− ( − ) + + + ( − )
where
L r2
z r2 a2 1 2mr, r 2 r2 a2cos2q (3.2)
3
L a2 L a2cos2q
k = (1 + ),( − c )1− = + (3.3)
3 3
= + = +
wherem 0and 0 a2 L −13 2 3 2.
As a function of r, z has one neg√ative root and three positive roots (possibly counted
≥ ≤ ≤ ( − )
with multiplicities). The greatest root, r , marks thecosmologicalhorizon, whilethe second
ch
greatest, r , markstheeventhorizon(from nowonsimplycalled “horizon”).
h
6
3 KERR-DESITTER 3.2 Extremehorizons
Thearea ofthehorizonis
4p r2 a2
A h (3.4)
k
( + )
and theinducedmetricon itreads =
m 2 k 2r 4
ds2 h h dq 2 sin2q df 2 (3.5)
k 2r 2 m 2c
h h
= es ( e2q + )
² ²
where m 2 r2 a2 2c and r r2 a2cos2q .
h h h h
Hence
=( + ) = + r2 a2 A
es +q H ec const. (3.6)
k 4p
+
and themetricisin the”canonical f=orm”(2.7=)of=[7] =
ds2 es e2qdq 2 sin2q df 2 (3.7)
withs q c const.. = ( + )
We now calculate the twist potential w h everywhere (not only on S), for h a ¶ df .
Adaptin+ga=kn=owncalculationinthecaseL 0(cf. e.g. AppendixAof[3]andomittingsome
( ) = /
intermediatesteps,wefind
=
z sinq
w ′ w q e qf rtgrrgtt¶ rh t e qf rtgrrgtf ¶ rh f k gtt¶ rgtf gtf ¶ rgff (3.8)
k 2masin3q 2r2
= c =sinq gff ¶ rgtf +gtf ¶ rgff k=2−r 2 r2( a2 r 2+ r2 a2 )= (3.9)
= −2ma ¶ ( − a)2c=o−sq sin4q [ − + ( + )]=
cos3q 3cosq (3.10)
k 2 ¶q r 2
= − ( − − )
It followsthat
2ma a2cosq sin4q
w cos3q 3cosq (3.11)
k 2 r 2
We note that compared to=th−e case(L 0,−w just g−ets an extra fact)or 1 k 2. Integrating and
using(2.8)weobtaininparticularthat
= /
J am k 2 (3.12)
which agrees withEqu. (2.10)of[5]and Eq=u. (1/8)of[4].
3.2 Extreme horizons
When at least two of the three non-negative roots of z r coincide, (one of which is neces-
sarily r ), the horizon is called extremal. When this happens the geometry near the horizon
h
( )
7
3 KERR-DESITTER 3.2 Extremehorizons
degeneratestoa“throat”. Wereferto[5]forafurtherdiscussion. Inwhatfollowswewilljust
need therelationbetween theparameters m,a,L ,Aand J whichwederiveexplicitly.
For extremal event horizons the radius of the limiting sphere r satisfies, in addition to
e
z r 0, theequation
e
( )= 1dz 2L r3 L a2
0 e r 1 m. (3.13)
e
2 dr 3 3
e
= ∣ =− + ( − )−
Here and henceforth a subscript e indicates extremality. Eliminating m from z r 0 and
e
(3.13)weobtain
L a2 ( )=
L r4 r2 1 a2 0. (3.14)
e e
3
For L 0 this equation has just a s+ing(le roo−t w)h+ich c=an be called extremal horizon, while
forL 0 thereare twosolutionsr r forgivenJ Explicitly,forL 0,
e ±
≤
> 1 =L a2 1 L a2 2 >
r2 1 1 4a2L . (3.15)
± 2L 3 2L ¿ 3
Á
ÁÀ
When r r , (and r is no=t a tr(ipl−eroot)), ±the first t(wo−positi)ve−roots meet and r r , which
e − e e ch
meansthatacosmologicalhorizonpersistsinspacetime. Ontheotherhandwhenr r ,then
e +
= <
the last two positiveroot coincide and the event and the cosmological horizons become both
=
extremal(and merge).
Using(3.14)toeliminatea2 from (3.4)wefind
8p r2
A e . (3.16)
1 L r2
e
=
On theotherhand, eliminatingre from (3.14) a+nd(3.4)gives
A 1 L A 4p
a2 . (3.17)
4p 1 L A 8p 1 L A 12p
− /
=
In equation (3.14) we eliminate n(ow− m u/sin)g((3−.13),/then)a2 using (3.14) and finally r2
e
using(3.14). Weobtainthefollowingsimplerelationbetweentheangularmomentumandthe
area forextremeK(a)dS
A L A L A
J E A 1 1 (3.18)
8p √ 4p 12p
which after a trivial reform∣u∣la=tio(n a)g∶r=ees with((2−.32))o(f [5−]. In )the case L 0 and J 0 the
zeros of the parentheses correspond to the black hole horizon and the cosmological horizon
> =
ofSchwarzschild-deSitter, respectively.
For L 0 we are only interested in the domain L A 4p 1 - recall that this bound can be
shown for all stable MOTS (irrespectively of spherical symmetry) [12]. In this range of A,
> / <
8
4 THESTRUCTUREANDTHEPROOFOFTHEMAINTHEOREM
(3.18)takeson amaximalvalue
3 2 1 0.17 6p 1
J 1 at A 1 (3.19)
max 8L √4 3 3 L max L 3
= √ ( −√ )≈ = ( −√ )
which is the value stated in (1.6). Moreover, for each J with J J there are two values
max
A J A J forthearea, cfFig 1.
− +
∣ ∣<
J
( )< ( )
J
max
J E A
= ( )
J
A
A J A A J 4p
− max + L
Figure1: Theshadedregionrepresents allpointssatisfying J A .
( ) ( )
∣ ∣≤E( )
Wearenowready to describetheproofofTheorem1.1.
4 The structure and the proof of the main theorem
Themaininequality
J E A (4.1)
with E given in (3.18) and L 0 will not be shown directly but it will follow from a related
∣ ∣≤ ( )
one. ThisisexplainedinthefollowingTheorem:
>
Theorem 4.1. For any given MOTS with area A, cosmological constant L and angular mo-
mentum J, there is a unique extreme KdS solution with area Aˆ constant L and angular mo-
mentum Jˆsuchthat
J Jˆ
, (4.2)
A2 Aˆ2
∣ ∣ ∣ ∣
and AˆL 4p . Moreover, theinequality J E=A isequivalentto theinequality
≤ ∣ ∣≤Aˆ( A). (4.3)
≥
9
4 THESTRUCTUREANDTHEPROOFOFTHEMAINTHEOREM
J
J const.A2
Jˆ =
J E A
= ( )
J
A Aˆ A
Figure2: Theconstruction described inTheorem4.1
Proof. Thefirst result, leading to equation(4.2), is intuitivelyclear from Fig 2 since through
any point A,J there is a unique parabola J A2 const., and any such parabola intersects
the “extreme” curve J E A precisely once apart from the trivial point 0,0 . To state this
rigorously,(let l) A Aˆ and hence Jˆ l 2 J a/nd A=ˆL 4p . Then the hatted version of (3.18)
givesaquadraticequati=on(for)l J,A . If32p 2 3J L A2 thisequationha(sau)niquesolution
∶= / ∣ ∣= ∣ ∣ ≤
other than 0,0 . Otherwise, there are two no√n-trivial solutions but only one of them lies in
theregionofinterestAˆL 4p . ( ) ∣ ∣>
Toprov(ethe)equivalencebetween (4.1)and (4.3),assumefirst thatAˆ A. Then
≤
A2 A2 E l A Aˆ2 ≥
A2 Aˆ2 Jˆ E Aˆ (4.4)
J J l 2 J
( )
≤ =∣ ∣ = ( ) =
where we have used (4.2), (3.18) and A∣ˆ∣ l A, re∣sp∣ectively. W∣ e∣ next use that the function
E(l A) is monotonically decreasing with l and therefore, as Aˆ A we bound the last term as
l 2
E(l A) E A . Puttingthistogetherwith(4=.4)wefind
l 2 ≥
≤ ( ) Aˆ2
Aˆ2 E A (4.5)
J
≤ ( )
which givesthedesired result,that is,that(4.3)im∣pl∣ies(4.1).
Toprovetheconverseassume J E A . Then Jˆ l 2J and(3.18)give
E ∣l ∣A≤ (Jˆ) l 2 J =l 2E A (4.6)
and therefore ( )=∣ ∣= ∣ ∣≤ ( )
E l A
E A . (4.7)
l 2
( )
Again,duetothemonotonicityofthelefthand≤sid(ew)ithrespecttol weobtainl A Awhich
≥
10