Area inequalities for stable marginally trapped surfaces

0.2 MB·English

by  José Luis Jaramillo

Area inequalities for stable marginally trapped
surfaces
Jose´ LuisJaramillo
2
1
0
2

n
a
J Abstract Wediscussafamilyofinequalitiesinvolvingthearea,angularmomentum

0 andchargesofstablyoutermostmarginallytrappedsurfacesingenericnon-vacuum
1 dynamicalspacetimes,withnon-negativecosmologicalconstantandmattersources

satisfyingthedominantenergycondition.Theseinequalitiesprovidelowerbounds
]
c for the area of spatial sections of dynamical trapping horizons, namely hypersur-
q faces offering quasi-local models of black hole horizons. In particular, these in-
-
r equalitiesrepresentparticularexamplesoftheextensionto aLorentziansettingof
g
toolsemployedinthediscussionofminimalsurfacesinRiemanniancontexts.
[

1
v
4 1 Introduction
5
0
2 TheLorentziannatureofspacetimegeometry,withitsinherentnotionofnullcone,
. controlstherichfeaturesoflightbendingingeneralrelativity.Thisincludesinpar-
1
0 ticularthepossibilityofcausaldisconnectionbetweenspacetimeregions,aswellas
2 theconvergencebehaviorof(trapped)lightrays.Bothaspects,relatedbythenotion
1 of (weak) cosmic censorship [44], lay at the basis of the conceptof black hole in
v: generalrelativity.Inspiteofthecomplexityofthegenericsituation,itisremarkable
i thatstationaryandvacuumblackholespacetimesarecompletelycharacterizedby
X
afewparameterswithphysicalinterpretation,namelythemassM(or,alternatively,
r
theareaAofthehorizon),theangularmomentumJ andcertainchargesQ.These
a i
parameters fulfill a class of geometric inequalities that bound the mass by below.
WhenusingthehorizonareaAinsteadofthemass,theypresentthegeneralform
(A/(4p ))2 (2J)2+((cid:229) Q2)2 , (1)
≥ i
i
Jose´LuisJaramillo
Max-Planck-Institutfu¨rGravitationsphysik,AlbertEinsteinInstitut,AmMu¨hlenberg1D-14476
PotsdamGermany,e-mail:[email protected]
1
2 Jose´LuisJaramillo
andprovidea familyofinequalitieswrittencompletelyin termsofthe quasi-local
geometry of the black hole horizon. As a second remarkable point, these quasi-
localgeometricinequalitiesdoextendtofullygenericdynamicalandnon-vacuum
situations,providinggenerallowerboundsfortheblackholehorizonarea.
In the stationary axisymmetriccase with matter surroundingthe horizon,these
quasi-localinequalitieshavebeenprovedtoholdfortheKillinghorizonin[6,40,
41, 5]. Regardingdynamicalsituations, and after the study in [25, 26, 27, 21, 22,
24,23]oftheglobalvacuumaxisymmetricinequalitiesinvolvingM,thequasi-local
vacuum case has been studied in [28, 1, 33, 31] using axisymmetric initial data.
Finally, in [43, 30, 34, 35] a purely spacetime (Lorentzian) perspective has been
adopted,permitting to identify and refine the key assumptions, this leading to the
extensionoftheinequalitiestogenericdynamicalscenarioswithmatter.Thestudy
oftheseinequalitiesinhigherdimensionshasbeenstartedin[42].Ageneralreview
ongeometricinequalitiesinaxiallysymmetricblackholesispresentedin[29].
Here we discuss these quasi-localinequalities, placing the emphasis on the in-
volvedLorentzianaspects,namelythenotionofstabilityofmarginallyoutertrapped
surfaces.The latter providea Lorentziananalogueto the notionof stable minimal
surfacesinRiemanniangeometry.Newresultsarepresentedregardingtheincorpo-
rationintheinequalitiesofYang-Millschargesanda geometricchargeforcertain
divergence-freevectorsonclosedsurfaces.We alsocommentontheinterpretation
oftheintegralcharacterizationofthestabilityconditionasanenergyfluxinequality.
2 Geometricand physicalelements
Let(M,g )bea4-dimensionalLorentzianmanifoldsatisfyingEinsteinequations
ab
G +L g =8p T , (2)
ab ab ab
whereG =R 1Rg istheEinsteintensorassociatedwiththeLevi-Civitacon-
nection(cid:209)ab,L iasbt−he2cosambologicalconstantandT thestress-energytensor.Unless
a ab
otherwise stated, in the following the stress-energy tensor is assumed to satisfy a
dominantenergycondition(namely,givena future-directedcausal vector va, then
Ta vbisafuture-orientedcausalvector)andthecosmologicalconstanttobenon-
b
n−egativeL 0.WeusethroughoutPenrose’sabstractindexnotation(e.g.[51]).
≥
2.1 Geometryof2-surfaces
Letusconsidera closedorientable2-surfaceS embeddedin (M,g )(inthe fol-
ab
lowing, we shall assume that surfaces S are closed and orientable, unless oth-
erwise stated). Let us denote the induced metric on S as q , with Levi-Civita
ab
connectionD , Ricci scalar 2R and volumeelemente (we will denoteby dS the
a ab
Areainequalitiesforstablemarginallytrappedsurfaces 3
areameasureonS).DecomposingthetangentplaneT M ateachpoint p S as
p
T M =T S T S, let us consider null vectors ℓa and ka spanning the∈normal
p p ⊕ p⊥
planeT S andnormalizedasℓak = 1.Thisleavesa(boost)rescalingfreedom
p⊥ a −
ℓa= fℓa,ka= f 1ka.Wecanwrite
′ ′ −
q =g +k ℓ +ℓ k . (3)
ab ab a b a b
Regardingtheextrinsiccurvatureelementsthatweneedinouranalysis,letuscon-
siderthedeformationtensorsQ (ℓ)andQ (k)alongℓa andka,respectively
ab ab
Q (ℓ) qc qd (cid:209) ℓ , Q (k) qc qd (cid:209) k . (4)
ab ≡ a b c d ab ≡ a b c d
TheydeterminethesecondfundamentalformK c of(S,q )into(M,g ),namely
ab ab ab
K c qd qe (cid:209) qc =kcQ (ℓ)+ℓcQ (k)(cf.Senovilla’scontributioninthisvolume).
ab≡ a b d e ab ab
Inparticular,theexpansionq (ℓ) andtheshears (ℓ) associatedwiththenullnormal
ab
ℓa,aregivenrespectivelybythetraceandtracelesspartsofQ (ℓ)
ab
1
q (ℓ) qabQ (ℓ)=qab(cid:209) ℓ , s (ℓ) Q (ℓ) q (ℓ)q . (5)
≡ ab a b ab ≡ ab −2 ab
Inaddition,weconsiderthenormalfundamentalformW (ℓ)
a
W (ℓ)= kcqd (cid:209) ℓ , (6)
a − a d c
that providesa connectionon the normalbundle T S. More specifically, consid-
∗
ering a form v T S, expressed as v =a ℓ +⊥b ℓ , we can write qc (cid:209) v =
a ∈ ∗ a a b a c b
Q (v)+D v ,whereQ⊥(v)=qc qd (cid:209) v =aQ (ℓ)+bQ (k)and
ab ⊥a b ab a b c d ab ab
D v =D (a ℓ +b ℓ )=(D a +W (ℓ)a )ℓ +(D b +W (ℓ)b )k . (7)
⊥a b ⊥a a b a a b a a b
Transformationrulesunderanullnormalrescalingℓa= fℓa,ka= f 1ka are
′ ′ −
q (ℓ′)= fq (ℓ) , s a(bℓ′)= fs a(bℓ) , W a(ℓ′)=W a(ℓ)+Da(lnf). (8)
2.1.1 Axisymmetry
TheintroductionofacanonicalangularmomentumJ onS requiresimposingax-
isymmetry.Inthiscontext,werequirethegeometryofS tobeaxisymmetricwith
axialKillingvectorh a onS.Morespecificallywerequire
Lh qab=0 , Lh W a(ℓ)=0 , Lh ℓa=Lh ka=0, (9)
where h a has closed integral curves, vanishes exactly at two points on S and is
normalizedsothatitsintegralcurveshaveanaffinelengthof2p .Weadoptatetrad
4 Jose´LuisJaramillo
(x a,h a,ℓa,ka)onS,wheretheunitvectorx atangenttoS satisfiesx ah =x aℓ =
a a
x ak =0,x ax =1.Wecanthenwriteq = 1h h +x x ,withh =h ah ,and
a a ab h a b a b a
W (ℓ)=W (h )+W (x ) , W (ℓ)W (ℓ)a=W (h )W (h )a+W (x )W (x )a , (10)
a a a a a a
withW (h )=h bW (ℓ)h /h andW (x )=x bW (ℓ)x .Wecanintroducenowacanonical
a b a a b a
(gravitational)angularmomentumas
1
J = W (ℓ)h adS, (11)
K 8p S a
Z
wherethe divergence-freecharacterof h a togetherwith the transformationsprop-
erties of W (ℓ) in (8) guarantee the invariance of J under a rescaling of the null
a
normals. This angular momentum on S coincides with the Komar one, namely
J = 1 (cid:209) h dSabwithdSab= 1(kaℓb ℓakb)dS,ifh acanbeextendedasa
KKiollminargve8cptorStoaaspbacetimeneighborh2oodof−S.
R
2.2 Electromagneticfield
Let us consider an electromagneticfield on (M,g ) with strength field (Faraday)
ab
tensorF .OnalocalchartwecanexpressF intermsofavectorpotentialA as
ab ab a
F =(cid:209) A (cid:209) A .Theelectromagneticstress-energytensorisgivenby
ab a b b a
−
1 1
TEM= F F c g F Fcd . (12)
ab 4p ac b −4 ab cd
(cid:18) (cid:19)
GivenS,wedenotetheelectricandmagneticfieldcomponentsnormaltoS as
E =Fabℓakb , B =∗Fabℓakb , (13)
⊥ ⊥
where F is thedualofF , namely F = 1e Fcd withe thevolumeele-
∗ ab ab ∗ ab 2 abcd abcd
mentofg .Electricandmagneticchargescanbeexpressedas(e.g.[10,17])
ab
1 1
Q = E dS , Q = B dS. (14)
E 4p S ⊥ M 4p S ⊥
Z Z
When discussing the angular momentum in the presence of an electro-magnetic
field,weaddLh Aa=0totheaxisymmetryrequirements(9).Then,thefollowing
canonicalnotionoftotalangularmomentumcanbeintroducedonS [19,48,8,29]
1 1
J=J +J = W (ℓ)h adS+ (A h a)E dS. (15)
K EM 8p S a 4p S a ⊥
Z Z
Areainequalitiesforstablemarginallytrappedsurfaces 5
2.2.1 Yang-Millsfields
GivenaYang-MillstheorywithLiegroupG,thedynamicalfieldsaregiveninterms
ofa1-formA evaluatedontheLiealgebraG ofG.Moreproperly,A isaconnec-
a a
tionon a principalG bundleP overthe spacetimeM. Denotingthegeneratorsin
G asT andwritingth−eLie-algebracommutationrelationsas
i
[T,T ]=CkT , (16)
i j ij k
theCartan-KillingquadraticformonG isgivenby
k =CkCl , (17)
ij il jk
which is non-degeneratefor semisimple Lie algebras. For real compact Lie alge-
bras,k isnon-degenerateandpositive-definite(usuallyabasis T ofG suchthat
ij i
k =d isemployed).Moregenerally,thenon-degeneratepositi{ve-}definitecharac-
ij ij
terofk holdsforLiegroupscorrespondingtoproductsofcompactrealLiegroups
ij
andU(1)factors.WritingtheYang-MillsconnectionasA =A iT,theYang-Mills
a a i
tensorF =F iT isgivenbythecurvatureofA ,thatisF =(dA) +A A =
ab ab i a ab ab a b
∧
(cid:209) A k (cid:209) A k+CkAiAj T .TheYang-Millsstress-energytensorcanbewritten
a b − b a ij a b k
(cid:16) (cid:17)
TYM= 1 k F iF cj 1g F iFcdj . (18)
ab 4p ij ac b −4 ab cd
(cid:18) (cid:19)
WecandefineYang-Millselectricandmagneticcharges[20,50,11]as
1 1
QYM= EYM dS , QYM= BYM dS, (19)
E 4p Z | ⊥ | M 4p Z | ⊥ |
where
1 1
EYM = F ikaℓb k F jkcℓd 2 , BYM = F ikaℓb k F jkcℓd 2(2.0)
ab ij cd ∗ ab ij ∗ cd
| ⊥ | | ⊥ |
h(cid:16) (cid:17) (cid:16) (cid:17)i h(cid:16) (cid:17) (cid:16) (cid:17)i
Electromagnetic theory corresponds to the commutative case G =U(1). In par-
ticular, the Yang-Mills principal fiber-bundle perspective sheds light on the topo-
logical nature of the magnetic charge Q , offering an understandingof magnetic
M
monopolesasassociatedwiththenon-trivialityoftheU(1)-bundleoverM(seee.g.
[52,47]),whereQ iscontrolledbythefirstChernclassoftheU(1)-bundle.
M
3 Stability ofmarginallyouter trapped surfaces
The stability for marginally trapped surfaces is the crucial element in the present
discussionoftheareainequalities.Thisnotionisextensivelyreviewedinthecontri-
butionbyM.Marsinthisvolume.Wediscussthebasicelementshereneeded.
6 Jose´LuisJaramillo
3.1 Basicdefinitions
First,wechooseconventionallyℓaastheoutgoingnullvectoratS,andrefertoS
as a marginallyoutertrappedsurface(MOTS)if q (ℓ)=0. Note thatnocondition
is requiredon the ingoingexpansionq (k). The stability of MOTSis introducedin
termsofthedeformationoperatord onS,thatcontrolstheinfinitesimalvariations
v
of geometric objects defined on S under an infinitesimal deformationof the sur-
facealonga vectorva onS (here,va willbealwaysnormalto S).Thisoperator
d ,discussedindetailin[3,4](seealsoM.Marscontributionand[16,18]),isthe
v
analogueintheLorentziansettingtothedeformationoperatoremployedinthedis-
cussion of minimal surfaces in Riemannian geometry.We require S to be stably
outermostinthesenseintroducedin[3,4](seealso[37,46]):
Definition1. GivenaclosedorientablemarginallyoutertrappedsurfaceS anda
vectorvaorthogonaltoit,wewillrefertoS asstablyoutermostwithrespecttothe
direction va iff there exists a functiony >0 on S such that the variation of q (ℓ)
withrespecttoy vafulfillstheconditiond y vq (ℓ) 0.
≥
Morespecifically,werequireS tobespacetimestablyoutermost[43,30].
Definition2. A closed orientable marginally outer trapped surface S is referred
to as spacetime stably outermostif there exists anoutgoing( ka-oriented)vector
Xa=g ℓa y ka,withg 0andy >0,withrespecttowhichS−isstablyoutermost:
− ≥
d q (ℓ) 0. (21)
X
≥
If,inaddition,Xa(i.e.g ,y )andW (ℓ)areaxisymmetric,wewillrefertod q (ℓ) 0
a X
≥
asan(axisymmetry-compatible)spacetimestablyoutermostcondition.
Alternatively,onecouldintroducethenotionofstabilityforMOTSintermsofthe
non-negativityoftheprincipaleigenvaluel ofthestabilityoperatorL associated
v v
withd v,namelyLvy =d y vq (ℓ).AlthoughLv isnotself-adjoint,itsprincipaleigen-
value(i.e.theeigenvaluewithsmallestrealpart)isindeedreal.Then,thecharacter-
izationinDefinition1canbeprovedasalemma[3,4].Thisisthestrategyfollowed
inthecontributionbyM.Marsinthisvolume.
Finally,notethatthe characterizationof MOTSsas spacetimestably outermost
isindependentofthechoiceoffuture-orientednullnormalsℓaandka.Indeed,given
f >0,forℓa= fℓa andka= f 1ka wecanwriteXa=g ℓa y ka=g ℓa y ka
′ ′ − ′ ′ ′ ′
(withg ′= f−1g 0andy ′= fy >0),anditholdsd Xq (ℓ′)=−f d Xq (ℓ)>0−.
≥ ·
3.2 Integral-inequalitycharacterizationsofMOTSstability
Thefirststepintheproofsofareainequalities(1)consistsincastingcondition(21)
as an integral geometric inequality over S. Condition (21) plays, for MOTS in a
Lorentzian context,a role analogousto that of the stability conditionfor minimal
Areainequalitiesforstablemarginallytrappedsurfaces 7
surfaces in Riemannian geometry.In the Riemannian case this refers to the mini-
mizationoftheareaofS withrespecttoarbitrarydeformationsalonga sa,where
sa is the normal to S in a given 3-slice and a is an arbitrary function on S. In
contrast,thestabilityconditioninDefinition2onlystatestheexistenceofapositive
function y (and, secondarily, of g 0). The proof of area inequalities involving
≥
the angular momentum requires writing (21) as an integral inequality in terms of
arbitrary(axisymmetric)functionsa .Thefollowinglemma[43]providesthis1
Lemma1. GivenaclosedorientablemarginallyoutertrappedsurfaceS satisfy-
ing the spacetime stably outermost conditionfor an axisymmetric Xa, then for all
axisymmetricfunctionsa onS
1
D a Daa + a 22R dS (22)
a
S 2 ≥
Z (cid:20) (cid:21)
a 2W (h )W (h )a+abs (ℓ)s (ℓ)ab+G a ℓa(a kb+b ℓb) dS,
a ab ab
S
Z
h i
whereb =ag /y .Ifinadditionweassumethattherighthandsideintheinequality
(22)isnotidenticallyzero,thenS hasaS2topology.
Proof. We basicallyfollowthediscussioninsection3.3.of[2](cf.Th.2.1in[36]
for a similar reasoning,essentially reducinga nontime-symmetriccase to a time-
symmetric one). First, we evaluate d q (ℓ)/y for Xa =g ℓa y ka in Definition 1,
X
withaxisymmetricg andy .Forthisweuse(e.g.Eqs.(2.23)−and(2.24)in[16])
d a ℓq (ℓ) =k (a ℓ)q (ℓ)−a s a(bℓ)s (ℓ)ab+Gabℓakb+12 q (ℓ) 2 ,
(cid:20) (cid:16) (cid:17) (cid:21)
d b kq (ℓ) =k (b k)q (ℓ)+2Db 2W a(ℓ)Dab
−
+b W (ℓ)W (ℓ)a DaW (ℓ) 12R+G kaℓb q (ℓ)q (k) ,
a a ab
− −2 −
(cid:20) (cid:21)
wherek (v)= vakb(cid:209) ℓ .Imposingq (ℓ)=0,wecanwriteforXa=g ℓa y ka
a b
− −
1d q (ℓ) = g s (ℓ)s (ℓ)ab+G ℓaℓb
y X −y ab ab
h i
2D lny D lny Dalny +2W (ℓ)Dalny (23)
a a
− −
DaW (ℓ)+W (ℓ)W (ℓ)c 12R+G kaℓb .
a c ab
− − −2
(cid:20) (cid:21)
We multiply by a 2 for arbitrary (axisymmetric) a and integrate on S. Using
a 2d q (ℓ)dS 0andintegratingbyparts,wecanwrite
S y X
≥
R
1 Alternatively,onecouldstartcharacterizingMOTSstabilityintermsoftheprincipaleigenvalue
l .Then, theexpression ofl inaRayleigh-Ritztypecharacterization leadsessentiallytothe
X X
integralinequality.SeeM.Marscontribution,wheretheroleofa isplayedbythefunctionu.
8 Jose´LuisJaramillo
0 ab s (ℓ)s (ℓ)ab G ℓaℓb dS
≤ S − ab − ab
Z
h i
+ a 2 W (ℓ)W (ℓ)a+12R G kaℓb dS
a ab
S − 2 −
Z (cid:20) (cid:21)
+ 2a D a Dalny a 2D lny Dalny dS
a a
S −
Z
+ (cid:2)2a 2W (ℓ)Dalny 2aW (ℓ)Daa dS(cid:3). (24)
a a
S −
Z
h i
Fromtheaxisymmetryofa andy ,W (h )aD a =W (h )aD y =0,andusing(10)
a a
0 ab s (ℓ)s (ℓ)ab G ℓaℓb dS
≤ S − ab − ab
Z
h i
+ a 2 W (h )W (h )a+12R G kaℓb dS
a ab
S − 2 −
Z (cid:20) (cid:21)
+ 2(Daa )(a D lny aW (x )) (25)
a a
S −
Z
(ahD lny aW (x ))(a Dalny aW (x )a) dS.
a a
− − −
i
MakinguseofthefollowingYoung’sinequalityinthelastintegral
Daa D a 2Daa (a D lny aW (x )) a Dlny aW (x ) 2 (26)
a a a
≥ − −| − |
inequality(22)followsforallaxisymmetrica .Finally,iftherighthandsideof(22)
doesnotvanish,thesphericityofS followsbyconsideringaconstanta in(22):it
impliesapositivevaluefortheEulercharacteristicofS.
Theproofofthearea-chargeinequality,resultingfromdroppingtheangularmo-
mentumJ in(1),requiresneitherasymmetryassumptionnorcasting(21)interms
ofanarbitrarya .Weusethefollowinglemma(slightlygeneralizingthatin[30]).
Lemma2. GivenaclosedorientablemarginallyoutertrappedsurfaceS satisfying
thespacetimestablyoutermostcondition,thenthefollowinginequalityholds
G ℓakb+N s (ℓ)s (ℓ)ab+G ℓaℓb dS 4p (1 g), (27)
S ab ab ab ≤ −
Z
h (cid:16) (cid:17)i
wheregisthegenusofS andN= g 0.Ifinadditionweassumethatthelefthand
y
≥
side in the inequality(27) is non-negativeandnotidentically zero, then it follows
thatg=0andhenceS hastheS2topology.
Proof. TheproofisslightlymoresimplethanthatinLemma1.Weintegratedirectly
expression(23) overS. On the lefthand side we use the stability condition(21).
Divergencetermsintherighthandsideintegratetozeroandwerearrangetermsas
(D lny W (ℓ))(Dalny W (ℓ)a)= D lny Dalny +2W (ℓ)Dalny W (ℓ)W (ℓ)c,
a a a a c
− − − − −
(28)
Areainequalitiesforstablemarginallytrappedsurfaces 9
sothattheintegralisnon-positive.FromGauss-Bonnettheorem,wewrite
1
2RdS=4p (1 g) . (29)
S2 −
Z
Collectingtheseobservations,theinequality(22)follows.Ifthelefthandsideofthe
inequality(22)isnon-negativeitfollowsthatgcanbe0or1.Ifitisnotidentically
zerotheng=0andhenceS hastheS2topology.
3.3 Variantsto thestablyoutermostcondition
3.3.1 OnanaveragedoutermoststablyconditionsforMOTS
Inequalities(22) and(27) do notrequirea point-likestability condition.We could
consideran(inprincipleweaker)averagedstabilityconditionforMOTS.
Definition3. Given a closed orientable MOTS S we will refer to it as (dipole)
weight-averagedstablyoutermostifthereexistsanoutgoing( ka-oriented)vector
xa=g ℓa ka,withg 0suchthat,forallfunctionsa onS,−thevariationsofq (ℓ)
withresp−ecttoXa=a≥xa fulfilltheintegralcondition
(Xaℓ )d q (ℓ)dS 0 . (30)
a X
S ≥
Z
Proofsofinequalitiesinvolvingtheangularmomentum(cf.sections4and6)could
start from this averaged condition.Note that (Xaℓ )=a =const providesan av-
a
eraged stably outermost condition,the element needed in proving area-chargein-
equalities(cf.section5).Finally,a(2n-moment)weight-averagedstablyoutermost
conditioncouldbeintroducedas (Xaℓ )nd q (ℓ)dS 0,forintegersn.
S a X
≥
R
3.3.2 Towardsaxisymmetryrelaxation
Let h a be a divergence-freevector on S, with squared-normh =h ah constant
a
alongitself,i.e.h aD h =0(fulfilled,inparticular,byKillingvectors).Asin2.1.1,
a
wewriteq = 1h h +x x ,withx ah =x aℓ =x ak =0,x ax =1,andW (h )=
ab h a b a b a a a a a
h bW (ℓ)h /h andW (x )=x bW (ℓ)x ,sorelations(10)hold.Thegeometricquantity
b a a b a
1 1
Q[h ]= W (ℓ)h adS , (31)
4p S √h a
Z
is well defined on S in the sense that:i) it doesnotdependon the normalization
of the null normal ℓa, and ii) there is no normalization ambiguity related to h a.
The first pointfollows from the transformationproperties(8) of W (ℓ) under ℓa =
a ′
10 Jose´LuisJaramillo
fℓa, together with the divergence-free character of hˆa =h a/√h , i.e. Da(hˆa)=
1 D h a 1 h aD h =0.Regardingthesecondpoint,Q[h ]isdefinedinterms
√h a −2h √h a
ofhˆa,withhˆahˆ =1.Thisistheanalogueofthe2p -orbitnormalizationforaxial
a
Killing vectors in expressions(11) and (15) for the angular momentum(note that
h a needsnottobeaxial).WecanthenadapttheMOTSstabilitycondition:
Definition4. Given a closed orientable marginally outer trapped surface S and
adivergencefreevectorh a onit, S issaidtobeh a-compatiblespacetimestably
outermostifthereexistsanoutgoing( ka-oriented)vectorXa= y ka,withy >
0 and h aD y =0, such that the vari−ation of q (ℓ) with respect t−o Xa fulfills the
a
conditiond q (ℓ) 0.
X
≥
Thefollowinglemmaholds.
Lemma3. GivenaclosedorientableMOTSS satisfyingtheh a-compatiblespace-
timestablyoutermostconditionforXa,thenforalla suchthath aD a =0,itholds
a
D a Daa +1a 22R dS a 2 W (h )W (h )a+G ℓakb dS, (32)
a a ab
S 2 ≥ S
Z (cid:20) (cid:21) Z h (cid:16) (cid:17)i
Proof. TheproofproceedsexactlyasinLemma1.Itisstraightforwardtogeneralize
itforXa=g ℓa y ka,g 0,sothattheshearandtheG ℓaℓbtermsareincorporated.
ab
− ≥
4 The area-angular momentum inequality
Wefirststatethemainresultinthissection(see[43]forfurtherdetails):
Theorem 1 (cf. Ref. [43]). Given an axisymmetric closed orientable marginally
outertrappedsurfaceS satisfyingthe(axisymmetry-compatible)spacetimestably
outermost condition,in a spacetime with non-negativecosmologicalconstantand
fulfillingthedominantenergycondition,itholdstheinequality
A 8p J , (33)
≥ | |
whereAandJ aretheareaandgravitational(Komar)angularmomentumofS.If
equalityholds,thenS hasthe geometryofanextreme Kerr throatsphere and,in
addition,ifthevectorXainthestabilityconditioncanbefoundtobespacelikethen
S isasectionofanon-expandinghorizon.
Theproofofthearea-angularmomentuminequality(33)hastwoparts.Thefirst
oneispurelygeometricandprovidesthelowerboundontheareaA
A 4p eM8−8 , (34)
≥
where M is a functionalon the sphere geometry.The second partsolves a varia-
tionalproblem,subjecttotheconstraintofkeepingconstanttheapriorigivenangu-
larmomentumJ.Inparticular,itisshown[1,33,31]theexistenceofaminimum