We prove that for sub-extremal axisymmetric and stationary black holes with
arbitrary surrounding matter the inequality $8\pi|J|<A$ holds, where $J$ is the
angular momentum and $A$ the horizon area of the black hole.Comment: 8 page

Ansorg, Marcus

Cederbaum, Carla

Hennig, Jörg

English

arXiv

We prove that for sub-extremal axisymmetric and stationary black holes with arbitrary surrounding matter the inequality $8\pi|J|<A$ holds, where $J$ is the angular momentum and $A$ the horizon area of the black hole

Hennig, J.

Ansorg, M.

Cederbaum, C.

MPG.PuRe

A universal inequality between angular momentum and horizon area for axisymmetric and stationary black holes with surrounding matter

arXiv:0805.4320v1  [gr-qc]  28 May 2008A universal inequality between angular momentumand horizon area for axisymmetric and stationaryblack holes with surrounding matterJörg Hennig, Marcus Ansorg and Carla CederbaumMax Planck Institute for Gravitational Physics,Am Mühlenberg 1, D-14476 Golm, GermanyE-mail: pjh@aei.mpg.de, mans@aei.mpg.de and Carla.Cederbaum@aei.mpg.deAbstract. We prove that for sub-extremal axisymmetric and stationary black holeswith arbitrary surrounding matter the inequality 8π|J | < A holds, where J is theangular momentum and A the horizon area of the black hole.PACS numbers: 04.70.Bw, 04.40.-b, 04.20.Cv preprint number: AEI-2008-0341. IntroductionA well-known property of the Kerr solution, describing a single rotating black hole invacuum, is given by|pJ | ≤ 1 with pJ :=8πJA, (1)where J and A denote the angular momentum and the horizon area of the black holerespectively. Equality in (1) holds if and only if the Kerr black hole is extreme. Aswas shown in [1], the equation |pJ | = 1 is even true more generally for axisymmetricand stationary black holes with surrounding matter in the degenerate limit (i.e. forvanishing surface gravity κ). Moreover, it was also conjectured in [1] that |pJ | ≤ 1 stillholds if the black hole is surrounded by matter. In this paper we prove this conjecture‡.We start by requiring that a physically relevant non-degenerate black hole becharacterized through the existence of trapped surfaces (i.e. surfaces with negativeexpansion θ(l) of outgoing null geodesics) in an interior vicinity of the event horizon.That is, in the terminology of [3], we concentrate on sub-extremal black holes. Inthe following we show that such surfaces cannot exist for |pJ | ≥ 1, provided that anappropriate functional I (to be defined below) cannot fall below 1. In turn, this can beproved by means of methods from the calculus of variations.‡ Note that in [1] a more general conjecture, incorporating the black hole’s electric charge Q, wasformulated. Here we prove this conjecture for the pure Einstein field, i.e. for Q = 0, and vanishingcosmological constant Λ = 0. (It should be noted that, for Λ 6= 0, the inequality |pJ | ≤ 1 can beviolated. An example is the Kerr-(A)dS family of black holes, see [3].)A universal inequality for axisymmetric and stationary black holes 22. Coordinate systems and Einstein equationsWe introduce suitable coordinates and metric functions by adopting our notation from[1], which is based on [2]. In spherical coordinates (r, θ, ϕ, t), an exterior vacuum vicinityof the event horizon§ can be described by the line elementds2 = e2µ(dr2 + r2dθ2) + r2B2e−2ν sin2θ (dϕ − ωdt)2 − e2νdt2. (2)The function B must be a solution of ∇·(r sin θ∇B) = 0, with Br sin θ = 0 at the eventhorizon H. Here ∇ is the nabla operator in three-dimensional flat space. In order to fixthe coordinates we chose the particular solution B = 1−r2h/r2, where rh = constant > 0.In this manner we obtain coordinates in which H is located at the coordinate spherer = rh. We now decompose the potential ν in the form ν = u + ln B, thereby obtainingthree regular metric functions µ, u and ω, depending on r and θ only.Following Bardeen [2], we introduce the new metric potentials µ̂ = r2e2µ, û =r2e−2u, which are positive and regular functions of r and cos θ, as well as the new radialcoordinate R = 12(r +r2hr). We arrive at the Boyer-Lindquist type line elementds2 = µ̂(dR2R2 − r2h+ dθ2)+ û sin2θ (dϕ − ωdt)2 − 4û(R2 − r2h)dt2, (3)which is singular at H (R = rh).In these coordinates, the vacuum Einstein equations read as follows:‖(R2 − r2h)ũ,RR + 2Rũ,R + ũ,θθ + ũ,θ cot θ = 1 −û28sin2θ(ω2,R +ω2,θR2 − r2h), (4)(R2 − r2h)(ω,RR + 4ω,Rũ,R) + ω,θθ + ω,θ(3 cot θ + 4ũ,θ) = 0, (5)(R2 − r2h)µ̃,RR + Rµ̃,R + µ̃,θθ =û216sin2θ(ω2,R +ω2,θR2 − r2h)+ Rũ,R− (R2 − r2h)ũ2,R − ũ,θ(ũ,θ + cot θ), (6)where ũ := 12ln(r−2h û)and µ̃ := 12ln(r−2h µ̂). At the horizon, the metric potentials obeythe boundary conditions [2]H : ω = constant = ωh,2rh√µ̂û= constant = κ, (7)with the horizon angular velocity ωh and the surface gravity κ. On the horizon’s northand south pole (R = rh and sin θ = 0), the following regularity conditions hold:µ̂(R = rh, θ = 0) = û(R = rh, θ = 0) = µ̂(R = rh, θ = π) = û(R = rh, θ = π) =2rhκ. (8)§ For a stationary space-time, the immediate vicinity of a black hole event horizon must be vacuum,see e. g. [2].‖ Throughout this paper we consider pure gravity, i.e. no electromagnetic fields, as well as vanishingcosmological constant, Λ = 0.A universal inequality for axisymmetric and stationary black holes 33. Necessary condition for the existence of trapped surfacesA crucial quantity for the following discussion is the expansion θ(l) = hab∇alb of outgoingnull rays for two-surfaces S in an interior vicinity of the horizon, where h is the interiormetric of S and l is the vector field describing outgoing null rays. In order to analyzeθ(l) inside the black hole, we introduce horizon-penetrating coordinates (R, θ, ϕ̃, t̃ ), inwhich the metric is regular at the horizon H:dt̃ = dt +T (R)∆dR, dϕ̃ = dϕ +Φ(R)∆dR, ∆ := 4(R2 − r2h). (9)The free functions T and Φ are chosen in such a way thata(R, θ) :=4µ̂û − T 2∆, b(R, θ) :=Φ − ωT∆(10)are regular at R = rh. Furthermore, we require a > 0 in order to guarantee thatt̃ = constant is a spacelike surface. As a consequence, T has to obey the conditionsT = 2√µ̂û =4rhκfor R = rh, T > 2√µ̂û for R < rh. (11)We thus obtain the regular line element¶ds2 =(aû+ ûb2 sin2θ)dR2 − ∆ûdt̃ 2 + û sin2θ(dϕ̃ − ωdt̃ )2 + µ̂dθ2+ 2(Tû+ ωûb sin2θ)dRdt̃ − 2ûb sin2θ dRdϕ̃. (12)In these coordinates, we calculate the expansion θ(l) for a surface S in a small interiorneighborhood of the horizon, described byS : R = rh − εr̂(θ), θ ∈ [0, π], ϕ̃ ∈ [0, 2π), t̃ = constant, (13)where ε > 0, r̂ > 0 and εr̂ ≪ rh. We obtainθ(l)(θ) =T̂2√2µ̂a sin θ[(r̂,θ sin θ),θr̂rh− (µ̂û),R2µ̂ûsin θ]∣∣∣∣Hε + O(ε2), (14)where we have used that T is of the form T = 2√µ̂û|H + εT̂ with T̂ = T̂ (θ) > 0 on S;see (11). [Note that θ(l) = 0 on the horizon (ε = 0).]Following Booth and Fairhurst [3], we study the criterion δn̄θ(l̄) < 0 (in theirnotation) for the existence of trapped surfaces, i.e. we characterize a physically relevant,sub-extremal black hole through a negative variation of the expansion on the horizon indirection of an ingoing null field n̄. In our formulation, this is equivalent to the existenceof a surface S with negative expansion θ(l). We first show the following lemma.Lemma 3.1. A necessary condition for the existence of trapped surfaces in the interiorvicinity of the event horizon of an axisymmetric and stationary black hole isπ∫0(µ̂û),R|H sin θ dθ > 0. (15)¶ Note that for the Kerr solution with mass M and angular momentum J , we obtain Kerr typecoordinates (where the slices t̃ = constant are spacelike) by choosing T = 4M(M + 2R) andΦ = constant = 2J/M .A universal inequality for axisymmetric and stationary black holes 4Proof. Let the surface S, defined in (13) for sufficiently small ε, be trapped. Then θ(l) isnegative everywhere on S, and so is the term in square brackets in (14) for all θ ∈ [0, π].Thus, the integral of this term along the horizon H is negative. Integrating by parts,and using that µ̂û = constant > 0 on H [see (7)], yields1rhπ∫0r̂2,θr̂2sin θ dθ − 12µ̂ûπ∫0(µ̂û),R sin θ dθ < 0. (16)Since the first integral is non-negative, we immediately obtain (15). 4. Calculation of pJFollowing the notation of [1], we express angular momentum J and horizon area A ofthe black hole byJ =18π∮Hma;bdSab = −116π∫0û2ω,R|H sin3θ dθ = −r2h41∫−1V e2U(1 − x2)dx, (17)A = 2ππ∫0√µ̂û|H sin θ dθ = 4π√µ̂û|H = 4πr2he2U(1), (18)where x := cos θ, U(x) :=[12ln(r−2h û)]∣∣H , V (x) :=[14û ω,R]∣∣H , and ma is the Killingvector corresponding to axisymmetry. Note that for this formulation we have usedconditions (7) and (8). As a consequence, pJ takes the formpJ ≡8πJA= −12e−2U(1)1∫−1V e2U(1 − x2)dx. (19)5. Reformulation in terms of a variational problemIn order to prove the inequality in question, we show that for |pJ | ≥ 1 Eq. (15) isviolated, i.e. that there are no trapped surfaces in the interior vicinity of the horizon.Using (4) and (6), the integrand in (15) can be expressed(µ̂û),R|H =2rhû2|θ=0[1 − 116ω2,Rû2 sin2θ − û,θ2û(û,θ2û+ 2 cot θ)]∣∣∣H. (20)Hence, we can write (15) in terms of U , V and x as follows:121∫−1[(V 2 + U ′ 2)(1 − x2) − 2xU ′]dx < 1 with U ′ :=dUdx. (21)From (19), the violation of (15) for |pJ | ≥ 1 can thus be formulated as the followingimplication, to be valid for any regular functions U, V defined on [−1, 1] and satisfyingthe condition U(−1) = U(1) [which follows from (8)]:∣∣∣∣∣∣1∫−1V e2U (1 − x2)dx∣∣∣∣∣∣≥ 2e2U(1) ⇒ 121∫−1[(V 2 + U ′ 2)(1 − x2) − 2xU ′]dx ≥ 1. (22)A universal inequality for axisymmetric and stationary black holes 5We now show that the validity of this implication holds provided that an appropriatefunctional I (defined below) cannot fall below 1.Applying the Cauchy-Schwarz inequality to the first inequality in (22) we obtain4e4U(1) ≤1∫−1V e2U (1 − x2)dx2≤1∫−1V 2(1 − x2)dx1∫−1e4U (1 − x2)dx. (23)Given this inequality, we replace the term∫V 2(1 − x2)dx in the second inequality in(22) and see thatI[U ] :=121∫−1[U ′ 2(x)(1 − x2) − 2xU ′(x)]dx +2e4U(1)1∫−1e4U(x)(1 − x2)dx≥ 1 (24)is as a sufficient condition for the validity of the implication, with the functionalI : W 1,2(−1, 1) → R defined on the Sobolev space W 1,2(−1, 1). Thus we have shownthe following.Lemma 5.1. The inequality |pJ | < 1 for any sub-extremal axisymmetric and stationaryblack hole with surrounding matter holds provided that the inequalityI[U ] ≥ 1 (25)is true for all U ∈ W 1,2(−1, 1) with U(−1) = U(1).This reformulation has led us to the following variational problem: Calculate theminimum of the functional I and show that it is not below 1.6. Complete solution of the variational problemConsider the functionalIε[U ] :=121−ε∫−1+ε[U ′ 2(x)(1 − x2) − 2xU ′(x)]dx +2e4U(1−ε)1−ε∫−1+εe4U(x)(1 − x2)dx(26)on W 1,2(−1+ ε, 1− ε), where 0 < ε ≪ 1 is a fixed real number. We use techniques fromthe calculus of variations to show that there exists a minimizer Uε for Iε in a suitableclass with sufficiently large value Iε[Uε]. Following this investigation, we take the limitε → 0 and see that the claim of lemma 5.1 follows.We now show the following statements:(i) The functional Iε is well-defined on the Sobolev spaceεW 1,2:= W 1,2(−1 + ε, 1 − ε)of functions U defined almost everywhere on (−1 + ε, 1 − ε). This is due to thewell-known proposition below.Proposition 6.1 (Thm 2.2 in Buttazzo-Giaquinta-Hildebrandt [4]). On anybounded interval J ⊆ R, W 1,2(J) →֒ C0(J) compactly. Moreover, the fundamentaltheorem of calculus holds inεW 1,2.A universal inequality for axisymmetric and stationary black holes 6Here, we use the adapted inner product∫ 1−ε−1+ε UV dµ +∫ 1−ε−1+ε U′V ′ dµ for U, V ∈εW 1,2, where dµ(x) := (1− x2) dx, which is equivalent to the ordinary one and thusmakesεW 1,2 a Hilbert space.For ease of notation set Xε := {U ∈εW 1,2 |U(1− ε) = 0}. Note that the functionalIε is invariant under addition of constants. We will use this in order to restrictour attention to the Hilbert subspace Xε on which (U, V ) :=∫ 1−ε−1+ε U′V ′ dµ is anequivalent inner product inducing the norm ‖U‖ := (∫ 1−ε−1+ε U′ 2 dµ)1/2.(ii) Iε is bounded from below. Using 0 ≤(x√1−x2 − U′(x)√1 − x2)2= x21−x2 − 2xU ′(x) +U ′ 2(x)(1 − x2) we conclude that Iε[U ] ≥ −12∫ 1−ε−1+εx21−x2 dx =: C(ε) > −∞.(iii) Applying the Cauchy-Schwarz inequality to∫ 1−ε−1+ε xU′(x)dx, we obtain that Iε[U ] ≥12‖U‖2 + 2C(ε)‖U‖ for any U ∈ Xε with C(ε) as in (ii). Hence, for every P ∈ Rthere exists a QP ∈ R such that Iε[U ] ≥ P whenever ‖U‖ ≥ QP . This is equivalentto coercivity of the functional Iε with respect to the weak topology on Xε.(iv) The functional Iε is sequentially lower semi-continuous (lsc) with respect to theweak topology in Xε. Recall that lower semi-continuity is additive and that the firstterms can be dealt with by standard theory (see e.g. [6]). For the last term, we useproposition 6.1 to deduce that Uk → U in C0([−1 + ε, 1− ε]). Whence there existsa uniform bound D > 0 of {Uk} so that by Lipschitz continuity of the exponentialmap on [−4D, 4D] with Lipschitz constant L, we have∣∣∣∣∫ 1−ε−1+εe4Ukdµ −∫ 1−ε−1+εe4Udµ∣∣∣∣≤ 4L∫ 1−ε−1+ε|Uk − U |dµ≤ 8L‖Uk − U‖C0([−1+ε,1−ε]) k→∞−→ 0.(27)The last term thus being lsc, we have shown Iε to be lsc.We can now show existence of a global minimizer for Iε in a suitable class :As we have seen in (ii), Iε is bounded from below on Xε. Hence for a ∈ R and c > 0 wecan choose a minimizing sequence {Uk} in the class Kεa,c := {U ∈ Xε |∫ 1−ε−1+ε e4U dµ =c, U(−1 + ε) = a}, with values tending to the infimum iεa,c of Iε in Kεa,c. By coercivity,{Uk} is bounded and we can extract a weakly converging subsequence with limitUεa,c ∈ Xε by Hilbert space techniques (theorem of Eberlein-Shmulyan [6]).The class Kεa,c is weakly sequentially closed by proposion 6.1, which can be shownas in (iv). Whence by (iv), Uεa,c ∈ Kεa,c satisfies Iε[Uεa,c] = iεa,c.SetεW 1,2a,0:= {U ∈εW 1,2 |U(−1 + ε) = a, U(1− ε) = 0} ⊂ Xε for any a ∈ R. By thetheory of Lagrange multipliers, each minimizer of Iε in the class Kεa,c is a critical pointof the functionalJε,c :εW 1,2a,0→ R : U 7→121−ε∫−1+ε[U ′ 2(x)(1 − x2) − 2xU ′(x)]dx +λ2(∫ 1−ε−1+εe4Udµ − c)(28)A universal inequality for axisymmetric and stationary black holes 7for some λ ∈ R, which is well-defined and sufficiently smooth by proposition 6.1. Inother words, there is λ := λεa,c ∈ R such that U := Uεa,c ∈ Kεa,c satisfies1−ε∫−1+ε[U ′(x)ϕ′(x)(1 − x2) − xϕ′(x)]dx + 2λ1−ε∫−1+εe4U(x)ϕ(x)(1 − x2)dx = 0 (29)for all ϕ ∈εW 1,20,0. This can be restated to say that U ∈εW 1,2 is a weak solution of−U ′′(x)(1 − x2) + 2xU ′(x) + 1 + 2λe4U(x)(1 − x2) = 0 ∀x ∈ (−1 + ε, 1 − ε),U(−1 + ε) = a, U(1 − ε) = 0,∫ 1−ε−1+ε e4Udµ = c.(30)Any weak solution U ∈εW 1,2 of (30) can be shown to be smooth and to satisfyequation (30) strongly: For all ϕ ∈εW 1,20,0, we can rewrite (29) as0 =1−ε∫−1+ε[U ′(x)(1 − x2) − x − 2λ∫ x−1+εe4Udµ]ϕ′(x) dx, (31)where we used integration by parts and proposition 6.1. By the fundamental lemma ofthe calculus of variations, there is a constant b ∈ R such thatU ′(x)(1 − x2) − x − 2λ∫ x−1+εe4Udµ = b (32)holds almost everywhere on (−1+ ε, 1− ε) as the integrand of (31) in square brackets isan L1-function. Solving for U ′, we deduce the smoothness of U by a bootstrap argument(similar to p. 462 in [5]). Differentiating Eq. (32), we get strong validity of (30). Inparticular, Uεa,c is a smooth classical solution of the Euler-Lagrange equation of Jε,c.Interestingly, there exists an integrating factor for Eq. (30) and it can be solvedexplicitly: ForF (x) := −(1 − x2)2U ′ 2(x) + 2x(1 − x2)U ′(x) + λe4U(x)(1 − x2)2 − x2 (33)we haveF ′(x) = 2[x − (1 − x2)U ′(x)][U ′′(x)(1 − x2) − 2xU ′(x) − 1 − 2λe4U(x)(1 − x2)]. (34)Thus, it suffices to solve the first order equation F = constant, which can be done withthe substitution W (x) := (1 − x2)2e4U(x).The unique smooth solution turns out to beW (x) = ε2(2 − ε)2(e2α artanh(1−ε) − βe−2α artanh(1−ε)e2α artanh x − βe−2α artanh x)2, (35)where the Lagrange multiplier λ was eliminated using the condition U(1 − ε) = 0. Thecomplex integration constants α and β are implicitly given in terms of a and c viaU(−1 + ε) = a and∫ 1−ε−1+ε e4Udµ = c. The corresponding value of Iε isIε[Uεa,c] =αβ2(2 − ε)4α − ε4α(1 + β2)[ε(2 − ε)]2α − β[(2 − ε)4α + ε4α] + 1 − ε +1 − α22lnε2 − ε+8α[ε(2 − ε)]2(α−1)[(2 − ε)2α − ε2α]2 ·(1 + β2)[ε(2 − ε)]2α − β[(2 − ε)4α + ε4α](2 − ε)4α − ε4α . (36)A universal inequality for axisymmetric and stationary black holes 8We now study the limit ε → 0. For any U ∈ W 1,2(−1, 1) set aε := U(−1 + ε) −U(1−ε), cε :=∫ 1−ε−1+ε e4U−4U(1−ε)dµ, c :=∫ 1−1 e4Udµ. Then U |(−1+ε,1−ε)−U(1−ε) ∈ Kεaε,cεand thus Iε[U ] ≥ iεaε,cε. As we have seen, there is a minimizer Uεaε,cε of Iε in Kεaε,cε andwe obtain Iε[U ] ≥ Iε[Uεaε,cε]. As discussed above, this minimizer is smooth and solvesthe Euler-Lagrange equation of the functional Jε,cε; the unique smooth solution of thisequation is given by (35). For ε → 0 we have αε → 1 (otherwise Iε diverges) and βε → −1[a consequence of aε → 0 by (25)]. Using these relations, we obtain Iε[Uεaε,cε] ≥ Cεaε withCεaε → 1 as ε → 0. By definition of I and Iε we see that|I(U) − Iε(U)| ≤12∫1−ε≤|x|<1|U ′ 2(x)(1−x2)−2xU ′(x)| dx+2∣∣∣∣∣e4U(1)∫ 1−1 e4Udµ− e4U(1−ε)∫ 1−ε−1+ε e4Udµ∣∣∣∣∣, (37)where the first term tends to 0 and the denominators of the latter tend to eachother as ε → 0 (the integrands are L1-functions, theorem of bounded convergence).The numerators of the latter term converge to each other by proposition 6.1. Thus,I[U ] ≥ 1.+Applying Lemma 5.1, we have shown |pJ | < 1 for sub-extremal black holes. Togetherwith the results about extremal black holes [1]∗ we arrive at the following.Theorem 6.2. Consider space-times with pure gravity (no electromagnetic fields) andvanishing cosmological constant. Then, for every axisymmetric and stationary sub-extremal black hole with arbitrary surrounding matter we have that 8π|J | < A. Theequality 8π|J | = A holds if the black hole is degenerate (extremal).AcknowledgmentsWe are indebted to José Luis Jaramillo and Herbert Pfister for many valuable hintsand comments. We wish to thank Paul T Allen for commenting on the manuscript.This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through theCollaborative Research Centre SFB/TR7 “Gravitational wave astronomy”.References[1] Ansorg M and Pfister H 2008 Class. Quantum Grav. 25 035009[2] Bardeen J M 1973 Rapidly rotating stars, disks, and black holes Black holes (Les Houches) ed CdeWitt and B deWitt (London: Gordon and Breach) pp 241-289[3] Booth I and Fairhurst S 2008 Phys. Rev. D 77 084005[4] Buttazzo G, Giaquinta M and Hildebrandt S 1998 One-dimensional variational problems (Oxford:Clarendon Press)[5] Evans L C 2002 Partial Differential Equations (Providence: American Mathematical Society)[6] Yosida K 1995 Functional Analysis (Berlin: Springer)+ Interestingly, equality in both inequalities of (22) [and hence I(U) = 1] is achieved only for thehorizon functions U and V of a degenerate black hole (e.g. extreme Kerr), cf. [1].∗ With the presented unique smooth solution (35) of the differential equation (30), the assumption ofequatorial symmetry as well as that of the existence of a continuous sequence in the proof presented in[1] can be abandoned, cf. equation (35) in [1].

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A universal inequality between the angular momentum and horizon area for axisymmetric and stationary black holes with surrounding matter

A universal inequality between angular momentum and horizon area for
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A universal inequality between angular momentum and horizon area for axisymmetric and stationary black holes with surrounding matter

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