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

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1 IOP PUBLISHING Class. Quantum Grav ) pp) CLASSICAL AND QUANTUM GRAVITY doi: / /5/16/1600 FAST TRACK COMMUNICATION A universal inequality between the angular momentum and horizon area for axisymmetric and stationary black holes with surrounding matter Jörg Hennig, Marcus Ansorg and Carla Cederbaum Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D Golm, Germany pjh@aei.mpg.de, mans@aei.mpg.de and Carla.Cederbaum@aei.mpg.de Received 9 May 008 Published 30 July 008 Online at stacks.iop.org/cqg/5/1600 Abstract We prove that for sub-extremal axisymmetric and stationary black holes with arbitrary surrounding matter the inequality 8π J < A holds, where J is the angular momentum and A the horizon area of the black hole. PACS numbers: Bw, b, 04.0.Cv 1. Introduction A well-known property of the Kerr solution, describing a single rotating black hole in vacuum, is given by p J 1 with p J := 8πJ A, 1) where J and A denote the angular momentum and the horizon area of the black hole respectively. Equality in 1) holds if and only if the Kerr black hole is extreme. Aswasshown in [1], the equation p J =1iseventrue more generally for axisymmetric and stationary black holes with surrounding matter in the degenerate limit i.e. for vanishing surface gravity κ). Moreover, it was also conjectured in [1] that p J 1 still holds if the black hole is surrounded by matter. In this paper we prove this conjecture 1. We start by requiring that a physically relevant non-degenerate black hole be characterized through the existence of trapped surfaces i.e. surfaces with a negative expansion θ l) of outgoing null geodesics) in every sufficiently small interior vicinity of the event horizon. That is, in the terminology of [3], we concentrate on sub-extremal black holes. In the following we show that such surfaces cannot exist for p J 1, provided that an appropriate functional I 1 Note that in [1] a more general conjecture, incorporating the black hole s electric charge Q, was formulated. Here we prove this conjecture for the pure Einstein field, i.e. for Q = 0, and vanishing cosmological constant = 0. It should be noted that, for 0, the inequality p J 1 can be violated. An example is the Kerr A)dS family of black holes, see [3].) /08/ $ IOP Publishing Ltd Printed in the UK 1

2 Class. Quantum Grav ) 1600 to be defined below) cannot fall below 1. In turn, this can be proved by means of methods from the calculus of variations.. Coordinate systems and Einstein equations We introduce suitable coordinates and metric functions by adopting our notation from [1], which is based on []. In spherical coordinates r,θ,ϕ,t), an exterior vacuum vicinity of the event horizon can be described by the line element ds = e µ dr + r dθ ) + r B e ν sin θdϕ ω dt) e ν dt. ) The function B mustbeasolutionof r sin θ B) = 0, with Br sin θ = 0 at the event horizon H. Here is the nabla operator in a three-dimensional / flat space. In order to fix the coordinates we chose the particular solution B = 1 rh r, where r h = constant > 0. In this manner, we obtain coordinates in which H is located at the coordinate sphere r = r h.wenow decompose the potential ν in the form ν = u +lnb, thereby obtaining three regular metric functions µ, u and ω, depending on r and θ only. Following Bardeen [], we introduce the new metric potentials ˆµ = r e µ, û = r e u, which are positive and regular functions of r and cos θ, as well as the new radial coordinate R = 1 r ) r + h r. We arrive at the Boyer Lindquist-type line element dr ds ) = ˆµ R rh +dθ + û sin θdϕ ω dt) û 4 R rh ) dt, 3) which is singular at H R = r h ). In these coordinates, the vacuum Einstein equations read as follows 3 : ) R rh )ũ,rr +Rũ,R + ũ,θθ + ũ,θ cot θ = 1 û 8 sin θ ω,r + ω,θ R rh, 4) R rh ) ω,rr +4ω,R ũ,r ) + ω,θθ + ω,θ 3 cot θ +4ũ,θ ) = 0, 5) ) R rh ) µ,rr + R µ,r + µ,θθ = û 16 sin θ ω,r + ω,θ R rh + Rũ,R R r h )ũ,r ũ,θ ũ,θ + cot θ), 6) where ũ := 1 ln r h û) and µ := 1 ln r h ˆµ ). At the horizon, the metric potentials obey the boundary conditions [] H: ω = constant = ω h, r h ˆµû = constant = κ, 7) with the horizon angular velocity ω h and the surface gravity κ. On the horizon s north and south pole R = r h and sin θ = 0), the following regularity conditions hold: ˆµR = r h,θ = 0) = ûr = r h,θ = 0) = ˆµR = r h,θ = π) = ûr = r h,θ = π) = r h κ. 8) For a stationary spacetime, the immediate vicinity of a black hole event horizon must be vacuum, see e.g. []. 3 Throughout this paper we consider pure gravity, i.e. no electromagnetic fields, as well as vanishing cosmological constant, = 0.

3 Class. Quantum Grav ) Necessary condition for the existence of trapped surfaces A crucial quantity for the following discussion is the expansion θ l) = h ab a l b of outgoing null rays for -surfaces S in an interior vicinity of the horizon, where h is the interior metric of S and l is the vector field describing outgoing null rays. In order to analyze θ l) inside the black hole, we introduce horizon-penetrating coordinates R, θ, ϕ, t), in which the metric is regular at the horizon H: d t = dt + TR) R) dr, d ϕ = dϕ + dr, := 4 R rh ). 9) The free functions T and are chosen in such a way that ar,θ) := 4ˆµû T ωt, br,θ) := 10) are regular at R = r h. Furthermore, we require a>0inorder to guarantee that t = constant is a spacelike surface. As a consequence, T has to obey the conditions T = ˆµû = 4r h for R = r h, T > ˆµû for R<r h. 11) κ We thus obtain the regular line element 4 a ds = û + ûb sin θ) dr û d t + û sin θd ϕ ω d t) +ˆµ dθ ) T + û + ωûb sin θ dr d t ûb sin θ dr d ϕ. 1) In these coordinates, we calculate the expansion θ l) for a surface S in a small interior neighborhood of the horizon, described by S: R = r h ˆrθ), θ [0,π], ϕ [0, π), t = constant, 13) where >0, ˆr >0 and ˆr r h. We obtain [ ˆT θ l) θ) = ˆr,θ sin θ),θ ˆµû) θ],r H ˆµa sin θ ˆrr h ˆµû sin + O ), 14) wherewehaveusedthatt is of the form T = ˆµû H + ˆT with ˆT = ˆTθ) > 0onS; see 11). Note that θ l) = 0 on the horizon = 0).) Following Booth and Fairhurst [3], we study the criterion δ n θ l) < 0 in their notation) 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 in the direction of an ingoing null field n. In our formulation, this is equivalent to the existence of a surface S with a negative expansion θ l). We first show the following lemma. Lemma 3.1. A necessary condition for the existence of trapped surfaces in the interior vicinity of the event horizon of an axisymmetric and stationary black hole is π 0 ˆµû),R H sin θ dθ >0. 15) Proof. Let the surface S, defined in 13) for sufficiently small, be trapped. Then θ l) is negative everywhere on S, and so is the term in square brackets in 14) for all θ [0,π]. 4 Note that for the Kerr solution with the mass M and the angular momentum J, we obtain Kerr-type coordinates where the slices t = constant are spacelike) by choosing T = 4MM +R) and = constant = J/M. 3

4 Class. Quantum Grav ) 1600 Thus, the integral of this term along the horizon H is negative. Integrating by parts, and using that ˆµû = constant > 0onH see 7)), yields 1 π ˆr,θ 1 π sin θ dθ ˆµû) r h 0 ˆr,R sin θ dθ <0. 16) ˆµû 0 Since the first integral is non-negative, we immediately obtain 15). 4. Calculation of p J Following the notation of [1], we express the angular momentum J and the horizon area A of the black hole by J = 1 8π A = π H π 0 m a;b ds ab = 1 π H 1 û ω,r sin 3 θ dθ = r h V e U 1 x ) dx, 17) ˆµû H sin θ dθ = 4π ˆµû H = 4πr h eu1), 18) where x := cos θ,ux) := [ 1 ln r h û)] H,Vx):= [ 1,R] 4ûω H and m a is the Killing vector corresponding to axisymmetry. Note that for this formulation we have used conditions 7) and 8). As a consequence, p J takes the form p J 8πJ 1 A = e U1) V e U 1 x ) dx. 19) 5. Reformulation in terms of a variational problem In order to prove the inequality in question, we show that for p J 1 equation 15) is violated, 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 as û,θ H û + cot. 0) Hence, we can write 15) in terms of U,V and x as follows: 1 1 [V + U )1 x ) xu ]dx<1 with U := du dx. 1) ˆµû),R H = r h û θ=0 [ ω,rû sin θ û,θ û From 19), the violation of 15) for p J 1 can thus be formulated as the following implication, to be valid for any regular functions U, V defined on [, 1] and satisfying the condition U) = U1) [which follows from 8)]: 1 V e U 1 x ) dx eu1) 1 1 [V + U )1 x ) xu ]dx 1. ) We now show that the validity of this implication holds provided that an appropriate functional I defined below) cannot fall below 1. Applying the Cauchy Schwarz inequality to the first inequality in ) we obtain e 4U1) V e U 1 x ) dx) V 1 x ) dx e 4U 1 x ) dx. 3) 4

5 Class. Quantum Grav ) 1600 Given this inequality, we replace the term V 1 x ) dx in the second inequality in ) and see that 1 I[U] := 1 [U x)1 x ) xu e 4U1) x)]dx ) e4ux) 1 x ) dx is as a sufficient condition for the validity of the implication, with the functional I: W 1,, 1) R defined on the Sobolev space W 1,, 1). Thus we have shown the following. Lemma 5.1. The inequality p J < 1 for any sub-extremal axisymmetric and stationary black hole with surrounding matter holds provided that the inequality I[U] 1 5) is true for all U W 1,, 1) with U) = U1). This reformulation has led us to the following variational problem: calculate the minimum of the functional I and show that it is not below Complete solution of the variational problem Consider the functional e 4U1 ) I [U] := 1 [U x)1 x ) xu x)]dx + 1 6) e4ux) 1 x ) dx on W 1,, 1 ), where 0 < 1 is a fixed real number. We use techniques from the calculus of variations to show that there exists a minimizer U for I in a suitable class with sufficiently large value I [U ]. Following this investigation, we take the limit 0 and see that the claim of lemma 5.1 follows. We now show the following statements: i) The functional I is well defined on the Sobolev space W 1, := W 1,, 1 ) of functions U defined almost everywhere on, 1 ). This is due to the well-known proposition below. Proposition 6.1 theorem. in Buttazzo Giaquinta Hildebrandt [4]). On any bounded interval J R,W 1, J ) C 0 J) compactly. Moreover, the fundamental theorem of calculus holds in W 1,. Here, we use the adapted inner product UV dµ + U V dµ for U,V W 1,, where dµx) := 1 x ) dx, which is equivalent to the ordinary one and thus makes W 1, a Hilbert space. For ease of notation set X :={U W 1, U1 ) = 0}. Note that the functional I is invariant under addition of constants. We will use this in order to restrict our attention to the Hilbert subspace X on which U, V ) := U V dµ is an equivalent inner product inducing the norm U := U dµ ) 1/. ii) I is bounded from below. Using 0 x U 1 x x) 1 x ) = x U x)1 x ) we conclude that I [U] 1 x 1 x dx =: C) >. 1 x xu x) + 5

6 Class. Quantum Grav ) 1600 iii) Applying the Cauchy Schwarz inequality to xu x) dx, we obtain that I [U] 1 U +C) U for any U X with C) as in ii). Hence, for every P R there exists a Q P R such that I [U] P whenever U Q P. This is equivalent to coercivity of the functional I with respect to the weak topology on X. iv) The functional I is sequentially lower semi-continuous lsc) with respect to the weak topology in X. Recall that lower semi-continuity is additive and that the first terms can be dealt with by standard theory see, e.g., [6]). For the last term, we use proposition 6.1 to deduce that U k U in C 0 [ +, 1 ]). Whence there exists a uniform bound D>0of{U k } so that by Lipschitz continuity of the exponential map on [ 4D, 4D] with Lipschitz constant L, wehave e 4U k dµ e 4U dµ 4L U k U dµ k 8L U k U C 0 [,1 ]) 0. 7) The last term thus being lsc, we have shown I to be lsc. We can now show the existence of a global minimizer for I in a suitable class: aswe have seen in ii), I is bounded from below on X. Hence for a R and c>0wecan choose a minimizing sequence {U k } in the class Ka,c := { U X e4u dµ = c, U) = a }, with values tending to the infimum ia,c of I in Ka,c. By coercivity, {U k} is bounded and we can extract a weakly converging subsequence with the limit Ua,c X by Hilbert space techniques theorem of Eberlein Shmulyan [6]). The class Ka,c is weakly sequentially closed by proposition 6.1, which can be shown as in iv). Whence by iv), Ua,c K a,c satisfies I ] [ U a,c = i a,c. Set W 1, a,0 :={U W 1, U +) = a,u1 ) = 0} X for any a R. By the theory of Lagrange multipliers, each minimizer of I in the class Ka,c is a critical point of the functional J,c : W 1, a,0 R : U 1 [U x)1 x ) xu x)]dx + λ ) e 4U dµ c for some λ R, which is well defined and sufficiently smooth by proposition 6.1. In other words, there is λ := λ a,c R such that U := U a,c K a,c satisfies [U x)ϕ x)1 x ) xϕ x)]dx +λ 8) e 4Ux) ϕx)1 x ) dx = 0 9) for all ϕ W 1, 0,0. This can be restated to say that U W 1, is a weak solution of U x)1 x ) +xu x) +1+λe 4Ux) 1 x ) = 0 x, 1 ), U) = a, U1 ) = 0, e 4U dµ = c. 30) Any weak solution U W 1, of 30) can be shown to be smooth and to satisfy equation 30) strongly: for all ϕ W 1, 0,0, we can rewrite 9)as [ x ] 0 = U x)1 x ) x λ e 4U dµ ϕ x) dx, 31) 6

7 Class. Quantum Grav ) 1600 where we used integration by parts and proposition 6.1. By the fundamental lemma of the calculus of variations, there is a constant b R such that U x)1 x ) x λ x e 4U dµ = b 3) holds almost everywhere on, 1 ) as the integrand of 31) in square brackets is an L 1 -function. Solving for U, we deduce the smoothness of U by a bootstrap argument similar to p 46 in [5]). Differentiating equation 3), we get strong validity of 30). In particular, Ua,c is a smooth classical solution of the Euler Lagrange equation of J,c. Interestingly, there exists an integrating factor for equation 30) and it can be solved explicitly: for Fx) := 1 x ) U x) +x1 x )U x) + λ e 4Ux) 1 x ) x 33) we have F x) = [x 1 x )U x)][u x)1 x ) xu x) 1 λ e 4Ux) 1 x )]. 34) Thus, it suffices to solve the first-order equation F = constant, which can be done with the substitution Wx) := 1 x ) e 4Ux). The unique smooth solution turns out to be e Wx) = ) α artanh1 ) α artanh1 ) ) β e, 35) e α artanh x β e α artanh x where the Lagrange multiplier λ was eliminated using the condition U1 ) = 0. The complex integration constants α and β are implicitly given in terms of a and c via U) = a and e4u dµ = c. The corresponding value of I is I [ U a,c ] = αβ ) 4α 4α 1 α +1 + ln 1+β )[ )] α β[ ) 4α + 4α ] 8α[ )]α) + [ ) α α ] 1+β )[ )] α β[ ) 4α + 4α ]. 36) ) 4α 4α We now study the limit 0. For any U W 1,, 1),seta := U) U1 ), c := e4u 4U1 ) dµ, c := 1 e4u dµ. Then U,1 ) U1 ) Ka,c and thus I [U] ia,c. As we have seen, there is a minimizer Ua,c of I in Ka,c and we obtain [ ] I [U] I U a,c. As discussed above, this minimizer is smooth and solves the Euler Lagrange equation of the functional J,c ; the unique smooth solution of this equation is given by 35). For 0wehaveα 1 otherwise I diverges) [ ] and β a consequence of a 0by5)). Using these relations, we obtain I U a,c C a with Ca 1as 0. By definition of I and I we see that IU) I U) 1 U x)1 x ) xu x) dx 1 x <1 e 4U1) + 1 e4u dµ e4u1 ) 1 e4u dµ, 37) where the first term tends to 0 and the denominators of the latter tend to each other as 0 the integrands are L 1 -functions, theorem of bounded convergence). The numerators of the latter term converge to each other by proposition 6.1. Thus, I[U] Interestingly, equality in both inequalities of ) and hence IU) = 1) is achieved only for the horizon functions U and V of a degenerate black hole e.g. extreme Kerr), cf [1]. 7

8 Class. Quantum Grav ) 1600 Applying lemma 5.1, wehaveshown p J < 1forsub-extremal black holes. Together with the results about extremal black holes [1] 6 we arrive at the following. Theorem 6.. Consider spacetimes with pure gravity no electromagnetic fields) and vanishing cosmological constant. Then, for every axisymmetric and stationary sub-extremal black hole with arbitrary surrounding matter we have that 8π J <A. The equality 8π J =A holds if the black hole is degenerate extremal). Acknowledgments We are indebted to José Luis Jaramillo and Herbert Pfister for many valuable hints and comments. We wish to thank Paul T Allen for commenting on the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft DFG) through the Collaborative Research Centre SFB/TR7 Gravitational Wave Astronomy. References [1] Ansorg M and Pfister H 008 Class. Quantum Grav [] Bardeen J M 1973 Rapidly rotating stars, disks, and black holes Black Holes Les Houches) ed C dewitt and B dewitt London: Gordon and Breach) pp [3] Booth I and Fairhurst S 008 Phys. Rev. D [4] Buttazzo G, Giaquinta M and Hildebrandt S 1998 One-Dimensional Variational Problems Oxford: Clarendon) [5] Evans L C 00 Partial Differential Equations Providence, RI: American Mathematical Society) [6] Yosida K 1995 Functional Analysis Berlin: Springer) 6 With the presented unique smooth solution 35) of the differential equation 30), the assumption of equatorial 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]. 8