A note on the Cardy formula and black holes in 3D (massive) bigravity
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(2) IOP PUBLISHING. CLASSICAL AND QUANTUM GRAVITY. doi:10.1088/0264-9381/30/4/045012. Class. Quantum Grav. 30 (2013) 045012 (6pp). A note on the Cardy formula and black holes in 3D (massive) bigravity Máximo Bañados and Miguel Pino Departamento de Fı́sica, P Universidad Católica de Chile, Casilla 306, Santiago 22, Chile E-mail: [email protected] and [email protected]. Received 12 November 2012, in final form 27 December 2012 Published 30 January 2013 Online at stacks.iop.org/CQG/30/045012 Abstract We study to what extent the Cardy formula is valid in the context of threedimensional massive gravity (bigravity). Since this theory has local bulk local excitations (a massive graviton) and black holes are not locally AdS, this example provides a non-trivial test of the robustness of the Cardy formula. PACS numbers: 04.70.Bw, 04.70.Dy, 04.50.Kd, 04.60.Kz. Since the successful application of Cardy’s formula [1, 2] to 3D black holes [3], countless examples of black holes with a conformal symmetry have been discussed and exact agreement with Bekenstein entropy is found in all cases (see, for example, [4–13] and references therein and thereof). The goal of this short note is to study another example, namely, three-dimensional bigravity. This is a non-trivial example in the sense that this theory does have propagating degrees of freedom (a massive graviton), and the black hole solutions are not locally AdS. Since the conformal symmetry lives at the asymptotic r → ∞ region, while the black hole properties are captured by the near horizon geometry, the validity of Cardy’s formula in this context is not a direct extension of [1, 2]. Bigravity, as first formulated by Isham et al [14], is a theory for two dynamical metrics gμν and fμν with action 1 √ (1) I[gμν , fμν ] = [ gR(g) + σ f R( f ) − U (g, f )]. 16π G U (g, f ) is an interaction potential. The dimensionless parameter σ measures the relative strengths of both Newton’s constants. Matter can be added coupled to either metric, or both. We shall work here in vacuum. We also assume that the potential does not contain derivatives of the metric fields. A specific example will be displayed below. Note that this family of theories include the recently much studied case of massive gravity [15]. The application of the Cardy formula to black hole physics require three ingredients: a black hole solution with an asymptotic conformal symmetry, the relation between the zero mode Virasoro field and black hole parameters and the corresponding Brown–Henneaux [16] central charge. 0264-9381/13/045012+06$33.00 © 2013 IOP Publishing Ltd. Printed in the UK & the USA. 1.
(3) Class. Quantum Grav. 30 (2013) 045012. M Bañados and M Pino. For bigravity all these ingredients can be read off from the action in a straightforward way. First, as discussed in [17], there exists solutions (for particular potentials, see below) that behave asymptotically as dg2 =. r2 2 2 2 dt + 2 dr + r2 dφ 2 + · · · 2 r. (2). . r2 2 2 2 2 2 dt + dr + r dφ + · · · (3) 2 r2 where N is a constant and is an AdS radius depending on the parameters in the potential (dg2 ≡ gμν dxμ dxν and d f 2 ≡ fμν dxμ dxν ). These solutions are asymptotically AdS, and have a conformal symmetry [17]1. The central charge is 3N 3 +σ . (4) c= 2G 2G This formula can be computed explicitly from the generators (see [17] for more details). It is more illuminating, however, to explain (4) directly from the action (1). The first term is the 1 √ expected contribution from 16πG gR. The second metric contributes in the same way. The only difference is that Newton’s constant for fμν appears in (1) divided by σ , and the AdS radius of (3) is not but N. The total central charge is then (4). Second, asymptotically AdS black holes in bigravity do exist. As recently discussed in [18, 19], the horizon for black holes in bigravity must be located at the same spacetime point. General arguments imply that the Bekenstein–Hawking entropy for black holes in bigravity is then A( f ) A(g) +σ (5) S= 4G 4G where A(g, f ) refers to the area of the horizon of each metric. Finally, since the potential does not have derivatives of the fields in the action (1), the total energy of any configuration is simply the sum of two ADM functionals, d f 2 = N2. E = EADM (g) + σ EADM ( f ).. (6). This allows a simple calculation of the total mass, and its relation to the conformal generators. With these ingredients at hand, we now proceed to check Cardy’s formula. At this point, we fix the potential so that an explicit solution is available. We consider the 2-parameter potential U (g, f ) = f (gμν − fμν )(gαβ − fαβ ) (7) × p21 ( f μα f νβ − f μν f αβ ) − p22 (gμα gνβ − gμν gαβ ) , where p1 , p2 are arbitrary real parameters2. Here f μν represents the inverse of fμν . The equations of motion following from the action (1) are solved by the spherically symmetric, static, black holes solution with metrics [18, 20] dr2 + r2 dφ 2 h d f 2 = − Xdt 2 + Y dr2 + 2Hdr dt + k2 r2 dφ 2 dg2 = − hdt 2 +. (8). 1 Note that both metrics are asymptotic to the same AdS space, with the same speed of light, in the same coordinate system. 2 This form of the potential, considered first in [18], shows a close relation with the recently proposed massive gravity theories [15]. In these theories, the potential is constructed as elementary symmetric polynomials of the matrix γ = g−1 f and shares common terms with potential (7).. 2.
(4) Class. Quantum Grav. 30 (2013) 045012. M Bañados and M Pino. where h, X, Y and H are given by h=. Qk2 σ κ α r2 r − 8M − g A2 α. X=. r2 Q − 8M f + rα B2 α . H=. 1 κ 2 k2. − XY. (9). (10). (11). while Y is best expressed by the relation, hY +. Qk2 σ 1 X = rα−2 + k2 + 2 4 . h κ k 4(2k2 − 1)p21. (12). This configuration has four integration constants: Mg, M f , k and Q. The constants κ, A and B are not independent but related to k and the action parameters p1 , p2 , σ by √ p2 k2 − 2 κ= (13) √ p1 1 − 2k2 4p2 (k − 1)(k + 1)(1 − k8 κ 2 + 2κ 2 k6 − 2k2 ) 1 = 1 2 A κk4 (k2 − 2) p21 (k − 1)(k + 1)(4k8 κ 2 − 11κ 2 k6 + 3κ 2 k4 + 3k4 + 5k2 − 4) 1 . = B2 κ 2 k6 σ (k2 − 2) The exponent α is related only to k by the relation. (14). (15). 4k4 − 7k2 + 4 . (16) (k2 − 2)(2k2 − 1) The constant k is arbitrary but we restrict its range by asymptotically AdS boundary conditions, i.e. demanding α to be negative. This implies √ 1/2 < k < 2. (17) α=. Since M f and Mg are independent, the horizons of gμν and fμν are located at different spacetime points. However, as discussed in [18, 19], each metric is singular at the horizon of the other. Thus, the only way to have a smooth and regular solution is to choose Mg and M f such that both horizons are located at the same point. Let r = r0 the location of the horizon. Mg and M f are then given by r02 Qk2 σ κ α r0 − A2 α r2 Q 8M f = 02 + r0α . B α. 8Mg =. (18). We now explore the asymptotic behavior at r → ∞. Assuming (17) is valid, the leading terms in both metrics are r2 A2 dg2 ≈ − 2 dt 2 + 2 dr2 + r2 dφ 2 A r 2 1 r 2 C 2 2 A2 A2 k2 2 2 k − 2 d f ≈ − 2 dt + 2 dr + − 2 dr dt + k2 r2 dφ 2 (19) B r k B κ 2 k2 B 3.
(5) Class. Quantum Grav. 30 (2013) 045012. where. M Bañados and M Pino. C = A k2 +. 1 A2 − 2 . κ 2 k4 B We demand both metrics to be proportional to some AdS background 2. (20). r2 2 2 2 dt + 2 dr + r2 dφ 2 (21) 2 r for some constant AdS radius . Directly from (21) and (19) one concludes that the only way to accomplish this condition is by imposing two conditions ds2 = −. k2 =. A2 B2. (22). B2C = A2 . (23) A2 These conditions3 put constraints on the action parameters and the value of k. But they do not restrict the horizon location which is still arbitrary, as it must be. Summarizing, if relations (22) and (23) hold, then both metrics are asymptotically AdS in the same coordinate system and the constant N entering in the central charge (4) is equal to A (24) N= . B Just as in the purely gravitational case [16], fluctuations around the AdS backgrounds are characterized by Virasoro fields with a non-zero classical central charge (4) [17]. The last step to apply the Cardy formula is to express the zero mode Virasoro charges L0 , L̄0 in terms of the black hole parameters. This is a straightforward, albeit long, calculation that we omit (see [17] for details). The result is 1 (AMg + σ BM f ). (25) L0 = L̄0 = 2G Note that these solutions are non-rotating hence L0 = L̄0 . We are now ready to apply Cardy formula cL0 cL̄0 + 2π (26) S = 2π 6 6 for the density of states. Plugging the central charge (4), the zero modes (25) and using (18), (22) and (23), we obtain, 2π r0 S= (1 + σ k), (27) 4G exactly equal to one-fourth of the sum of both horizon areas (see (5)), as promised. The (log) number of CFT states with L0 and L̄0 fixed is then exactly equal to the Bekenstein–Hawking entropy. We now study to what extent the conditions imposed to have an asymptotic CFT (equations (22) and (23)) are consistent with the near horizon geometry. As discussed in [18], absence of conical singularities at the horizon require that β, h0 and X0 are related by βh0 = 4π 3. 4. It is useful to note that relation (23) can be written as:. (28) 1 κ. = k3.
(6) Class. Quantum Grav. 30 (2013) 045012. M Bañados and M Pino. X0 (29) κk where h0 and X0 are the derivatives of the functions defined at (8) evaluated at the horizon r = r0 . These conditions are equivalent to imposing that surface gravities of both metrics coincide and it is a necessary condition in order for solution (8) to be regular at the horizon, see [18, 19]. Equations (28) and (29) also appear when deriving the semiclassical formula (27) for the entropy. In the Hamiltonian formalism, for example, the entropy is identified with the boundary term needed to regularize the action at the horizon. Since there are no derivatives in (7), these boundary terms will be just the sum of two ADM functionals, one for each metric. For the solutions (8) this gives X0 β h δr0 + σ δ(r0 k) + term at infinity, δI = − 8G 0 κk 2π r0 + term at infinity, (30) = − δ (1 + σ k) 4G where in the second line we have used (28) and (29). We then find that the term at the horizon is exactly equal to (5). Now, conditions (28) and (29) (making the horizon regular) together with (22) and (23) (making the geometry asymptotically AdS) imply Q = 0. In this situation, gμν and fμν become proportional describing the same geometry. It is an intriguing result that Cardy’s formula gives the correct semiclassical result without requiring horizon regularity. It only requires asymptotic AdS states. This is the main result of this short note. Demanding both, asymptotic AdS plus horizon regularity, implies that the solution collapses to proportional black holes. This result can be generalized to black holes with angular momentum, for which L0 = L̄0 , so this result is not a peculiarity of this solution. The interpretation of solutions having only an asymptotic AdS structure without insisting on horizon regularity has escaped us. We hope to come back to this problem elsewhere. h0 =. Acknowledgments We would like to thank A Gomberoff for many illuminating discussions on the Cardy formula. MP thanks H A González for useful discussions. MB was partially supported by Fondecyt (Chile) grants #1100282 and # 1090753. MP was supported by CONICYT grant (Chile) and VRAID grant (PUC, Chile). References [1] Strominger A 1998 Black hole entropy from near-horizon microstates J. High Energy Phys. JHEP02(1998)009 [2] Birmingham D, Sachs I and Sen S 1998 Entropy of three-dimensional black holes in string theory Phys. Lett. B 424 275–80 [3] Bañados M, Teitelboim C and Zanelli J 1992 The black hole in three-dimensional space-time Phys. Rev. Lett. 69 1849–51 [4] Guica M, Hartman T, Song W and Strominger A 2009 The Kerr/CFT correspondence Phys. Rev. D 80 124008 [5] Carlip S 1999 Entropy from conformal field theory at killing horizons Class. Quantum Grav. 16 3327–48 [6] Carlip S 1999 Black hole entropy from conformal field theory in any dimension Phys. Rev. Lett. 82 2828–31 [7] Bousso R, Maloney A and Strominger A 2002 Conformal vacua and entropy in de Sitter space Phys. Rev. D 65 104039 [8] Solodukhin S N 1999 Conformal description of horizon’s states Phys. Lett. B 454 213–22 5.
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