Abstract
The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
F. Denef, D. Gaiotto, A. Strominger, D. Van den Bleeken and X. Yin, Black hole deconstruction, JHEP 03 (2012) 071 [hep-th/0703252] [INSPIRE].
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
G.H. Hardy and S. Ramanujan, Asymptotic formulaæ in combinatory analysis, Proc. Lond. Math. Soc. 2 (1918) 75.
H. Rademacher, On the partition function p(n), Proc. Lond. Math. Soc. 2 (1938) 241.
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].
V. Jejjala and S. Nampuri, Cardy and Kerr, JHEP 02 (2010) 088 [arXiv:0909.1110] [INSPIRE].
A. Curir, Spin entropy of a rotating black hole, Nuovo Cim. B 51 (1979) 262.
A. Curir and M. Francaviglia, Spin thermodynamics of a Kerr black hole, Nuovo Cim. B 52 (1979) 165.
M. Cvetič and F. Larsen, General rotating black holes in string theory: grey body factors and event horizons, Phys. Rev. D 56 (1997) 4994 [hep-th/9705192] [INSPIRE].
M. Cvetič and F. Larsen, Black hole horizons and the thermodynamics of strings, Nucl. Phys. Proc. Suppl. 62 (1998) 443 [hep-th/9708090] [INSPIRE].
R. Penrose, Structure of space-time, in Battelle Rencontres: 1967 Lectures in Mathematics and Physics, C. de Witt-Morette and J. Wheeler eds., W.A. Benjamin, New York U.S.A. (1968), pg. 121 [INSPIRE].
J. McNamara, Instability of black hole inner horizons, Proc. Roy. Soc. Lond. A 358 (1978) 499.
E. Poisson and W. Israel, Internal structure of black holes, Phys. Rev. D 41 (1990) 1796 [INSPIRE].
G. Dotti, R.J. Gleiser, I.F. Ranea-Sandoval and H. Vucetich, Gravitational instabilities in Kerr space times, Class. Quant. Grav. 25 (2008) 245012 [arXiv:0805.4306] [INSPIRE].
D. Marolf, The dangers of extremes, Gen. Rel. Grav. 42 (2010) 2337 [arXiv:1005.2999] [INSPIRE].
D. Marolf and A. Ori, Outgoing gravitational shock-wave at the inner horizon: the late-time limit of black hole interiors, Phys. Rev. D 86 (2012) 124026 [arXiv:1109.5139] [INSPIRE].
F. Larsen, A string model of black hole microstates, Phys. Rev. D 56 (1997) 1005 [hep-th/9702153] [INSPIRE].
M. Cvetič, G.W. Gibbons and C.N. Pope, Universal area product formulaæ for rotating and charged black holes in four and higher dimensions, Phys. Rev. Lett. 106 (2011) 121301 [arXiv:1011.0008] [INSPIRE].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
A. Castro and M.J. Rodriguez, Universal properties and the first law of black hole inner mechanics, Phys. Rev. D 86 (2012) 024008 [arXiv:1204.1284] [INSPIRE].
S. Detournay, Inner mechanics of 3d black holes, Phys. Rev. Lett. 109 (2012) 031101 [arXiv:1204.6088] [INSPIRE].
M. Visser, Area products for stationary black hole horizons, Phys. Rev. D 88 (2013) 044014 [arXiv:1205.6814] [INSPIRE].
P.K. Townsend and B. Zhang, Thermodynamics of “exotic” Bañados-Teitelboim-Zanelli black holes, Phys. Rev. Lett. 110 (2013) 241302 [arXiv:1302.3874] [INSPIRE].
A. Castro, N. Dehmami, G. Giribet and D. Kastor, On the universality of inner black hole mechanics and higher curvature gravity, JHEP 07 (2013) 164 [arXiv:1304.1696] [INSPIRE].
B. Chen, S.-X. Liu and J.-J. Zhang, Thermodynamics of black hole horizons and Kerr/CFT correspondence, JHEP 11 (2012) 017 [arXiv:1206.2015] [INSPIRE].
B. Chen and J.-J. Zhang, Holographic descriptions of black rings, JHEP 11 (2012) 022 [arXiv:1208.4413] [INSPIRE].
B. Chen, Z. Xue and J.-J. Zhang, Note on thermodynamic method of black hole/CFT correspondence, JHEP 03 (2013) 102 [arXiv:1301.0429] [INSPIRE].
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [arXiv:1208.4371] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [gr-qc/9302012] [INSPIRE].
B. Chen and J.-J. Zhang, Electromagnetic duality in dyonic RN/CFT correspondence, Phys. Rev. D 87 (2013) 081505 [arXiv:1212.1960] [INSPIRE].
B. Chen, J.-J. Zhang, J.-D. Zhang and D.-L. Zhong, Aspects of warped AdS 3 /CFT 2 correspondence, JHEP 04 (2013) 055 [arXiv:1302.6643] [INSPIRE].
S. Ferrara, R. Kallosh and A. Strominger, N = 2 extremal black holes, Phys. Rev. D 52 (1995) 5412 [hep-th/9508072] [INSPIRE].
A. Strominger, Macroscopic entropy of N = 2 extremal black holes, Phys. Lett. B 383 (1996) 39 [hep-th/9602111] [INSPIRE].
S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D 54 (1996) 1514 [hep-th/9602136] [INSPIRE].
S. Ferrara and R. Kallosh, Universality of supersymmetric attractors, Phys. Rev. D 54 (1996) 1525 [hep-th/9603090] [INSPIRE].
S. Ferrara, G.W. Gibbons and R. Kallosh, Black holes and critical points in moduli space, Nucl. Phys. B 500 (1997) 75 [hep-th/9702103] [INSPIRE].
A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [INSPIRE].
K. Goldstein, N. Iizuka, R.P. Jena and S.P. Trivedi, Non-supersymmetric attractors, Phys. Rev. D 72 (2005) 124021 [hep-th/0507096] [INSPIRE].
D. Astefanesei, K. Goldstein, R.P. Jena, A. Sen and S.P. Trivedi, Rotating attractors, JHEP 10 (2006) 058 [hep-th/0606244] [INSPIRE].
A. Dabholkar, A. Sen and S.P. Trivedi, Black hole microstates and attractor without supersymmetry, JHEP 01 (2007) 096 [hep-th/0611143] [INSPIRE].
A. Sen, Black hole entropy function, attractors and precision counting of microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [INSPIRE].
K. Goldstein, R.P. Jena, G. Mandal and S.P. Trivedi, A C-function for non-supersymmetric attractors, JHEP 02 (2006) 053 [hep-th/0512138] [INSPIRE].
C. Toldo and S. Vandoren, Static nonextremal AdS 4 black hole solutions, JHEP 09 (2012) 048 [arXiv:1207.3014] [INSPIRE].
D. Astefanesei, K. Goldstein and S. Mahapatra, Moduli and (un)attractor black hole thermodynamics, Gen. Rel. Grav. 40 (2008) 2069 [hep-th/0611140] [INSPIRE].
M. Cvetič and F. Larsen, Conformal symmetry for general black holes, JHEP 02 (2012) 122 [arXiv:1106.3341] [INSPIRE].
M. Cvetič and F. Larsen, Conformal symmetry for black holes in four dimensions, JHEP 09 (2012) 076 [arXiv:1112.4846] [INSPIRE].
M. Cvetič and G.W. Gibbons, Conformal symmetry of a black hole as a scaling limit: a black hole in an asymptotically conical box, JHEP 07 (2012) 014 [arXiv:1201.0601] [INSPIRE].
M. Baggio, J. de Boer, J.I. Jottar and D.R. Mayerson, Conformal symmetry for black holes in four dimensions and irrelevant deformations, JHEP 04 (2013) 084 [arXiv:1210.7695] [INSPIRE].
H. Lü, C.N. Pope and J.F. Vazquez-Poritz, From AdS black holes to supersymmetric flux branes, Nucl. Phys. B 709 (2005) 47 [hep-th/0307001] [INSPIRE].
C.M. Miller, K. Schalm and E.J. Weinberg, Nonextremal black holes are BPS, Phys. Rev. D 76 (2007) 044001 [hep-th/0612308] [INSPIRE].
L. Andrianopoli, R. D’Auria, E. Orazi and M. Trigiante, First order description of black holes in moduli space, JHEP 11 (2007) 032 [arXiv:0706.0712] [INSPIRE].
G.L. Cardoso and V. Grass, On five-dimensional non-extremal charged black holes and FRW cosmology, Nucl. Phys. B 803 (2008) 209 [arXiv:0803.2819] [INSPIRE].
J. Perz, P. Smyth, T. Van Riet and B. Vercnocke, First-order flow equations for extremal and non-extremal black holes, JHEP 03 (2009) 150 [arXiv:0810.1528] [INSPIRE].
P. Galli, T. Ortín, J. Perz and C.S. Shahbazi, Non-extremal black holes of N = 2, D = 4 supergravity, JHEP 07 (2011) 041 [arXiv:1105.3311] [INSPIRE].
S. Barisch, G. Lopes Cardoso, M. Haack, S. Nampuri and N.A. Obers, Nernst branes in gauged supergravity, JHEP 11 (2011) 090 [arXiv:1108.0296] [INSPIRE].
P. Meessen, T. Ortín, J. Perz and C.S. Shahbazi, Black holes and black strings of N = 2, D = 5 supergravity in the H-FGK formalism, JHEP 09 (2012) 001 [arXiv:1204.0507] [INSPIRE].
A. Gnecchi and C. Toldo, First order flow for non-extremal AdS black holes and mass from holographic renormalization, JHEP 1410 (2014) 75 [arXiv:1406.0666] [INSPIRE].
K. Goldstein, S. Nampuri and Ã. Véliz-Osorio, Heating up branes in gauged supergravity, JHEP 08 (2014) 151 [arXiv:1406.2937] [INSPIRE].
E.J. Martinec, The Cheshire cap, arXiv:1409.6017 [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].
V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries and D1-D5-P bound states, Phys. Rev. D 71 (2005) 124030 [hep-th/0504181] [INSPIRE].
I. Bena, S. Giusto, C. Ruef and N.P. Warner, A (running) bolt for new reasons, JHEP 11 (2009) 089 [arXiv:0909.2559] [INSPIRE].
G. Compere, K. Copsey, S. de Buyl and R.B. Mann, Solitons in five dimensional minimal supergravity: local charge, exotic ergoregions and violations of the BPS bound, JHEP 12 (2009) 047 [arXiv:0909.3289] [INSPIRE].
G. Dall’Agata, S. Giusto and C. Ruef, U-duality and non-BPS solutions, JHEP 02 (2011) 074 [arXiv:1012.4803] [INSPIRE].
N. Bobev, B. Niehoff and N.P. Warner, Hair in the back of a throat: non-supersymmetric multi-center solutions from Kähler manifolds, JHEP 10 (2011) 149 [arXiv:1103.0520] [INSPIRE].
B.E. Niehoff, Non-supersymmetric, multi-center solutions with topological flux, JHEP 10 (2014) 168 [arXiv:1308.6335] [INSPIRE].
R. Kallosh, A.D. Linde, T. Ortín, A.W. Peet and A. Van Proeyen, Supersymmetry as a cosmic censor, Phys. Rev. D 46 (1992) 5278 [hep-th/9205027] [INSPIRE].
G.W. Gibbons and K.-I. Maeda, Black holes and membranes in higher dimensional theories with dilaton fields, Nucl. Phys. B 298 (1988) 741 [INSPIRE].
H. Leutwyler, Solution statique à symmetrie sphérique en théorie penta-dimensionnelle, Arch. Sci. 13 (1960) 549.
P. Dobiasch and D. Maison, Stationary, spherically symmetric solutions of Jordan’s unified theory of gravity and electromagnetism, Gen. Rel. Grav. 14 (1982) 231 [INSPIRE].
A. Chodos and S.L. Detweiler, Spherically symmetric solutions in five-dimensional general relativity, Gen. Rel. Grav. 14 (1982) 879 [INSPIRE].
D. Pollard, Antigravity and classical solutions of five-dimensional Kaluza-Klein theory, J. Phys. A 16 (1983) 565 [INSPIRE].
G.W. Gibbons and D.L. Wiltshire, Black holes in Kaluza-Klein theory, Annals Phys. 167 (1986) 201 [Erratum ibid. 176 (1987) 393] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1410.3478
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Goldstein, K., Jejjala, V. & Nampuri, S. Hot attractors. J. High Energ. Phys. 2015, 75 (2015). https://doi.org/10.1007/JHEP01(2015)075
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2015)075