# Effective Conformal Descriptions of Black Hole Entropy

## Abstract

**:**

## 1. Introduction

- –
- The Bekenstein–Hawking entropy$$\begin{array}{c}\hfill S=\frac{\mathcal{A}}{4\hslash G}\end{array}$$
- –
- While the “correct” microscopic origin of black hole entropy is not settled, we now have a number of candidates, coming from string theory, loop quantum gravity, holographic entanglement entropy, Sakharov-style induced gravity, and several other approaches. (See [4] for a more detailed discussion.) But although these methods count very different states—or, in the case of Hawking’s original derivation [2], seem to know nothing about quantum gravitational states at all—they all yield the same entropy. To be sure, no approach is yet complete, and each requires additional assumptions. But within its domain of validity, each seems to work fairly well, and the universal agreement with (1) remains deeply mysterious.

## 2. Building a Black Hole/CFT Correspondence in Four Easy Steps

- Find an appropriate boundary and impose boundary conditions that specify properties of the black hole;
- Determine how these boundary conditions affect the symmetries of general relativity, the algebra of diffeomorphisms or “surface deformations”;
- Look for a preferred subalgebra of diffeomorphisms of the circle ($Diff{S}^{1}$), or perhaps two copies ($Diff{S}^{1}\times Diff{S}^{1}$);

#### 2.1. Find an Appropriate Boundary and Impose Boundary Conditions

#### 2.2. Determine how These Boundary Conditions Affect the Symmetries

#### 2.3. Look for a Preferred Subalgebra of Diffeomorphisms of the Circle

#### 2.4. Use Conformal Field Theory Methods to Extract Physical Information

## 3. Central Terms in the ADM Formalism

## 4. Some Examples

#### 4.1. The BTZ Black Hole

#### 4.2. The Extremal Kerr Black Hole

#### 4.3. A Generic (3 + 1)-Dimensional Black Hole

#### 4.4. Two-Dimensional Dilaton Gravity

## 5. What Does This Tell Us about the States?

## 6. Conclusions

## Acknowledgements

## References and Notes

- Bekenstein, J.D. Black holes and entropy. Phys. Rev.
**1973**, D7, 2333–2346. [Google Scholar] [CrossRef] - Hawking, S.W. Black hole explosions. Nature
**1974**, 248, 30–31. [Google Scholar] [CrossRef] - Iyer, V.; Wald, R.M. Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev.
**1994**, D50, 846–864. [Google Scholar] [CrossRef] - Carlip, S. Black hole thermodynamics and statistical mechanics. In Physics of Black Holes: A Guided Tour; Papantonopoulos, E., Ed.; Springer: Berlin, Germany, 2009; pp. 89–123. [Google Scholar]
- Carlip, S. Statistical mechanics and black hole entropy. In Field Theory, Integrable Systems and Symmetries; Khanna, F., Vinet, L., Eds.; Les Publications CRM: Montreal, Canada, 1997; pp. 11–19. [Google Scholar]
- Banados, M.; Teitelboim, C; Zanelli, J. The black hole in three-dimensional space-time. Phys. Rev. Lett.
**1992**, 69, 1849–1851. [Google Scholar] [CrossRef] [PubMed] - Strominger, A. Black hole entropy from near horizon microstates. J. High Energy Phys.
**1998**, 9802, 009:1–009:11. [Google Scholar] [CrossRef] - Birmingham, D.; Sachs, I.; Sen, S. Entropy of three-dimensional black holes in string theory. Phys. Lett.
**1998**, B424, 275–280. [Google Scholar] [CrossRef] - Carlip, S. Conformal field theory, (2+1)-dimensional gravity, and the BTZ black hole. Classical Quantum Gravity
**2005**, 22, R85–R124. [Google Scholar] [CrossRef] - Witten, E. Three-dimensional gravity revisited, 2007. arXiv:0706.3359. arXiv.org e-Print archive. Available online: http://arXiv.org/abs/arXiv:0706.3359 (accessed on 15 July 2011).
- Maloney, A.; Witten, E. Quantum gravity partition functions in three dimensions. J. High Energy Phys.
**2010**, 1002, 029:1–029:58. [Google Scholar] [CrossRef] - Carlip, S. Black hole entropy from conformal field theory in any dimension. Phys. Rev. Lett.
**1999**, 82, 2828–2831. [Google Scholar] [CrossRef] - Solodukhin, S.N. Conformal description of horizon’s states. Phys. Lett.
**1999**, B454, 213–222. [Google Scholar] [CrossRef] - Carlip, S. Entropy from conformal field theory at Killing horizons. Classical Quantum Gravity
**1999**, 16, 3327–3348. [Google Scholar] [CrossRef] - Guica, M.M.; Hartman, T.; Song, W.; Strominger, A. The Kerr/CFT correspondence. Phys. Rev.
**2009**, D80, 124008:1–124008:9. [Google Scholar] [CrossRef] - Cardy, J.L. Operator content of two-dimensional conformally invariant theories. Nucl. Phys.
**1986**, B 270, 186–204. [Google Scholar] [CrossRef] - Blöte, H.W.J.; Cardy, J.L.; Nightingale, M.P. Conformal invariance, the central charge, and universal finite size amplitudes at criticality. Phys. Rev. Lett.
**1986**, 56, 742–745. [Google Scholar] [CrossRef] [PubMed] - Carlip, S. Horizon constraints and black hole entropy. Classical Quantum Gravity
**2005**, 22, 1303–1312. [Google Scholar] [CrossRef] - Carlip, S. Black hole thermodynamics from Euclidean horizon constraints. Phys. Rev. Lett.
**2007**, 99, 021301:1–021301:4. [Google Scholar] [CrossRef] - Brown, J. D.; Henneaux, M. Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity. Commun. Math. Phys.
**1986**, 104, 207–226. [Google Scholar] [CrossRef] - Skenderis, K. Black holes and branes in string theory. Lect. Notes Phys.
**2000**, 541, 325–364. [Google Scholar] - Bardeen, J; Horowitz, G.T. The extreme Kerr throat: A vacuum analog of AdS
_{2}× S^{2}. Phys. Rev.**1999**, D60, 104030:1–10403:10. [Google Scholar] - Carlip, S.; Clements, M.; Della Pietra, S.; Della Pietra, V. Sewing Polyakov amplitudes I: Sewing at a fixed conformal structure. Commun. Math. Phys.
**1990**, 127, 253–271. [Google Scholar] [CrossRef] - Witten, E. On holomorphic factorization of WZW and coset models. Commun. Math. Phys.
**1992**, 144, 189–212. [Google Scholar] [CrossRef] - Thorne, K.S.; MacDonald, D.A.; Price, R.H. Black Holes: The Membrane Paradigm; Yale University Press: New Haven, CT, USA, 1986. [Google Scholar]
- Carlip, S. Extremal and nonextremal Kerr/CFT correspondences. J. High Energy Phys.
**2011**, 1104, 076:1–076:17. [Google Scholar] [CrossRef] - Kiefer, C. Quantum Gravity; Clarendon Press: Oxford, UK, 2004. [Google Scholar]
- Wald, R.M. Black hole entropy is the Noether charge. Phys. Rev.
**1993**, D48, 3427–3431. [Google Scholar] [CrossRef] - Ashtekar, A.; Bombelli, L.; Reula, O. The covariant phase space of asymptotically flat gravitational fields. In Mechanics, Analysis and Geometry: 200 Years after Lagrange; Francaviglia, M., Ed.; North-Holland: Amsterdam, The Netherlands, 1991; pp. 417–450. [Google Scholar]
- Crnkovic, C.; Witten, E. Covariant description of canonical formalism in geometrical theories. In Three Hundred Years of Gravitation; Hawking, S.W., Israel, W., Eds.; Cambridge University Press: Cambridge, UK, 1987; pp. 676–684. [Google Scholar]
- Arnowitt, R.; Deser, S.; Misner, C.W. The dynamics of general relativity. In Gravitation: an introduction to current research; Witten, L., Ed.; Wiley: New York, NY, USA, 1962; pp. 227–265. [Google Scholar]
- Teitelboim, C. How commutators of constraints reflect the space-time structure. Ann. Phys.
**1973**, 79, 542–557. [Google Scholar] [CrossRef] - Guven, J.; Ryan, M.P. Functional integrals and canonical quantum gravity: Facts and fancies. Phys. Rev.
**1992**, D45, 3559–3576. [Google Scholar] [CrossRef] - Regge, T.; Teitelboim, C. Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys.
**1974**, 88, 286–318. [Google Scholar] [CrossRef] - Solovev, V.O. Boundary values as Hamiltonian variables. I. New Poisson brackets. J. Math. Phys.
**1993**, 34, 5747–5769. [Google Scholar] [CrossRef] - Koga, J. Asymptotic symmetries on killing horizons. Phys. Rev.
**2001**, D64, 124012:1–124012:19. [Google Scholar] [CrossRef] - Silva, S. Black hole entropy and thermodynamics from symmetries. Classical Quantum Gravity
**2002**, 19, 3947–3962. [Google Scholar] [CrossRef] - Barnich, G.; Brandt, F. Covariant theory of asymptotic symmetries, conservation laws and central charges. Nucl. Phys.
**2002**, B 633, 3–82. [Google Scholar] [CrossRef] - Compere, G. Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions. arXiv:0708.3153. arXiv.orge-Print archive. Available online: http://arXiv.org/abs/arXiv:0708.3153 (accessed on 15 July 2011).
- Gel’fand, I.M.; Fuks, D.B. Cohomology of Lie algebra of the vector fields on the circle. Funct. Anal. Appl.
**1968**, 2, 342–343. [Google Scholar] [CrossRef] - Di Francesco, P.; Mathieu, P.; Sénéchal, D. Conformal Field Theory; Springer: New York, NY, USA, 1997. [Google Scholar]
- Rasmussen, J. A near-NHEK/CFT correspondence. arXiv:1004.4773. arXiv.org e-Print archive. Available online: http://arXiv.org/abs/arXiv:1004.4773 (accessed on 15 July 2011).
- Ashtekar, A.; Krishnan, B. Isolated and dynamical horizons and their applications. Living Rev. Relativ.
**2004**, 7, 10:1–10:91. [Google Scholar] [CrossRef] - Date, G. Isolated horizon, Killing horizon, and event horizon. Classical Quantum Gravity
**2001**, 18, 5219–5226. [Google Scholar] [CrossRef] - Wall, A.C. A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices. arXiv:1105.3445. arXiv.org e-Print archive. Available online: http://arXiv.org/abs/arXiv:1105.3445 (accessed on 15 July 2011).
- Carlip, S. Logarithmic corrections to black hole entropy from the Cardy formula. Classical Quantum Gravity
**2000**, 17, 4175–4186. [Google Scholar] [CrossRef] - Birmingham, D.; Sachs, I.; and Sen, S. Exact results for the BTZ black hole. Int. J. Mod. Phys.
**2001**, D10, 833–858. [Google Scholar] [CrossRef] - Loran, F.; Sheikh-Jabbari, M.M.; Vincon, M. Beyong logarithmic corrections to Cardy formula. J. High Energy Phys.
**2011**, 1101, 110:1–110:26. [Google Scholar] - Carlip, S. What we don’t know about BTZ black hole entropy. Classical Quantum Gravity
**1998**, 15, 3609–3625. [Google Scholar] [CrossRef] - Bousso, R.; Maloney, A.; and Strominger, A. Conformal vacua and entropy in de Sitter space. Phys. Rev.
**2002**, D65, 104039:1–104039:24. [Google Scholar] [CrossRef] - Maldacena, J.M.; Strominger, A. Universal low-energy dynamics for rotating black holes. Phys. Rev.
**1997**, D56, 4975–4983. [Google Scholar] [CrossRef] - Bredberg, I.; Hartman, T.; Song, W.; Strominger, A. Black hole superradiance from Kerr/CFT. J. High Energy Phys.
**2010**, 019:1–019:32. [Google Scholar] [CrossRef] - Emparan, R.; Sachs, I. Quantization of AdS
_{3}black holes in external fields. Phys. Rev. Lett.**1998**, 81, 2408–2411. [Google Scholar] [CrossRef] - I use the sign conventions of [99], and units 16πG = 1, although I will occasionally restore factors of G.
- Brown, J.D.; Lau, S.R.; York, J.W. Action and energy of the gravitational field. Ann. Phys.
**2002**, 297, 175–218. [Google Scholar] [CrossRef] - Bañados, M.; Henneaux, M.; Teitelboim, C.; Zanelli, J. Geometry of the 2+1 black hole. Phys. Rev.
**1993**, D4, 1506–1525. [Google Scholar] [CrossRef] - The full AdS/CFT correspondence involves many additional degrees of freedom; it remains unclear whether (2 + 1)-dimensional gravity alone contains enough degrees of freedom to fully account for black hole entropy [9,10,11].
- Bredberg, I.; Keeler, C.; Lysov, V.; Strominger, A. Cargese lectures on the Kerr/CFT correspondence. arXiv:1103.2355. arXiv.org e-Print archive. Available online: http://arXiv.org/abs/arXiv:1103.2355 (accessed on 15 July 2011).
- Frolov, V.P.; Novikov, I.D. Black Hole Physics; Kluwer: Dordrecht, The Netherlands, 1998. [Google Scholar]
- Castro, A.; Maloney, A.; Strominger, A. Hidden conformal symmetry of the Kerr black hole. Phys. Rev.
**2010**, D82, 024008:1–024008:7. [Google Scholar] [CrossRef] - Frolov, V.P.; Thorne, K.S. Renormalized stress-energy tensor near the horizon of a slowly evolving, rotating black hole. Phys. Rev.
**1989**, D39, 2125–2154. [Google Scholar] [CrossRef] - Hartman, T.; Murata, K.; Nishioka, T.; Strominger, A. CFT duals for extreme black holes. J. High Energy Phys.
**2009**, 0904, 019:1–019:17. [Google Scholar] [CrossRef] - Lu, H.; Mei, J.; Pope, C.N. Kerr/CFT correspondence in diverse dimensions. J. High Energy Phys.
**2009**, 0904, 054:1–054:17. [Google Scholar] - Castro, A.; Larsen, F. Near extremal Kerr entropy from AdS
_{2}quantum gravity. J. High Energy Phys.**2009**, 0912, 037:1–037:24. [Google Scholar] - Hawking, S.W. Black holes in general relativity. Commun. Math. Phys.
**1972**, 25, 152–166. [Google Scholar] [CrossRef] - Medved, A.J.M.; Martin, D.; Visser, M. Dirty black holes: Symmetries at stationary nonstatic horizons. Phys. Rev.
**2004**, D70, 024009:1–024009:8. [Google Scholar] [CrossRef] - Here and in what follows, there is a subtlety regarding the radial derivatives ∂
_{ρ}, coming from the fact that the proper distance ρ is metric-dependent. This issue is discussed in detail in the appendices of [26]. - Jing, J.; Yan, M.L. Statistical entropy of the static dilaton black holes from the Cardy formulas. Phys. Rev.
**2001**, D63, 024003:1–024003:6. [Google Scholar] - Cvitan, M.; Pallua, S.; Prester, P. Entropy of Killing horizons from Virasoro algebra in D-dimensional extended Gauss-Bonnet gravity. Phys. Lett.
**2003**, B555, 248–254. [Google Scholar] [CrossRef] - Cvitan, M.; Pallua, S. Conformal entropy for generalised gravity theories as a consequence of horizon properties. Phys. Rev.
**2005**, D71, 104032:1–104032:9. [Google Scholar] - Jackiw, R. Liouville theory: a two-dimensional model for gravity? In Quantum Theory of Gravity; Christensen, S.M., Ed.; Adam Hilger Ltd.: Bristol, UK, 1984; pp. 403–420. [Google Scholar]
- Teitelboim, C. The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly. In Quantum Theory of Gravity; Christensen, S.M., Ed.; Adam Hilger Ltd.: Bristol, UK, 1984; pp. 327–344. [Google Scholar]
- Cadoni, M.; Mignemi, S. Nonsingular four-dimensional black holes and the Jackiw-Teitelboim theory. Phys. Rev.
**1995**, D51, 4319–4329. [Google Scholar] [CrossRef] - Hotta, M. Asymptotic isometry and two-dimensional anti-de Sitter gravity. arXiv:gr-qc/9809035. arXiv.org e-Print archive. Avalable online: http://arXiv.org/abs/arXiv:gr-qc/9809035 (accessed on 15 July 2011).
- Cadoni, M.; Mignemi, S. Entropy of 2-D black holes from counting microstates. Phys. Rev.
**1999**, D59, 081501:1–081501:5. [Google Scholar] [CrossRef] - Cadoni, M.; Mignemi, S. Asymptotic symmetries of AdS
_{2}and conformal group in d = 1. Nucl. Phys.**1999**, B557, 165–180. [Google Scholar] [CrossRef] - Catelani, G.; Vanzo, L. On the 2
^{1/2}puzzle in AdS_{2}/CFT_{1}. arXiv:hep-th/0009186. arXiv.orge-Print archive. Avalable online: http://arXiv.org/abs/arXiv:hep-th/0009186 (accessed on 15 July 2011). - Chamon, C.; Jackiw, R.; Pi, S.-Y.; Santos, L. Conformal quantum mechanics as the CFT
_{1}dual to AdS_{2}. Phys. Lett.**2011**, B70, 503–507. [Google Scholar] [CrossRef] - Navarro-Salas, J.; Navarro, P. AdS
_{2}/CFT_{1}correspondence and near extremal black hole entropy. Nucl. Phys.**2000**, B579, 250–266. [Google Scholar] [CrossRef] - Castro, A.; Grumiller, D.; Larsen, F.; McNees, R. Holographic description of AdS
_{2}black holes. J. High Energy Phys.**2008**, 0811, 052:1–052:28. [Google Scholar] - Castro, A.; Keeler, C.; Larsen, F. Three dimensional origin of AdS
_{2}gravity. J. High Energy Phys.**2010**, 1007, 033:1–033:25. [Google Scholar] [CrossRef] - Izquierdo, J.M.; Navarro-Salas, J.; Navarro, P. Kaluza-Klein theory, AdS/CFT correspondence and black hole entropy. Classical Quantum Gravity
**2002**, 19, 563–570. [Google Scholar] [CrossRef] - Balasubramanian, V.; de Boer, J.; Sheikh-Jabbari, M.M.; Simon, J. What is a chiral 2d CFT? And what does it have to do with extremal black holes? J. High Energy Phys.
**2010**, 1002, 017:1–017:20. [Google Scholar] - This analogy was suggested to me by Nemanja Kaloper and John Terning.
- Goldstone, J.; Salam, A.; Weinberg, S. Broken symmetries. Phys. Rev.
**1962**, 127, 965–970. [Google Scholar] [CrossRef] - Carlip, S. Statistical mechanics and black hole thermodynamics. In Constrained Dynamics and Quantum Gravity 1996; de Alfaro, V., Nelson, J.E., Bandelloni, G., Blasi, A., Cavaglia, M., Filippov, A.T., Eds.; North-Holland: Amsterdam, The Netherlands, 1997; pp. 8–12. [Google Scholar]
- Carlip, S. Symmetries, horizons, and black hole entropy. Gen. Rel. Grav.
**2007**, 39, 1519–1523. [Google Scholar] [CrossRef] - Carlip, S. Dynamics of asymptotic diffeomorphisms in (2+1)-dimensional gravity. Classical Quantum Gravity
**2005**, 22, 3055–3060. [Google Scholar] [CrossRef] - Chen, Y. Quantum Liouville theory and BTZ black hole entropy. Classical Quantum Gravity
**2004**, 21, 1153–1180. [Google Scholar] [CrossRef] - Robinson, S.P.; Wilczek, F. A relationship between Hawking radiation and gravitational anomalies. Phys. Rev. Lett.
**2005**, 95, 011303:1–011303:4. [Google Scholar] [CrossRef] - Iso, S.; Umetsu, H.; Wilczek, F. Hawking radiation from charged black holes via gauge and gravitational anomalies. Phys. Rev. Lett.
**2006**, 96, 151302:1–151302:4. [Google Scholar] [CrossRef] - Banerjee, R.; Kulkarni, S. Hawking radiation and covariant anomalies. Phys. Rev.
**2008**, D77, 024018:1–024018:5. [Google Scholar] - Iso, S.; Morita, T.; Umetsu, H. Hawking radiation via higher-spin gauge anomalies. Phys. Rev.
**2008**, D 77, 045007:1–045007:8. [Google Scholar] [CrossRef] - Bonora, L.; Cvitan, M.; Pallua, S.; Smolić, I. Hawking radiation, W
_{∞}algebra and trace anomalies. J. High Energy Phys.**2008**, 0805, 071:1–071:22. [Google Scholar] [CrossRef] - Bonora, L.; Cvitan, M.; Pallua, S.; Smolić, I. Hawking fluxes, W
_{∞}algebras and anomalies. J. High Energy Phys.**2008**, 0812, 021:1–021:32. [Google Scholar] [CrossRef] - Bonora, L.; Cvitan, M.; Pallua, S.; Smolić, I. Hawking fluxes, fermionic currets, W
_{1+∞}algebras and anomalies. Phys. Rev.**2009**, D80, 084034:1–084034:12. [Google Scholar] - Giddings, S.B. The black hole information paradox. arXiv:hep-th/9508151. arXiv.org e-Printarchive. Available online: http://arXiv.org/abs/arXiv:hep-th/9508151 (accessed on 15 July 2011).
- Zamolodchikov, A.B. Irreversibility of the flux of the renormalization group in a 2D field theory. JETP Lett.
**1986**, 43, 730–732. [Google Scholar] - Wald, R.M. General Relativity; University of Chicago Press: Chicago, IL, USA, 1984. [Google Scholar]

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Carlip, S.
Effective Conformal Descriptions of Black Hole Entropy. *Entropy* **2011**, *13*, 1355-1379.
https://doi.org/10.3390/e13071355

**AMA Style**

Carlip S.
Effective Conformal Descriptions of Black Hole Entropy. *Entropy*. 2011; 13(7):1355-1379.
https://doi.org/10.3390/e13071355

**Chicago/Turabian Style**

Carlip, Steven.
2011. "Effective Conformal Descriptions of Black Hole Entropy" *Entropy* 13, no. 7: 1355-1379.
https://doi.org/10.3390/e13071355