Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 005, 30 pages      arXiv:1201.6102
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Entropy of Quantum Black Holes

Romesh K. Kaul
The Institute of Mathematical Sciences, CIT Campus, Chennai-600 113, India

Received September 14, 2011, in final form February 03, 2012; Published online February 08, 2012

In the Loop Quantum Gravity, black holes (or even more general Isolated Horizons) are described by a SU(2) Chern-Simons theory. There is an equivalent formulation of the horizon degrees of freedom in terms of a U(1) gauge theory which is just a gauged fixed version of the SU(2) theory. These developments will be surveyed here. Quantum theory based on either formulation can be used to count the horizon micro-states associated with quantum geometry fluctuations and from this the micro-canonical entropy can be obtained. We shall review the computation in SU(2) formulation. Leading term in the entropy is proportional to horizon area with a coefficient depending on the Barbero-Immirzi parameter which is fixed by matching this result with the Bekenstein-Hawking formula. Remarkably there are corrections beyond the area term, the leading one is logarithm of the horizon area with a definite coefficient −3/2, a result which is more than a decade old now. How the same results are obtained in the equivalent U(1) framework will also be indicated. Over years, this entropy formula has also been arrived at from a variety of other perspectives. In particular, entropy of BTZ black holes in three dimensional gravity exhibits the same logarithmic correction. Even in the String Theory, many black hole models are known to possess such properties. This suggests a possible universal nature of this logarithmic correction.

Key words: black holes; micro-canonical entropy; topological field theories; SU(2) Chern-Simons theory; Isolated Horizons; Bekenstein-Hawking formula; logarithmic correction; Barbero-Immirzi parameter; conformal field theories; Cardy formula; BTZ black hole; canonical entropy.

pdf (606 kb)   tex (64 kb)


  1. Bekenstein J., Black holes and entropy, Phys. Rev. D 7 (1973), 2333-2346.
    Bekenstein J., Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974), 3292-3300.
    Bardeen J.M., Carter B., Hawking S.W., The four laws of black hole mechanics, Comm. Math. Phys. 31 (1973), 161-170.
    Hawking S.W., Particle creation by black holes, Comm. Math. Phys. 43 (1975), 199-220.
    Page D., Particle emission rates from a black hole: massless particles from an uncharged, non-rotating hole, Phys. Rev. D 13 (1976), 198-206.
    Unruh W.G., Notes on black-hole evaporation, Phys. Rev. D 14 (1976), 870-892.
  2. Ashtekar A., Beetle C., Dreyer O., Fairhurst S., Krishnan B., Lewandowski J., Wisniewski J., Generic isolated horizons and their applications, Phys. Rev. Lett. 85 (2000), 3564-3567, gr-qc/0006006.
    Ashtekar A., Beetle C., Fairhurst S., Mechanics of isolated horizons, Classical Quantum Gravity 17 (2000), 253-298, gr-qc/9907068.
    Ashtekar A., Fairhurst S., Krishnan B., Isolated horizons: Hamiltonian evolution and the first law, Phys. Rev. D 62 (2000), 104025, 29 pages, gr-qc/0005083.
  3. Kaul R.K., Majumdar P., Schwarzschild horizon dynamics and SU(2) Chern-Simons theory, Phys. Rev. D 83 (2011), 024038, 10 pages, arXiv:1004.5487.
  4. Birmingham D., Blau M., Rakowski M., Thompson G., Topological field theory, Phys. Rep. 209 (1991), 129-340.
  5. Kaul R.K., Govindarajan T.R., Ramadevi P., Schwarz type topological quantum field theories, in Encyclopedia of Mathematical Physics, Elsevier, Amsterdam, 2006, 494-503, hep-th/0504100.
  6. Rovelli C., Quantum Gravity, Cambridge University Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
    Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
    Theimann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
    Sahlmann H., Loop quantum gravity - a short review, arXiv:1001.4188.
  7. Basu R., Kaul R.K., Majumdar P., Entropy of isolated horizons revisited, Phys. Rev. D 82 (2010), 024007, 5 pages, arXiv:0907.0846.
  8. Engle J., Noui K., Perez A., Black hole entropy and SU(2) Chern-Simons theory, Phys. Rev. Lett. 105 (2010), 031302, 4 pages, arXiv:0905.3168.
    Engle J., Noui K., Perez A., Pranzetti D., Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons, Phys. Rev. D 82 (2010), 044050, 23 pages, arXiv:1006.0634.
  9. Ashtekar A., Baez J., Corichi A., Krasnov K., Quantum geometry and black hole entropy, Phys. Rev. Lett. 80 (1998), 904-907, gr-qc/9710007;
    Ashtekar A., Corichi A., Krasnov K., Isolated horizons: the classical phase space, Adv. Theor. Math. Phys. 3 (2000), 419-478, gr-qc/9905089;
    Ashtekar A., Baez J., Krasnov K., Quantum geometry of isolated horizons and black hole entropy, Adv. Theor. Math. Phys. 4 (2000), 1-94, gr-qc/0005126.
  10. Smolin L., Linking topological quantum field theory and nonperturbative quantum gravity, J. Math. Phys. 36 (1995), 6417-6455, gr-qc/9505028.
  11. Krasnov K.V., On quantum statistical mechanics of Schwarzschild black hole, Gen. Relativity Gravitation 30 (1998), 53-68, gr-qc/9605047.
  12. Rovelli C., Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77 (1996), 3288-3291, gr-qc/9603063.
  13. Kaul R.K., Majumdar P., Quantum black hole entropy, Phys. Lett. B 439 (1998), 267-270, gr-qc/9801080.
  14. Kaul R.K., Majumdar P., Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000), 5255-5257, gr-qc/0002040.
  15. Das S., Kaul R.K., Majumdar P., A new holographic entropy bound from quantum geometry, Phys. Rev. D 63 (2001), 044019, 4 pages, hep-th/0006211.
  16. Kaul R.K., Kalyana Rama S., Black hole entropy from spin one punctures, Phys. Rev. D 68 (2003), 024001, 4 pages, gr-qc/0301128.
  17. Fursaev D.V., Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51 (1995), R5352-R5355, hep-th/9412161.
    Mann R.B., Solodukhin S.N., Universality of quantum entropy for extreme black holes, Nuclear Phys. B 523 (1998), 293-307, hep-th/9709064.
  18. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.
  19. Witten E., On holomorphic factorization of WZW and coset models, Comm. Math. Phys. 144 (1992), 189-212.
    Blau M., Thompson G., Derivation of the Verlinde formula from Chern-Simons theory and the G/G model, Nuclear Phys. B 408 (1993), 345-390, hep-th/9305010.
  20. Kaul R.K., Chern-Simons theory, colored-oriented braids and link invariants, Comm. Math. Phys. 162 (1994), 289-320, hep-th/9305032.
    Kaul R.K., Chern-Simons theory, knot invariants, vertex models and three manifold invariants, in Frontiers of Field Theory, Quantum Gravity and Strings, Horizons in World Physics, Vol. 227, Nova Science Publishers, New York, 1999, 45-63, hep-th/9804122.
    Kaul R.K., Ramadevi P., Three-manifold invariants from Chern-Simons field theory with arbitrary semi-simple gauge groups, Comm. Math. Phys. 217 (2001), 295-314, hep-th/0005096.
  21. Di Francesco P., Mathieu P., Senechal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer, Berlin, 1997.
  22. Livine E.R., Terno D.R., Quantum black holes: entropy and entanglement on the horizon, Nuclear Phys. B 741 (2006), 131-161, gr-qc/0508085.
  23. Domagala M., Lewandowski J., Black-hole entropy from quantum geometry, Classical Quantum Gravity 21 (2004), 5233-5243, gr-qc/0407051.
    Meissner K.A., Black-hole entropy in loop quantum gravity, Classical Quantum Gravity 21 (2004), 5245-5251, gr-qc/0407052.
    Khriplovich I.B., Quantized black holes, correspondence principle, and holographic bound, gr-qc/0409031.
  24. Ghosh A., Mitra P., An improved lower bound on black hole entropy in the quantum geometry approach, Phys. Lett. B 616 (2005), 114-117, gr-qc/0411035.
  25. Aref'eva I.Ya., Non-Abelian Stokes theorem, Theoret. and Math. Phys. 43 (1980), 353-356.
  26. Sahlmann H., Thiemann T., Chern-Simons theory, Stokes' theorem, and the Duflo map, J. Geom. Phys. 61 (2011), 1104-1121, arXiv:1101.1690.
    Sahlmann H., Thiemann T., Chern-Simons expectation values and quantum horizons from LQG and the Duflo map, arXiv:1109.5793.
  27. Duflo M., Opérateurs differéntiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265-288.
  28. Carlip S., Logarithmic corrections to black hole entropy from the Cardy formula, Classical Quantum Gravity 17 (2000), 4175-4186, gr-qc/0005017.
  29. Cardy J.L., Operator content of two-dimensional conformally invariant theories, Nuclear Phys. B 270 (1986), 186-204,
    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. 56 (1986), 742-745.
  30. Brown J.D., Henneaux M., Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Comm. Math. Phys. 104 (1986), 207-226.
    Larsen F., A string model of black hole microstates, Phys. Rev. D 56 (1997), 1005-1008, hep-th/9702153.
    Maldacena J.M., Strominger A., Universal low-energy dynamics for rotating black holes, Phys. Rev. D 56 (1997), 4975-4983, hep-th/9702015.
  31. Strominger A., Vafa C., Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996), 99-104, hep-th/9601029;
    Horowitz G.T,, Lowe D.A., Maldacena J.M., Statistical entropy of non extremal four-dimensional black holes and U duality, Phys. Rev. Lett. 77 (1996), 430-433, hep-th/9603195.
  32. Birmingham D., Sen S., An exact black hole entropy bound, Phys. Rev. D 63 (2001), 047501, 3 pages, hep-th/0008051.
    Birmingham D., Sachs I., Sen S., Exact results for the BTZ black hole, Internat. J. Modern Phys. D 10 (2001), 833-857, hep-th/0102155.
  33. Govindarajan T.R,, Kaul R.K., Suneeta V., Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole, Classical Quantum Gravity 18 (2001), 2877-2885, gr-qc/0104010.
  34. Achúcarro A., Towensend P.K., A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986), 89-92.
    Witten E., (2+1)-dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988), 46-78.
    Carlip S., Entropy from conformal field theory at Killing horizons, Classical Quantum Gravity 16 (1999), 3327-3348, gr-qc/9906126.
  35. Carlip S., Teitelboim C., Aspects of black hole quantum mechanics and thermodynamics in 2+1 dimensions, Phys. Rev. D 51 (1995), 622-631, gr-qc/9405070.
  36. Labastida J.M.F., Ramallo A.V., Operator formalism for Chern-Simons theories, Phys. Lett. B 227 (1989), 92-102.
    Isidro J.M., Labastida J.M.F., Ramallo A.V., Polynomials for torus links from Chern-Simons gauge theories, Nuclear Phys. B 398 (1993), 187-236, hep-th/9210124.
    Hayashi N., Quantum Hilbert space of GC Chern-Simons-Witten theory and gravity, Prog. Theor. Phys. Suppl. (1993), no. 114, 125-147.
  37. Carlip S., The statistical mechanics of the three-dimensional Euclidean black hole, Phys. Rev. D 55 (1997), 878-882, gr-qc/9606043.
    Suneeta V., Kaul R.K., Govindarajan T.R., BTZ black hole entropy from Ponzano-Regge gravity, Modern Phys. Lett. A 14 (1999), 349-358, gr-qc/9811071.
  38. Govindarajan T.R., Kaul R.K., Suneeta V., Quantum gravity on dS3, Classical Quantum Gravity 19 (2002), 4195-4205, hep-th/0203219.
  39. 't Hooft G., The black hole interpretation of string theory, Nuclear Phys. B 335 (1990), 138-154.
  40. Susskind L., Some speculations about black hole entropy in string theory, hep-th/9309145.
  41. Halyo E., Rajaraman A., Susskind L., Braneless black holes, Phys. Lett. B 392 (1997), 319-322, hep-th/9605112.
    Halyo E., Kol B., Rajaraman A., Susskind L., Counting Schwarzschild and charged black holes, Phys. Lett. B 401 (1997), 15-20, hep-th/9609075.
    Horowitz G.T., Polchinski J., A correspondence principle for black holes and strings, Phys. Rev. D 55 (1997), 6189-6197, hep-th/9612146.
    Damour T., Veneziano G., Selfgravitating fundamental strings and black holes, Nuclear Phys. B 568 (2000), 93-119, hep-th/9907030.
    Halyo E., Universal counting of black hole entropy by strings on the stretched horizon, J. High Energy Phys. 2001 (2001), no. 12, 005, 15 pages, hep-th/0108167.
  42. Kaul R.K., Black hole entropy from a highly excited elementary string, Phys. Rev. D 68 (2003), 024026, 4 pages, hep-th/0302170.
  43. Kalyana Rama S., Asymptotic density of open p-brane states with zero-modes included, Phys. Lett. B 566 (2003), 152-156, hep-th/0304152.
  44. Gour G., Algebraic approach to quantum black holes: logarithmic corrections to black hole entropy, Phys. Rev. D 66 (2002), 104022, 8 pages, gr-qc/0210024.
    Gupta K.S., Sen S., Further evidence for the conformal structure of a Schwarzschild black hole in an algebraic approach, Phys. Lett. B 526 (2002), 121-126, hep-th/0112041.
    Bianchi E., Black hole entropy, loop gravity, and polymer physics, Classical Quantum Gravity 28 (2011), 114006, 12 pages, arXiv:1011.5628.
  45. Davidson A., Holographic shell model: stack data structure inside black holes, arXiv:1108.2650.
  46. Agulló I., Barbero G. J.F., Borja E.F., Diaz-Polo J., Villaseñor E.J.S., Combinatorics of the SU(2) black hole entropy in loop quantum gravity, Phys. Rev. D 80 (2009), 084006, 3 pages, arXiv:0906.4529.
    Agulló I., Barbero G. J.F., Borja E.F., Diaz-Polo J., Villaseñor E.J.S., Detailed black hole state counting in loop quantum gravity, Phys. Rev. D 82 (2010), 084029, 31 pages, arXiv:1101.3660;
    Engle J., Noui K., Perez A., Pranzetti D., The SU(2) black hole entropy revisited, J. High Energy Phys. 2011 (2011), no. 05, 016, 30 pages, arXiv:1103.2723.
  47. Freidel L., Livine E.R., The fine structure of SU(2) intertwiners from U(N) representations, J. Math. Phys. 51 (2010), 082502, 19 pages, arXiv:0911.3553.
  48. Sahlmann H., Black hole horizons from within loop quantum gravity, Phys. Rev. D 84 (2011), 044049, 12 pages, arXiv:1104.4691.
  49. Krasnov K., Rovelli C., Black holes in full quantum gravity, Classical Quantum Gravity 26 (2009), 245009, 8 pages, arXiv:0905.4916.
  50. Agulló I., Barbero G. J.F., Díaz-Polo J., Borja E.F., Villaseñor E.J.S., Black hole state counting in LQG: a number theoretical approach, Phys. Rev. Lett. 100 (2008), 211301, 4 pages, arXiv:0802.4077.
    Barbero G. J.F., Villaseñor E.J.S., Generating functions for black hole entropy in loop quantum gravity, Phys. Rev. D 77 (2008), 121502, 5 pages, arXiv:0804.4784.
    Barbero G.J.F., Villaseñor E.J.S., On the computation of black hole entropy in loop quantum gravity, Classical Quantum Gravity 26 (2009), 035017, 22 pages, arXiv:0810.1599.
  51. Caravelli F., Modesto L., Holographic actions from black hole entropy, Phys. Lett. B 702 (2011), 307-311, arXiv:1001.4364.
  52. Das S., Majumdar P., Bhaduri R.K., General logarithmic corrections to black hole entropy, Classical Quantum Gravity 19 (2002), 2355-2367, hep-th/0111001.
  53. Gour G., Medved A.J.M., Thermal fluctuations and black hole entropy, Classical Quantum Gravity 20 (2003), 3307-3326, gr-qc/0305018.
  54. Chatterjee A., Majumdar P., Universal canonical black hole entropy, Phys. Rev. Lett. 92 (2004), 141301, 4 pages, gr-qc/0309026.
    Chatterjee A., Majumdar P., Universal criterion for black hole stability, Phys. Rev. D 72 (2005), 044005, 3 pages, gr-qc/0504064.
    Majumdar P., Generalized Hawking-Page phase transition, Classical Quantum Gravity 24 (2007), 1747-1753, gr-qc/0701014.
  55. Easson D.A., Frampton P.H., Smoot G.F., Entropic inflation, arXiv:1003.1528.

Previous article  Next article   Contents of Volume 8 (2012)