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


SIGMA 8 (2012), 048, 58 pages      arXiv:1112.0291      http://dx.doi.org/10.3842/SIGMA.2012.048
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity

Jacobo Diaz-Polo a and Daniele Pranzetti b
a) Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA
b) Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany

Received December 02, 2011, in final form July 18, 2012; Published online August 01, 2012

Abstract
We review the black hole entropy calculation in the framework of Loop Quantum Gravity based on the quasi-local definition of a black hole encoded in the isolated horizon formalism. We show, by means of the covariant phase space framework, the appearance in the conserved symplectic structure of a boundary term corresponding to a Chern-Simons theory on the horizon and present its quantization both in the U(1) gauge fixed version and in the fully SU(2) invariant one. We then describe the boundary degrees of freedom counting techniques developed for an infinite value of the Chern-Simons level case and, less rigorously, for the case of a finite value. This allows us to perform a comparison between the U(1) and SU(2) approaches and provide a state of the art analysis of their common features and different implications for the entropy calculations. In particular, we comment on different points of view regarding the nature of the horizon degrees of freedom and the role played by the Barbero-Immirzi parameter. We conclude by presenting some of the most recent results concerning possible observational tests for theory.

Key words: black hole entropy; quantum gravity; isolated horizons.

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References

  1. 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.
  2. 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.
  3. Agulló I., Barbero G. J.F., Diaz-Polo J., Borja E.F., Villaseñor E.J.S., Black hole state counting in loop quantum gravity: a number-theoretical approach, Phys. Rev. Lett. 100 (2008), 211301, 4 pages, arXiv:0802.4077.
  4. Agulló I., Borja E.F., Diaz-Polo J., Computing black hole entropy in loop quantum gravity from a conformal field theory perspective, J. Cosmol. Astropart. Phys. 2009 (2009), 016, 9 pages, arXiv:0903.1667.
  5. Agulló I., Diaz-Polo J., Borja E.F., Black hole state degeneracy in loop quantum gravity, Phys. Rev. D 77 (2008), 104024, 11 pages, arXiv:0802.3188.
  6. Archer F., Williams R.M., The Turaev-Viro state sum model and three-dimensional quantum gravity, Phys. Lett. B 273 (1991), 438-444.
  7. Ashtekar A., Baez J., Corichi A., Krasnov K., Quantum geometry and black hole entropy, Phys. Rev. Lett. 80 (1998), 904-907, gr-qc/9710007.
  8. Ashtekar A., Baez J.C., Krasnov K., Quantum geometry of isolated horizons and black hole entropy, Adv. Theor. Math. Phys. 4 (2000), 1-94, gr-qc/0005126.
  9. 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.
  10. Ashtekar A., Beetle C., Fairhurst S., Isolated horizons: a generalization of black hole mechanics, Classical Quantum Gravity 16 (1999), L1-L7, gr-qc/9812065.
  11. Ashtekar A., Beetle C., Fairhurst S., Mechanics of isolated horizons, Classical Quantum Gravity 17 (2000), 253-298, gr-qc/9907068.
  12. Ashtekar A., Beetle C., Lewandowski J., Geometry of generic isolated horizons, Classical Quantum Gravity 19 (2002), 1195-1225, gr-qc/0111067.
  13. Ashtekar A., Beetle C., Lewandowski J., Mechanics of rotating isolated horizons, Phys. Rev. D 64 (2001), 044016, 17 pages, gr-qc/0103026.
  14. Ashtekar A., Bojowald M., Black hole evaporation: a paradigm, Classical Quantum Gravity 22 (2005), 3349-3362, gr-qc/0504029.
  15. Ashtekar A., Bojowald M., Quantum geometry and the Schwarzschild singularity, Classical Quantum Gravity 23 (2006), 391-411, gr-qc/0509075.
  16. Ashtekar A., Corichi A., Krasnov K., Isolated horizons: the classical phase space, Adv. Theor. Math. Phys. 3 (1999), 419-478, gr-qc/9905089.
  17. Ashtekar A., Engle J., Pawlowski T., Van Den Broeck C., Multipole moments of isolated horizons, Classical Quantum Gravity 21 (2004), 2549-2570, gr-qc/0401114.
  18. Ashtekar A., Engle J., Van Den Broeck C., Quantum horizons and black-hole entropy: inclusion of distortion and rotation, Classical Quantum Gravity 22 (2005), L27-L34, gr-qc/0412003.
  19. 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.
  20. Ashtekar A., Krishnan B., Dynamical horizons and their properties, Phys. Rev. D 68 (2003), 104030, 25 pages, gr-qc/0308033.
  21. Ashtekar A., Krishnan B., Dynamical horizons: energy, angular momentum, fluxes, and balance laws, Phys. Rev. Lett. 89 (2002), 261101, 4 pages, gr-qc/0207080.
  22. Ashtekar A., Krishnan B., Isolated and dynamical horizons and their applications, Living Rev. Relativ. 7 (2004), 10, 91 pages, gr-qc/0407042.
  23. Ashtekar A., Lewandowski J., Background independent quantum gravity: a status report, Classical Quantum Gravity 21 (2004), R53-R152, gr-qc/0404018.
  24. Ashtekar A., Taveras V., Varadarajan M., Information is not lost in the evaporation of 2D black holes, Phys. Rev. Lett. 100 (2008), 211302, 4 pages, arXiv:0801.1811.
  25. 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.
  26. 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.
  27. Barbero G. J.F., Villaseñor E.J.S., Statistical description of the black hole degeneracy spectrum, Phys. Rev. D 83 (2011), 104013, 21 pages, arXiv:1101.3662.
  28. Barbero G. J.F., Villaseñor E.J.S., The thermodynamic limit and black hole entropy in the area ensemble, Classical Quantum Gravity 28 (2011), 215014, 15 pages, arXiv:1106.3179.
  29. Barrau A., Cailleteau T., Cao X., Diaz-Polo J., Grain J., Probing loop quantum gravity with evaporating black holes, Phys. Rev. Lett. 107 (2011), 251301, 5 pages, arXiv:1109.4239.
  30. Beetle C., Engle J., Generic isolated horizons in loop quantum gravity, Classical Quantum Gravity 27 (2010), 235024, 13 pages, arXiv:1007.2768.
  31. Bekenstein J.D., Black holes and entropy, Phys. Rev. D 7 (1973), 2333-2346.
  32. Bianchi E., Black hole entropy, loop gravity, and polymer physics, Classical Quantum Gravity 28 (2011), 114006, 12 pages, arXiv:1011.5628.
  33. Bojowald M., Nonsingular black holes and degrees of freedom in quantum gravity, Phys. Rev. Lett. 95 (2005), 061301, 4 pages, gr-qc/0506128.
  34. Bojowald M., Kastrup H.A., Symmetry reduction for quantized diffeomorphism-invariant theories of connections, Classical Quantum Gravity 17 (2000), 3009-3043, hep-th/9907042.
  35. Booth I., Black hole boundaries, Can. J. Phys. 83 (2005), 1073-1099, gr-qc/0508107.
  36. Broderick A.E., Loeb A., Narayan R., The event horizon of sagittarius A*, Astrophys. J. 701 (2009), 1357-1366, arXiv:0903.1105.
  37. Burton D.M., Elementary number theory, McGraw-Hill, New York, 2002.
  38. Carlip S., Black hole entropy from conformal field theory in any dimension, Phys. Rev. Lett. 82 (1999), 2828-2831, hep-th/9812013.
  39. Carlip S., Black hole thermodynamics and statistical mechanics, in Physics of Black Holes, Lecture Notes in Phys., Vol. 769, Springer, Berlin, 2009, 89-123, arXiv:0807.4520.
  40. Carlip S., Entropy from conformal field theory at Killing horizons, Classical Quantum Gravity 16 (1999), 3327-3348, gr-qc/9906126.
  41. Carlip S., Logarithmic corrections to black hole entropy, from the Cardy formula, Classical Quantum Gravity 17 (2000), 4175-4186, gr-qc/0005017.
  42. Chandrasekhar S., The mathematical theory of black holes, International Series of Monographs on Physics, Vol. 69, The Clarendon Press Oxford University Press, New York, 1992.
  43. Chary V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  44. Corichi A., Diaz-Polo J., Borja E.F., Black hole entropy quantization, Phys. Rev. Lett. 98 (2007), 181301, 4 pages, gr-qc/0609122.
  45. Corichi A., Diaz-Polo J., Borja E.F., Quantum geometry and microscopic black hole entropy, Classical Quantum Gravity 24 (2007), 243-251, gr-qc/0605014.
  46. Corichi A., Wilson-Ewing E., Surface terms, asymptotics and thermodynamics of the Holst action, Classical Quantum Gravity 27 (2010), 205015, 14 pages, arXiv:1005.3298.
  47. Crnkovic C., Witten E., Covariant description of canonical formalism in geometrical theories, in Three Hundred Years of Gravitation, Cambridge University Press, Cambridge, 1987, 676-684.
  48. Das S., Kaul R.K., Majumdar P., New holographic entropy bound from quantum geometry, Phys. Rev. D 63 (2001), 044019, 4 pages, hep-th/0006211.
  49. De Raedt H., Michielsen K., De Raedt K., Miyashita S., Number partitioning on a quantum computer, Phys. Lett. A 290 (2001), 227-233, quant-ph/0010018.
  50. DeBenedictis A., Kloster S., Brannlund J., A note on the symmetry reduction of SU(2) on horizons of various topologies, Classical Quantum Gravity 28 (2011), 105023, 11 pages, arXiv:1101.4631.
  51. Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.
  52. Diaz-Polo J., Borja E.F., Black hole radiation spectrum in loop quantum gravity: isolated horizon framework, Classical Quantum Gravity 25 (2008), 105007, 8 pages, arXiv:0706.1979.
  53. Domagala M., Lewandowski J., Black-hole entropy from quantum geometry, Classical Quantum Gravity 21 (2004), 5233-5243, gr-qc/0407051.
  54. Durka R., Kowalski-Glikman J., Gravity as a constrained BF theory: Noether charges and Immirzi parameter, Phys. Rev. D 83 (2011), 124011, 6 pages, arXiv:1103.2971.
  55. 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.
  56. Engle J., Noui K., Perez A., Pranzetti D., Black hole entropy from the SU(2)-invariant formulation of type I isolated horizons, Phys. Rev. D 82 (2010), 044050, 23 pages, arXiv:1006.0634.
  57. Engle J., Noui K., Perez A., Pranzetti D., The SU(2) black hole entropy revisited, J. High Energy Phys. 2011 (2011), no. 5, 016, 30 pages, arXiv:1103.2723.
  58. Flajolet P., Sedgewick R., Analytic combinatorics, Cambridge University Press, Cambridge, 2009.
  59. Fleischhack C., Representations of the Weyl algebra in quantum geometry, Comm. Math. Phys. 285 (2009), 67-140, math-ph/0407006.
  60. 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.
  61. Frodden E., Ghosh A., Perez A., A local first law for isolated horizons, arXiv:1110.4055.
  62. Geroch R.P., Held A., Penrose R., A space-time calculus based on pairs of null directions, J. Math. Phys. 14 (1973), 874-881.
  63. Ghosh A., Mitra P., An improved estimate of black hole entropy in the quantum geometry approach, Phys. Lett. B 616 (2005), 114-117, gr-qc/0411035.
  64. Ghosh A., Mitra P., Counting black hole microscopic states in loop quantum gravity, Phys. Rev. D 74 (2006), 064026, 5 pages, hep-th/0605125.
  65. Ghosh A., Mitra P., Fine-grained state counting for black holes in loop quantum gravity, Phys. Rev. Lett. 102 (2009), 141302, 4 pages, arXiv:0809.4170.
  66. Ghosh A., Mitra P., Log correction to the black hole area law, Phys. Rev. D 71 (2005), 027502, 3 pages, gr-qc/0401070.
  67. Ghosh A., Perez A., Black hole entropy and isolated horizons thermodynamics, Phys. Rev. Lett. 107 (2011), 241301, 5 pages, arXiv:1107.1320.
  68. 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.
  69. Hawking S.W., Particle creation by black holes, Comm. Math. Phys. 43 (1975), 199-220.
  70. Hawking S.W., Ellis G.F.R., The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, London, 1973.
  71. Hayward S.A., General laws of black-hole dynamics, Phys. Rev. D 49 (1994), 6467-6474, gr-qc/9303006.
  72. Hayward S.A., Spin coefficient form of the new laws of black hole dynamics, Classical Quantum Gravity 11 (1994), 3025-3035, gr-qc/9406033.
  73. Jacobson T., A note on renormalization and black hole entropy in loop quantum gravity, Classical Quantum Gravity 24 (2007), 4875-4879, arXiv:0707.4026.
  74. Kaul R.K., Majumdar P., Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000), 5255-5257, gr-qc/0002040.
  75. Kaul R.K., Majumdar P., Quantum black hole entropy, Phys. Lett. B 439 (1998), 267-270, gr-qc/9801080.
  76. Kerr R.P., Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963), 237-238.
  77. Kloster S., Brannlund J., DeBenedictis A., Phase space and black-hole entropy of higher genus horizons in loop quantum gravity, Classical Quantum Gravity 25 (2008), 065008, 18 pages, gr-qc/0702036.
  78. Krasnov K., On quantum statistical mechanics of a Schwarzschild black hole, Gen. Relativity Gravitation 30 (1998), 53-68, gr-qc/9605047.
  79. Krasnov K., Quantum geometry and thermal radiation from black holes, Classical Quantum Gravity 16 (1999), 563-578, gr-qc/9710006.
  80. Krasnov K., Rovelli C., Black holes in full quantum gravity, Classical Quantum Gravity 26 (2009), 245009, 8 pages, arXiv:0905.4916.
  81. Lee J., Wald R.M., Local symmetries and constraints, J. Math. Phys. 31 (1990), 725-743.
  82. Lewandowski J., Spacetimes admitting isolated horizons, Classical Quantum Gravity 17 (2000), L53-L59, gr-qc/9907058.
  83. Lewandowski J., Okoów A., Sahlmann H., Thiemann T., Uniqueness of diffeomorphism invariant states on holonomy-flux algebras, Comm. Math. Phys. 267 (2006), 703-733, gr-qc/0504147.
  84. Livine E.R., Terno D.R., Bulk entropy in loop quantum gravity, Nuclear Phys. B 794 (2008), 138-153, arXiv:0706.0985.
  85. 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.
  86. Livine E.R., Terno D.R., The entropic boundary law in BF theory, Nuclear Phys. B 806 (2009), 715-734, arXiv:0805.2536.
  87. Lochan K., Vaz C., Canonical partition function of loop black holes, Phys. Rev. D 85 (2012), 104041, 9 pages, arXiv:1202.2301.
  88. Lochan K., Vaz C., Statistical analysis of entropy correction from topological defects in loop black holes, arXiv:1205.3974.
  89. Massar S., Parentani R., How the change in horizon area drives black hole evaporation, Nuclear Phys. B 575 (2000), 333-356, gr-qc/9903027.
  90. Meissner K.A., Black-hole entropy in loop quantum gravity, Classical Quantum Gravity 21 (2004), 5245-5251, gr-qc/0407052.
  91. Mitra P., Area law for black hole entropy in the SU(2) quantum geometry approach, Phys. Rev. D 85 (2012), 104025, 4 pages, arXiv:1107.4605.
  92. Modesto L., Disappearance of the black hole singularity in loop quantum gravity, Phys. Rev. D 70 (2004), 124009, 5 pages, gr-qc/0407097.
  93. Modesto L., Loop quantum black hole, Classical Quantum Gravity 23 (2006), 5587-5601, gr-qc/0509078.
  94. Müller A., Experimental evidence of black holes, PoS Proc. Sci. (2006), PoS(P2GC), 017, 30 pages, astro-ph/0701228.
  95. Newman E.T., Couch E., Chinnapared K., Exton A., Prakash A., Torrence R., Metric of a rotating, charged mass, J. Math. Phys. 6 (1965), 918-919.
  96. Ooguri H., Sasakura N., Discrete and continuum approaches to three-dimensional quantum gravity, Modern Phys. Lett. A 6 (1991), 3591-3600, hep-th/9108006.
  97. Perez A., Introduction to loop quantum gravity and spin foams, gr-qc/0409061.
  98. Perez A., Pranzetti D., Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy, Entropy 13 (2011), 744-777, arXiv:1011.2961.
  99. Pranzetti D., Radiation from quantum weakly dynamical horizons in loop quantum gravity, Phys. Rev. Lett. 109 (2012), 011301, 5 pages, arXiv:1204.0702.
  100. Reid M.J., Is there a supermassive black hole at the center of the milky way?, Internat. J. Modern Phys. D 18 (2009), 889-910, arXiv:0808.2624.
  101. Rendall A.D., Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations, Proc. Roy. Soc. London Ser. A 427 (1990), 221-239.
  102. Rovelli C., Black hole entropy from loop quantum gravity, Phys. Rev. Lett. 77 (1996), 3288-3291, gr-qc/9603063.
  103. Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004.
  104. Rovelli C., Thiemann T., Immirzi parameter in quantum general relativity, Phys. Rev. D 57 (1998), 1009-1014, gr-qc/9705059.
  105. Sahlmann H., Black hole horizons from within loop quantum gravity, Phys. Rev. D 84 (2011), 044049, 12 pages, arXiv:1104.4691.
  106. Sahlmann H., Entropy calculation for a toy black hole, Classical Quantum Gravity 25 (2008), 055004, 14 pages, arXiv:0709.0076.
  107. Sahlmann H., Toward explaining black hole entropy quantization in loop quantum gravity, Phys. Rev. D 76 (2007), 104050, 7 pages, arXiv:0709.2433.
  108. Sahlmann H., Thiemann T., Chern-Simons expectation values and quantum horizons from loop quantum gravity and the Duflo map, Phys. Rev. Lett. 108 (2012), 111303, 5 pages, arXiv:1109.5793.
  109. Smolin L., Linking topological quantum field theory and nonperturbative quantum gravity, J. Math. Phys. 36 (1995), 6417-6455, gr-qc/9505028.
  110. Strominger A., Black hole entropy from near-horizon microstates, J. High Energy Phys. 1998 (1998), no. 2, 009, 11 pages, hep-th/9712251.
  111. Strominger A., Vafa C., Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996), 99-104, hep-th/9601029.
  112. Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
  113. Thiemann T., Quantum spin dynamics. VIII. The master constraint, Classical Quantum Gravity 23 (2006), 2249-2265, gr-qc/0510011.
  114. Thiemann T., The Phoenix Project: master constraint programme for loop quantum gravity, Classical Quantum Gravity 23 (2006), 2211-2247, gr-qc/0305080.
  115. Wald R.M., Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993), R3427-R3431, gr-qc/9307038.
  116. Wald R.M., General relativity, University of Chicago Press, Chicago, IL, 1984.
  117. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.

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