|
SIGMA 8 (2012), 048, 58 pages arXiv:1112.0291
https://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.
pdf (1101 kb)
tex (638 kb)
References
- 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.
- 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.
- 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.
- 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.
- Archer F., Williams R.M., The Turaev-Viro state sum model and
three-dimensional quantum gravity, Phys. Lett. B 273
(1991), 438-444.
- 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., 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.
- 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., Isolated horizons: a generalization of
black hole mechanics, Classical Quantum Gravity 16 (1999),
L1-L7, gr-qc/9812065.
- Ashtekar A., Beetle C., Fairhurst S., Mechanics of isolated horizons,
Classical Quantum Gravity 17 (2000), 253-298,
gr-qc/9907068.
- Ashtekar A., Beetle C., Lewandowski J., Geometry of generic isolated horizons,
Classical Quantum Gravity 19 (2002), 1195-1225,
gr-qc/0111067.
- Ashtekar A., Beetle C., Lewandowski J., Mechanics of rotating isolated
horizons, Phys. Rev. D 64 (2001), 044016, 17 pages,
gr-qc/0103026.
- Ashtekar A., Bojowald M., Black hole evaporation: a paradigm, Classical
Quantum Gravity 22 (2005), 3349-3362, gr-qc/0504029.
- Ashtekar A., Bojowald M., Quantum geometry and the Schwarzschild singularity,
Classical Quantum Gravity 23 (2006), 391-411,
gr-qc/0509075.
- Ashtekar A., Corichi A., Krasnov K., Isolated horizons: the classical phase
space, Adv. Theor. Math. Phys. 3 (1999), 419-478,
gr-qc/9905089.
- 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.
- 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.
- 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.
- Ashtekar A., Krishnan B., Dynamical horizons and their properties,
Phys. Rev. D 68 (2003), 104030, 25 pages,
gr-qc/0308033.
- Ashtekar A., Krishnan B., Dynamical horizons: energy, angular momentum, fluxes,
and balance laws, Phys. Rev. Lett. 89 (2002), 261101,
4 pages, gr-qc/0207080.
- Ashtekar A., Krishnan B., Isolated and dynamical horizons and their
applications, Living Rev. Relativ. 7 (2004), 10, 91 pages,
gr-qc/0407042.
- Ashtekar A., Lewandowski J., Background independent quantum gravity: a status
report, Classical Quantum Gravity 21 (2004), R53-R152,
gr-qc/0404018.
- 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.
- 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.
- 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.
- 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.
- 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.
- Beetle C., Engle J., Generic isolated horizons in loop quantum gravity,
Classical Quantum Gravity 27 (2010), 235024, 13 pages,
arXiv:1007.2768.
- Bekenstein J.D., Black holes and entropy, Phys. Rev. D 7
(1973), 2333-2346.
- Bianchi E., Black hole entropy, loop gravity, and polymer physics,
Classical Quantum Gravity 28 (2011), 114006, 12 pages,
arXiv:1011.5628.
- Bojowald M., Nonsingular black holes and degrees of freedom in quantum gravity,
Phys. Rev. Lett. 95 (2005), 061301, 4 pages,
gr-qc/0506128.
- Bojowald M., Kastrup H.A., Symmetry reduction for quantized
diffeomorphism-invariant theories of connections, Classical Quantum
Gravity 17 (2000), 3009-3043, hep-th/9907042.
- Booth I., Black hole boundaries, Can. J. Phys. 83 (2005),
1073-1099, gr-qc/0508107.
- Broderick A.E., Loeb A., Narayan R., The event horizon of sagittarius A*,
Astrophys. J. 701 (2009), 1357-1366, arXiv:0903.1105.
- Burton D.M., Elementary number theory, McGraw-Hill, New York, 2002.
- Carlip S., Black hole entropy from conformal field theory in any dimension,
Phys. Rev. Lett. 82 (1999), 2828-2831,
hep-th/9812013.
- 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.
- Carlip S., Entropy from conformal field theory at Killing horizons,
Classical Quantum Gravity 16 (1999), 3327-3348,
gr-qc/9906126.
- Carlip S., Logarithmic corrections to black hole entropy, from the Cardy
formula, Classical Quantum Gravity 17 (2000), 4175-4186,
gr-qc/0005017.
- 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.
- Chary V., Pressley A., A guide to quantum groups, Cambridge University Press,
Cambridge, 1994.
- Corichi A., Diaz-Polo J., Borja E.F., Black hole entropy quantization,
Phys. Rev. Lett. 98 (2007), 181301, 4 pages,
gr-qc/0609122.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory,
Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.
- 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.
- Domagala M., Lewandowski J., Black-hole entropy from quantum geometry,
Classical Quantum Gravity 21 (2004), 5233-5243,
gr-qc/0407051.
- 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.
- 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 the
SU(2)-invariant formulation of type I isolated horizons, Phys.
Rev. D 82 (2010), 044050, 23 pages, arXiv:1006.0634.
- 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.
- Flajolet P., Sedgewick R., Analytic combinatorics, Cambridge University Press,
Cambridge, 2009.
- Fleischhack C., Representations of the Weyl algebra in quantum geometry,
Comm. Math. Phys. 285 (2009), 67-140,
math-ph/0407006.
- 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.
- Frodden E., Ghosh A., Perez A., A local first law for isolated horizons,
arXiv:1110.4055.
- Geroch R.P., Held A., Penrose R., A space-time calculus based on pairs of null
directions, J. Math. Phys. 14 (1973), 874-881.
- 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.
- Ghosh A., Mitra P., Counting black hole microscopic states in loop quantum
gravity, Phys. Rev. D 74 (2006), 064026, 5 pages,
hep-th/0605125.
- 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.
- Ghosh A., Mitra P., Log correction to the black hole area law, Phys.
Rev. D 71 (2005), 027502, 3 pages, gr-qc/0401070.
- Ghosh A., Perez A., Black hole entropy and isolated horizons thermodynamics,
Phys. Rev. Lett. 107 (2011), 241301, 5 pages,
arXiv:1107.1320.
- 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.
- Hawking S.W., Particle creation by black holes, Comm. Math. Phys.
43 (1975), 199-220.
- Hawking S.W., Ellis G.F.R., The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, Cambridge
University Press, London, 1973.
- Hayward S.A., General laws of black-hole dynamics, Phys. Rev. D
49 (1994), 6467-6474, gr-qc/9303006.
- Hayward S.A., Spin coefficient form of the new laws of black hole dynamics,
Classical Quantum Gravity 11 (1994), 3025-3035,
gr-qc/9406033.
- Jacobson T., A note on renormalization and black hole entropy in loop quantum
gravity, Classical Quantum Gravity 24 (2007), 4875-4879,
arXiv:0707.4026.
- Kaul R.K., Majumdar P., Logarithmic correction to the Bekenstein-Hawking
entropy, Phys. Rev. Lett. 84 (2000), 5255-5257,
gr-qc/0002040.
- Kaul R.K., Majumdar P., Quantum black hole entropy, Phys. Lett. B
439 (1998), 267-270, gr-qc/9801080.
- Kerr R.P., Gravitational field of a spinning mass as an example of
algebraically special metrics, Phys. Rev. Lett. 11 (1963),
237-238.
- 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.
- Krasnov K., On quantum statistical mechanics of a Schwarzschild black hole,
Gen. Relativity Gravitation 30 (1998), 53-68,
gr-qc/9605047.
- Krasnov K., Quantum geometry and thermal radiation from black holes,
Classical Quantum Gravity 16 (1999), 563-578,
gr-qc/9710006.
- Krasnov K., Rovelli C., Black holes in full quantum gravity, Classical
Quantum Gravity 26 (2009), 245009, 8 pages, arXiv:0905.4916.
- Lee J., Wald R.M., Local symmetries and constraints, J. Math. Phys.
31 (1990), 725-743.
- Lewandowski J., Spacetimes admitting isolated horizons, Classical
Quantum Gravity 17 (2000), L53-L59, gr-qc/9907058.
- 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.
- Livine E.R., Terno D.R., Bulk entropy in loop quantum gravity, Nuclear
Phys. B 794 (2008), 138-153, arXiv:0706.0985.
- 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.
- Livine E.R., Terno D.R., The entropic boundary law in BF theory,
Nuclear Phys. B 806 (2009), 715-734, arXiv:0805.2536.
- Lochan K., Vaz C., Canonical partition function of loop black holes,
Phys. Rev. D 85 (2012), 104041, 9 pages,
arXiv:1202.2301.
- Lochan K., Vaz C., Statistical analysis of entropy correction from topological
defects in loop black holes, arXiv:1205.3974.
- Massar S., Parentani R., How the change in horizon area drives black hole
evaporation, Nuclear Phys. B 575 (2000), 333-356,
gr-qc/9903027.
- Meissner K.A., Black-hole entropy in loop quantum gravity, Classical
Quantum Gravity 21 (2004), 5245-5251, gr-qc/0407052.
- 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.
- Modesto L., Disappearance of the black hole singularity in loop quantum
gravity, Phys. Rev. D 70 (2004), 124009, 5 pages,
gr-qc/0407097.
- Modesto L., Loop quantum black hole, Classical Quantum Gravity
23 (2006), 5587-5601, gr-qc/0509078.
- Müller A., Experimental evidence of black holes, PoS Proc. Sci.
(2006), PoS(P2GC), 017, 30 pages, astro-ph/0701228.
- 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.
- Ooguri H., Sasakura N., Discrete and continuum approaches to three-dimensional
quantum gravity, Modern Phys. Lett. A 6 (1991), 3591-3600,
hep-th/9108006.
- Perez A., Introduction to loop quantum gravity and spin foams,
gr-qc/0409061.
- 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.
- Pranzetti D., Radiation from quantum weakly dynamical horizons in loop quantum
gravity, Phys. Rev. Lett. 109 (2012), 011301, 5 pages,
arXiv:1204.0702.
- 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.
- 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.
- Rovelli C., Black hole entropy from loop quantum gravity, Phys. Rev.
Lett. 77 (1996), 3288-3291, gr-qc/9603063.
- Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics,
Cambridge University Press, Cambridge, 2004.
- Rovelli C., Thiemann T., Immirzi parameter in quantum general relativity,
Phys. Rev. D 57 (1998), 1009-1014, gr-qc/9705059.
- Sahlmann H., Black hole horizons from within loop quantum gravity,
Phys. Rev. D 84 (2011), 044049, 12 pages,
arXiv:1104.4691.
- Sahlmann H., Entropy calculation for a toy black hole, Classical
Quantum Gravity 25 (2008), 055004, 14 pages, arXiv:0709.0076.
- Sahlmann H., Toward explaining black hole entropy quantization in loop quantum
gravity, Phys. Rev. D 76 (2007), 104050, 7 pages,
arXiv:0709.2433.
- 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.
- Smolin L., Linking topological quantum field theory and nonperturbative quantum
gravity, J. Math. Phys. 36 (1995), 6417-6455,
gr-qc/9505028.
- Strominger A., Black hole entropy from near-horizon microstates,
J. High Energy Phys. 1998 (1998), no. 2, 009, 11 pages,
hep-th/9712251.
- Strominger A., Vafa C., Microscopic origin of the Bekenstein-Hawking
entropy, Phys. Lett. B 379 (1996), 99-104,
hep-th/9601029.
- Thiemann T., Modern canonical quantum general relativity, Cambridge Monographs
on Mathematical Physics, Cambridge University Press, Cambridge, 2007.
- Thiemann T., Quantum spin dynamics. VIII. The master constraint,
Classical Quantum Gravity 23 (2006), 2249-2265,
gr-qc/0510011.
- Thiemann T., The Phoenix Project: master constraint programme for loop
quantum gravity, Classical Quantum Gravity 23 (2006),
2211-2247, gr-qc/0305080.
- Wald R.M., Black hole entropy is the Noether charge, Phys. Rev. D
48 (1993), R3427-R3431, gr-qc/9307038.
- Wald R.M., General relativity, University of Chicago Press, Chicago, IL, 1984.
- Witten E., Quantum field theory and the Jones polynomial, Comm. Math.
Phys. 121 (1989), 351-399.
|
|