We give an efficient method, combining number-theoretic and combinatorial ideas, to exactly compute black hole entropy in the framework of loop quantum gravity. Along the way we provide a complete characterization of the relevant sector of the spectrum of the area operator, including degeneracies, and explicitly determine the number of solutions to the projection constraint. We use a computer implementation of the proposed algorithm to confirm and extend previous results on the detailed structure of the black hole degeneracy spectrum.

Ever since the pioneering works of Bekenstein and Hawking, black hole entropy has been known to have a quantum origin. Furthermore, it has long been argued by Bekenstein that entropy should be quantized in discrete (equidistant) steps given its identification with horizon area in (semi-)classical general relativity and the properties of area as an adiabatic invariant. This lead to the suggestion that the black hole area should also be quantized in equidistant steps to account for the discrete black hole entropy. Here we shall show that loop quantum gravity, in which area is not quantized in equidistant steps, can nevertheless be consistent with Bekenstein's equidistant entropy proposal in a subtle way. For that we perform a detailed analysis of the number of microstates compatible with a given area and show consistency with the Bekenstein framework when an oscillatory behavior in the entropy-area relation is properly interpreted. Black hole entropy is one of the most intriguing constructs of modern theoretical physics. On the one hand, it has a correspondence with the black hole horizon area through the laws of (classical) black hole mechanics. On the other hand, it is assumed to have a quantum statistical origin given that the proper identification between entropy and area S A=4' 2 p came only after an analysis of quantum fields on a fixed background [1].Furthermore, it has long been argued by Bekenstein that the proportionality between entropy and area, for large, classical black holes, can be justified from the adiabatic invariance properties of horizon area when subject to different scenarios (see [2,3] for a review). Further heuristic quantization arguments lead to the suggestion that area, when quantized, should have a discrete, equidistant spectrum in the large horizon limitwith a parameter and n integer. The relation between area and entropy that one expects to encounter in the large horizon radius is then extrapolated to the full spectrum. This would imply that entropy too would have a discrete spectrum, a property that might also be expected if entropy is to be associated with (the logarithm of) the number of microstates compatible with a given macrostate. When this condition is imposed, then the area is expected to have an spectrum of the formwith k and n integers [4]. Even when appealing and physically well motivated, these arguments remain somewhat heuristic and have no detailed microscopic quantum gravity formalism to support them. A quantum canonical description of black holes that has offered a detailed description of the quantum horizon degrees of freedom is given by loop quantum gravity (LQG) [5]. This formalism allows the inclusion of several matter couplings (including nonminimal couplings) and black holes far from extremality, in four dimensions. There is no restriction in the values of the matter charges. The approach uses as a starting point isolated horizon (IH) boundary conditions at the classical level, where the interior of the black hole is excluded from the region under consideration...

Quantum black holes within the loop quantum gravity (LQG) framework are considered. The number of microscopic states that are consistent with a black hole of a given horizon area A 0 are counted and the statistical entropy, as a function of the area, is obtained for A 0 up to 550 ℓ 2 Pl . The results are consistent with an asymptotic linear relation and a logarithmic correction with a coefficient equal to −1/2. The Barbero-Immirzi parameter that yields the asymptotic linear relation compatible with the Bekenstein-Hawking entropy is shown to coincide with a value close to γ = 0.274, which has been previously obtained analytically. However, a new and oscillatory functional form for the entropy is found for small, Planck size, black holes that calls for a physical interpretation.

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The Loop Quantum Gravity (LQG) program is briefly reviewed and one of its main applications, namely the counting of black hole entropy within the framework is considered. In particular, recent results for Planck size black holes are reviewed. These results are consistent with an asymptotic linear relation (that fixes uniquely a free parameter of the theory) and a logarithmic correction with a coefficient equal to −1/2. The account is tailored as an introduction to the subject for non-experts.