Combinatorics of theSU(2)black hole entropy in loop quantum gravity

Abstract

We use the combinatorial and number-theoretical methods developed in previous work by the authors to study black hole entropy in the new proposal put forward by Engle, Noui and Pérez. Specifically we give the generating functions relevant for the computation of the entropy and use them to derive its asymptotic behavior including the value of the Immirzi parameter and the coefficient of the logarithmic correction. In this brief note we want to study some of the physical consequences that follow from the black hole entropy definition proposed, in the context of loop quantum gravity, by Engle, Noui and Pérez in [1]. The main reason to do this is to check wether this new definition satisfies the obvious physical requirement of reproducing the Bekenstein-Hawking formula for large black holes. Without going into the details of the theoretical foundations of this new proposal, this analysis can be seen as a straightforward consistency check. We also want to obtain corrections to this formula that can be eventually compared with equivalent results found in different approaches [2,3,4,5]. An additional reason to perform this study is to show the power of the combinatorial methods developed by the authors in [6,7,8,9].The problem of interest can be enunciated in the following way [1]. Given a value of the black hole area a H = 4πγℓ 2 P κ (where κ ∈ N is the level of the SU(2) Chern-Simons theory on the horizon, 1 ℓ P denotes the Planck length, and γ the Immirzi parameter), we have to determine the number of states labeled by spins j 1 , . . . , j n satisfying an inequality of the * Ivan.Agullo@uv.es † fbarbero@iem.cfmac.csic.es ‡ Enrique.Fernandez@uv.es § Jacobo.Diaz@uv.es ¶ ejsanche@math.uc3m.es 1 For an earlier treatment, based on different considerations, of the SU (2) Chern-Simons theory in this framework see [10].

The SU(2) black hole entropy revisited

“…We adapt the techniques used in [18][19][20][21][22] and firstly introduced in [11,12] to compute the entropy of a black hole. We are not going into the mathematical details of these techniques which has been very well exposed in [18][19][20][21][22] and which are in fact very well-known in the domain of probabilities and used to understand some properties of random walks. We recover that in the spherically symmetric and distorted black holes, the leading term of the entropy is proportional to the area and the first corrections are still logarithmic: S(a) ∼ αa + β log a.…”

“…In this paper we study the finite k counting problem by means of simple asymptotic methods. The powerful methods that have been developed for the resolution of the counting problem in the k = ∞ [18][19][20][21][22][23][24] are perhaps generalizable to the finite k case. Here we follow a less rigorous and more physical approach.…”

“…In this case it is natural (although not necessary) to consider SU(2) (or U(1)) Chern-Simons theory with a level that scales with the macroscopic classical area k ∝ a H . This makes the state-counting (necessary for the computation of the entropy) a combinatorial problem which can be entirely formulated in terms of the representation theory of the classical group SU(2) (or U(1)): for practical purposes one can take k = ∞ from the starting point [11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

A note on the symmetry reduction of SU(2) on horizons of various topologies

“…It is found that on the isolated horizon (inner boundary) the SU(2) theory produces a Chern-Simons theory, which in turn reduces to only U(1) true degrees of freedom [12], and this has ramifications for the sub-leading correction coefficient (for example, see [14], [15], and [16], the last reference utilising a combinatoric approach). There has been some ambiguity regarding this reduction.…”

Spin Foams

“…Furthermore, here and throughout the rest of this section we use the Einstein summation convention: when a given index appears twice in an expression, once up and once down, summation shall be implied over all possible values of the given index. In the final expression above, we have defined B (Note that in the spin foam literature, sometimes B IJ µν is defined to be only the first term of the expression in parentheses in (16). )…”