Entropy calculation for a toy black hole

Abstract

In this note we carry out the counting of states for a black hole in loop quantum gravity, however assuming an equidistant area spectrum. We find that this toy-model is exactly solvable, and we show that its behavior is very similar to that of the correct model. Thus this toy-model can be used as a nice and simplifying 'laboratory' for questions about the full theory.

Flux-area operator and black hole entropy

“…In fact, as mentioned in Appendix B, some of the results that we have used to compute the entropy actually appear in [13].…”

“…It is important to mention that some operators similar to the one proposed here have been considered in the literature [10][11][12][13]. In these papers the authors have suggested to use area operators with eigenvalues defined in terms of the spins j i .…”

“…Another such study is due to Sahlmann [13]. In that paper the author starts from the Domagala-Lewandowski prescription [3] to obtain the entropy by studying only sequences of third spin components.…”

“…3. h The problem of finding generating functions for the combinatorial problems posed above was considered also by H. Sahlmann in [13]. His approach was based on the use of paths in the integer lattice Z.…”

The SU(2) black hole entropy revisited

“…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].…”