Toward explaining black hole entropy quantization in loop quantum gravity

In a remarkable numerical analysis of the spectrum of states for a spherically symmetric black hole in loop quantum gravity, Corichi, Diaz-Polo and Fernandez-Borja found that the entropy of the black hole horizon increases in what resembles discrete steps as a function of area. In the present article we reformulate the combinatorial problem of counting horizon states in terms of paths through a certain space. This formulation sheds some light on the origins of this steplike behavior of the entropy. In particular, using a few extra assumptions we arrive at a formula that reproduces the observed step length to a few tenths of a percent accuracy. However, in our reformulation the periodicity ultimately arises as a property of some complicated process, the properties of which, in turn, depend on the properties of the area spectrum in loop quantum gravity in a rather opaque way. Thus, in some sense, a deep explanation of the observed periodicity is still lacking.

Figure 1

Paths that end below $\mathcal{R}=4$ (left) and below $\mathcal{R}=8$ (right).

Figure 2

Density (left) and logarithm of density (right) of paths that end below $\mathcal{R}=8$ (in fiducial units).

Figure 3

Logarithm of density of physical states, in a small interval around $a$ (left) and in an interval adapted to the periodicity (right).

Figure 4

The possible steps starting at 0 (left) and a set of these steps superimposed onto the spectrum of paths (right).

Figure 5

The set of paths ending below $\mathcal{R}=8$, with a distorted area spectrum $a^{\prime }(m)$.