Black hole state degeneracy in loop quantum gravity

The combinatorial problem of counting the black hole quantum states within the isolated horizon framework in loop quantum gravity is analyzed. A qualitative understanding of the origin of the band structure shown by the degeneracy spectrum, which is responsible for the black hole entropy quantization, is reached. Even when motivated by simple considerations, this picture allows to obtain analytical expressions for the most relevant quantities associated to this effect.

Figure 1

The ${\widehat{n}}_s$ given by the MDD is plotted as a function of $s$. The relevant contribution of the lower values of $s$ and the exponential decrease as $s$ grows are observed.

Figure 2

The value of $\lambda$ as a function of $s$ is plotted, and compared with the asymptotic value.

Figure 3

Plot of the degeneracy (number of different horizon states in each area interval of $0.01{\ell }_{\mathrm{P}}^2$). States accumulate around some equidistant values of area, exhibiting a band structure.

Figure 4

Plot of the results for the entropy as a function of area (in Planck units) obtained with the computational counting. The stairlike behavior, with a step width corresponding to $\Delta A$, is observed.

Figure 5

Plot of the sum $S$ of spin labels vs the number $p$ of punctures. All the discrete configurations are placed in the marked points. The thin lines represent “constant area surfaces” ($K$-lines) while the thick line contains the values of ($S$, $p$) that satisfy the quotient ${\widehat{s}}_{md}$ (MD-line).

Figure 6

Comparison between the analytical values for the area of the peaks (dashed vertical lines) and the actual peaks obtained from the computational data.

Figure 7

The mean values of ${\widehat{n}}_s$ obtained from the computational results (points) are compared with the exponential decrease expected from the MDD.