Detailed black hole state counting in loop quantum gravity

TL;DR

This work develops a comprehensive, number-theoretic framework for counting black hole microstates in loop quantum gravity across ABCK/DL and related schemes. By solving Pell-type equations, organizing area spectra through squarefree integers, and encoding degeneracies with generating functions, the authors reveal a robust band structure in the horizon degeneracy and connect microscopic periodicities to macroscopic entropy growth. They demonstrate that, across multiple countings, the entropy scales linearly with area with model-dependent logarithmic corrections and derive exact generating-function expressions that isolate contributions from specific spectral substructures. The results provide both a detailed microscopic picture for small black holes and a solid pathway toward asymptotic large-area behavior, with implications for the universality of the Immirzi parameter and potential connections to conformal-field-theory techniques.

Abstract

We give a complete and detailed description of the computation of black hole entropy in loop quantum gravity by employing the most recently introduced number-theoretic and combinatorial methods. The use of these techniques allows us to perform a detailed analysis of the precise structure of the entropy spectrum for small black holes, showing some relevant features that were not discernible in previous computations. The ability to manipulate and understand the spectrum up to the level of detail that we describe in the paper is a crucial step towards obtaining the behavior of entropy in the asymptotic (large horizon area) regime.

Figures (16)

• Figure 1: Plot of the black hole degeneracy spectrum $D^{\rm DL}$ (in units of $10^{19}$) in terms of the area (in units of $4\pi\gamma\ell^2_P$) for a range of area values. The periodicity can be traced all the way back to the smaller values of the area.
• Figure 2: Plot of $S_{\leq}(a)$ in terms of the area (in units of $4\pi\gamma\ell^2_P$). The computation of $S_{\leq}(a)$ has been done by using the algorithm based in number-theoretical method discussed in the paper.
• Figure 3: Plot of the black hole degeneracy spectrum $D^{\rm ENP}$ (in units of $10^{21}$) in terms of the area (in units of $4\pi\gamma\ell^2_P$) for a range of area values.
• Figure 4: Plot of $S^{\rm ENP}_{\leq}(a)$ in terms of the area (in units of $4\pi\gamma\ell^2_P$). This plot is essentially the same for $S^{\rm GM}_{\leq}(a)$.
• Figure 5: Plot of $\log D_*^{\rm DL}(a)$ and $\log D^{\rm DL}(a)$ in terms of the area (in units of $4\pi\gamma\ell^2_P$). The figure shows the values associated with area eigenvalues multiples of $\sqrt{3}$, $\sqrt{2}$ and $\sqrt{15}$ respectively. The solid lines correspond to the asymptotic approximations given by straight lines with the slopes $s_2$, $s_1$ and $s_{10}$ appearing in table \ref{['tablasquarefree']}. When the projection constraint is taken into account (lower plot) the asymptotic approximations have also a logarithmic correction $-(\log a)/2$.