Statistical Entropy of a BTZ Black Hole from Loop Quantum Gravity

TL;DR

This paper calculates the statistical entropy of a BTZ black hole within 3D loop quantum gravity, showing it reproduces Bekenstein-Hawking entropy in the classical limit through a model based on spin network states and an analytic continuation of the cosmological constant.

Contribution

It introduces a method to estimate BTZ black hole microstates in loop quantum gravity that reproduces classical entropy, extending previous 4D results to 3D with a cosmological constant.

Findings

Reproduces Bekenstein-Hawking entropy in the classical limit.

Shows independence of the graph choice under certain conditions.

Uses analytic continuation from positive to negative cosmological constant.

Abstract

We compute the statistical entropy of a BTZ black hole in the context of three-dimensional Euclidean loop quantum gravity with a cosmological constant $\Lambda$. As in the four-dimensional case, a quantum state of the black hole is characterized by a spin network state. Now however, the underlying colored graph $\Gamma$ lives in a two-dimensional spacelike surface $\Sigma$, and some of its links cross the black hole horizon, which is viewed as a circular boundary of $\Sigma$. Each link $\ell$ crossing the horizon is colored by a spin $j_\ell$ (at the kinematical level), and the length $L$ of the horizon is given by the sum $L=\sum_\ell L_\ell$ of the fundamental length contributions $L_\ell$ carried by the spins $j_\ell$ of the links $\ell$. We propose an estimation for the number $N^\text{BTZ}_\Gamma(L,\Lambda)$ of the Euclidean BTZ black hole microstates (defined on a fixed graph $\Gamma$) based on an analytic continuation from the case $\Lambda>0$ to the case $\Lambda<0$. In our model, we show that $N^\text{BTZ}_\Gamma(L,\Lambda)$ reproduces the Bekenstein-Hawking entropy in the classical limit. This asymptotic behavior is independent of the choice of the graph $\Gamma$ provided that the condition $L=\sum_\ell L_\ell$ is satisfied, as it should be in three-dimensional quantum gravity.