Statistical Entropy of a BTZ Black Hole from Loop Quantum Gravity

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 $Λ$. 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 $Γ$ lives in a two-dimensional spacelike surface $Σ$, and some of its links cross the black hole horizon, which is viewed as a circular boundary of $Σ$. 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 $N^\text{BTZ}_Γ(L,Λ)$ of the Euclidean BTZ black hole microstates (defined on a fixed graph $Γ$) based on an analytic continuation from the case $Λ>0$ to the case $Λ<0$. In our model, we show that $N^\text{BTZ}_Γ(L,Λ)$ reproduces the Bekenstein-Hawking entropy in the classical limit. This asymptotic behavior is independent of the choice of the graph $Γ$ provided that the condition $L=\sum_\ell L_\ell$ is satisfied, as it should be in three-dimensional quantum gravity.