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

14 pages. 1 figure. Paragraph added on page 7 to clarify the horizon condition