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 $\Lambda$. 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 $N_{\Gamma }^{\text{BTZ}}(L,\Lambda )$ 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 $N_{\Gamma }^{\text{BTZ}}(L,\Lambda )$ 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.

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

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