# 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 $N^\text{BTZ}_\Gamma(L,\Lambda)$ 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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00957632
Contributor : Alejandro Perez <>
Submitted on : Monday, March 10, 2014 - 4:56:17 PM
Last modification on : Friday, February 19, 2021 - 4:10:02 PM

### Citation

Ernesto Frodden, Marc Geiller, Karim Noui, Alejandro Perez. Statistical Entropy of a BTZ Black Hole from Loop Quantum Gravity. Journal of High Energy Physics, Springer, 2013, 2013 (5), pp.139. ⟨10.1007/JHEP05(2013)139⟩. ⟨hal-00957632⟩

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