Statistical entropy of a BTZ black hole from loop quantum gravity

Abstract

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 ℓ crossing the horizon is colored by a spin j ℓ (at the kinematical level), and the length L of the horizon is given by the sum L = ∑ ℓ L ℓ of the fundamental length contributions L ℓ carried by the spins j ℓ of the links ℓ. We propose an estimation for the number $ N_{\varGamma}^{\mathrm{BTZ}}\left( {L,\Lambda} \right) $ 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_{\varGamma}^{\mathrm{BTZ}}\left( {L,\Lambda} \right) $ 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 = ∑ ℓ L ℓ is satisfied, as it should be in three-dimensional quantum gravity.