Statistical entropy of a BTZ black hole from loop quantum gravity

  • Ernesto Frodden
  • Marc GeillerEmail author
  • Karim Noui
  • Alejandro Perez
Open Access


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 crossing the horizon is colored by a spin j (at the kinematical level), and the length L of the horizon is given by the sum L = ∑ L of the fundamental length contributions L carried by the spins j of the links . 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 is satisfied, as it should be in three-dimensional quantum gravity.


Quantum Groups Models of Quantum Gravity Black Holes 


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Copyright information

© SISSA 2013

Authors and Affiliations

  • Ernesto Frodden
    • 1
  • Marc Geiller
    • 2
    Email author
  • Karim Noui
    • 2
    • 3
  • Alejandro Perez
    • 1
  1. 1.Centre de Physique ThéoriqueMarseilleFrance
  2. 2.Laboratoire APC — Astroparticule et CosmologieUniversité Paris Diderot Paris 7ParisFrance
  3. 3.Laboratoire de Mathématique et Physique ThéoriqueToursFrance

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