Statistical entropy of a BTZ black hole from loop quantum gravity

• Published in: The journal of high energy physics Vol. 2013; no. 5; pp. 1 - 17
• Main Authors: Frodden, Ernesto, Geiller, Marc, Noui, Karim, Perez, Alejandro
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-05-2013; Springer Nature B.V; Springer
• Subjects: Asymptotic properties; Black holes (astronomy); Boundaries; Classical and Quantum Gravitation; Elementary Particles; Entropy; General Relativity and Quantum Cosmology; Graphs; High energy physics; High Energy Physics - Theory; Horizon; Links; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum gravity; Quantum Physics; Relativity Theory; String Theory; Quantum Groups; Black Holes; Models of Quantum Gravity
• ISSN: 1029-8479; 1126-6708; 1029-8479
• DOI: 10.1007/JHEP05(2013)139

A
bstract
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
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
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.