[Paper Review] The Entropy of BTZ Black Hole from Loop Quantum Gravity

Jingbo Wang|arXiv (Cornell University)|Jan 14, 2014

Black Holes and Theoretical Physics24 references3 citations

TL;DR

This paper calculates the entropy of the BTZ black hole using loop quantum gravity, showing that horizon degrees of freedom are described by a 2D SO(1,1) BF theory. It derives the Bekenstein-Hawking area law with the 1/4 factor, confirming the entropy-area proportionality via a combinatorial count of physical states under quantized flux assumptions.

ABSTRACT

In this paper, we calculated the entropy of the BTZ black hole in the framework of loop quantum gravity. We got the result that the horizon degrees of freedom can be described by the 2D SO(1,1) punctured BF theory. Finally we got the area law for the entropy of BTZ black hole.

Motivation & Objective

• To derive the entropy of the BTZ black hole within the loop quantum gravity framework.
• To identify the effective field theory describing horizon degrees of freedom in 3D quantum gravity.
• To establish whether the Bekenstein-Hawking area law emerges from LQG in 2+1 dimensions.
• To explore the role of physical state constraints and flux quantization in entropy counting.

Proposed method

• Decompose the symplectic structure of the isolated horizon into bulk and boundary terms.
• Show that the boundary term matches the symplectic form of a 2D SO(1,1) BF theory via gauge fixing.
• Assume quantized magnetic quantum numbers $ l_m = \alpha m $ with $ m \in \mathbb{N}^+ $, introducing a parameter $ \alpha $ analogous to the Barbero-Immirzi parameter.
• Reduce the entropy calculation to a combinatorics problem: counting ordered positive integer partitions of $ a = L_H / (8\pi l_{\text{Pl}} \alpha) $.
• Use the known result that the number of such partitions is $ \mathcal{N} = 2^{a-1} $, leading to entropy $ S = \log \mathcal{N} $.

Experimental results

Research questions

• RQ1 Can the entropy of the BTZ black hole be derived from loop quantum gravity in 2+1 dimensions using the ABCK method?
• RQ2 What effective field theory describes the horizon degrees of freedom in the isolated horizon formalism for 3D gravity?
• RQ3 How does flux quantization affect the counting of horizon states and the resulting entropy?
• RQ4 Does the derived entropy obey the Bekenstein-Hawking area law with the 1/4 factor in 3D quantum gravity?
• RQ5 What role do dynamical constraints, such as the Hamiltonian constraint, play in restricting physical states beyond the kinematical level?

Key findings

• The horizon degrees of freedom in the BTZ black hole are described by a 2D SO(1,1) BF theory after gauge fixing.
• The number of physical horizon states is $ \mathcal{N} = 2^{a-1} $, where $ a = L_H / (8\pi l_{\text{Pl}} \alpha) $, under the assumption of quantized flux $ l_m = \alpha m $.
• The entropy is $ S = \log \mathcal{N} \approx \frac{\log 2 \cdot L_H}{8\pi \alpha l_{\text{Pl}}} - \log 2 $, which reduces to the Bekenstein-Hawking area law when $ \alpha = \log 2 / (2\pi) $.
• With this choice of $ \alpha $, the entropy reproduces the 1/4 factor in the area law, matching the classical result.
• The spectrum of the horizon length operator is found to be $ L_n = 4 \log 2 \cdot n \cdot l_{\text{Pl}} $, indicating discrete quantization.
• The work highlights that dynamics—particularly through physical state constraints—are essential for deriving correct entropy, not just kinematical state counting.