[Paper Review] The Entropy of the BTZ Black Hole and AdS/CFT Correspondence

Taejin Lee|ArXiv.org|Jun 15, 1998

Black Holes and Theoretical Physics3 citations

TL;DR

This paper constructs a quantum action for the BTZ black hole using an SL(2,R) Wess-Zumino-Witten model on the boundary, coupled to an anti-de Sitter background. It shows that the coupling via a one-cocycle condition dominates the Virasoro algebra's central charge, explaining the Bekenstein-Hawking entropy in the context of AdS/CFT correspondence through conformal field theory methods.

ABSTRACT

We construct an action, which governs the dynamics of the Bañados-Teitelboim-Zanelli (BTZ) black hole and perform the canonical quantization. The quantum action is given by a $SL(2,R)$ Wess-Zumino-Witten model on the boundary coupled to the classical anti-de Sitter background, representing a massless BTZ black hole. The coupling, determined by a one-cocyle condition, is found to give dominant contribution to the central charge of Virasoro algebra. The entropy of the BTZ black hole is discussed from the point view of the AdS/CFT correspondence and an explanation is given to the puzzle of black hole entropy in the BTZ case. The BTZ black hole is a quantum object and the BTZ black hole with finite mass should be considered as a quantum excitation of the massless one.

Motivation & Objective

• To understand the quantum nature of the BTZ black hole in three-dimensional anti-de Sitter space.
• To resolve the puzzle of black hole entropy in the BTZ case using the AdS/CFT correspondence.
• To derive the entropy of the BTZ black hole from a boundary conformal field theory perspective.
• To show that the finite-mass BTZ black hole arises as a quantum excitation of the massless one.

Proposed method

• Formulating a canonical action for the BTZ black hole dynamics using the SL(2,R) Wess-Zumino-Witten model on the boundary.
• Coupling the boundary WZW model to a classical anti-de Sitter background representing a massless BTZ black hole.
• Applying a one-cocycle condition to determine the coupling strength between the boundary CFT and the bulk geometry.
• Using the resulting coupling to compute the central charge of the Virasoro algebra in the boundary CFT.
• Applying the Cardy formula to compute the entropy from the central charge and conformal weights.
• Treating the finite-mass BTZ black hole as a quantum excitation of the massless one in the quantum framework.

Experimental results

Research questions

• RQ1 How can the entropy of the BTZ black hole be derived from a boundary conformal field theory in the AdS/CFT framework?
• RQ2 What role does the SL(2,R) WZW model play in describing the quantum dynamics of the BTZ black hole?
• RQ3 How does the one-cocycle condition determine the coupling that contributes to the central charge?
• RQ4 Why does the BTZ black hole with finite mass arise as a quantum excitation of the massless BTZ solution?
• RQ5 What is the precise relation between the central charge of the Virasoro algebra and the Bekenstein-Hawking entropy in this model?

Key findings

• The coupling between the boundary SL(2,R) WZW model and the classical anti-de Sitter background, determined by the one-cocycle condition, gives the dominant contribution to the central charge of the Virasoro algebra.
• The central charge derived from this coupling matches the value required to reproduce the Bekenstein-Hawking entropy formula for the BTZ black hole.
• The finite-mass BTZ black hole is interpreted as a quantum excitation of the massless BTZ solution within the quantum framework.
• The entropy of the BTZ black hole is successfully explained via the Cardy formula applied to the boundary CFT with the computed central charge.
• The AdS/CFT correspondence provides a consistent quantum description of the BTZ black hole, resolving the entropy puzzle through conformal symmetry.