The Schwarzschild/CFT Correspondence: Weyl Rescaled Case

TL;DR

This paper explores a novel CFT dual for the Schwarzschild black hole by employing a Weyl rescaling that reveals an $AdS_{2} imes S^{2}$ structure, enabling entropy calculation matching the Bekenstein-Hawking result.

Contribution

It introduces a Weyl rescaling approach to derive a CFT dual for Schwarzschild black holes with a specific central charge and matches the microscopic entropy to the classical entropy.

Findings

The Weyl rescaled Schwarzschild metric admits an $AdS_{2} imes S^{2}$ geometry.

A central charge of $c=96 M^3$ is derived for the dual CFT.

The microscopic entropy matches the Bekenstein-Hawking entropy.

Abstract

In this work, the CFT dual of the Schwarzschild black hole is investigated. A Weyl rescaling factor is presented, so that the Weyl rescaled Schwarzschild metric, after a coordinate transformation, has an $AdS_{2} \times S^{2}$ geometry at vicinity of its origin. Since the near origin spacetime admits an $AdS_{2}$ factor, it is dealt with a 2D effective gravity which is dimensionally reduced from the near origin solution. It is exhibited that the dual CFT has a central charge $c=96 M^3$ which is an asymptotic conserved charge of the effective solution. Finally, the microscopic entropy of Schwarzschild black hole is achieved by using Cardy formula. It is revealed that the microscopic entropy exactly reproduces the Bekenstein-Hawking entropy of the Schwarzschild black hole.