Note on Thermodynamics Method of Black Hole/CFT Correspondence

TL;DR

This paper refines the thermodynamics approach to black hole/CFT correspondence, showing how to derive central charges from entropy products and exploring conditions for hidden conformal symmetry in black holes.

Contribution

It demonstrates that the central charges can be obtained directly from entropy products and clarifies the relation between thermodynamics and hidden conformal symmetry in black holes.

Findings

Central charges derived from entropy product $S_+S_-$ when mass-independent.

Thermodynamics and ASG analysis yield consistent results in the extremal limit.

Condition $T_+A_+=T_-A_-$ is necessary for hidden conformal symmetry.

Abstract

In the paper we further refine the thermodynamics method of black hole/CFT correspondence. We show that one can derive the central charges of different holographic pictures directly from the entropy product $S_+S_-$ if it is mass-independent, for a black hole in the Einstein gravity or the gravity without diffeomorphism anomaly. For a general black hole in the Einstein gravity that admits holographic descriptions, we show that the thermodynamics method and asymptotic symmetry group (ASG) analysis can always give consistent results in the extreme limit. Furthermore, we discuss the relation between black hole thermodynamics and the hidden conformal symmetry. We show that the condition $T_+A_+=T_-A_-$, with $A_\pm$ being the outer and inner horizon areas, is the necessary, but not sufficient, condition for a black hole to have the hidden conformal symmetry. In particular, for the Einstein(-Maxwell) gravity $T_+A_+=T_-A_-$ is just the condition $T_+S_+=T_-S_-$, with $S_\pm$ being the outer and inner horizon entropies, which is the condition for the entropy product $S_+S_-$ being mass-dependent. When there exists the hidden conformal symmetry in the low-frequency scattering off the generic non-extremal black hole, it always leads to the same temperatures of dual CFT as the ones got from the thermodynamics method.