Thermodynamics of Black Hole Horizons and Kerr/CFT Correspondence

TL;DR

This work shows that for stationary black holes with two horizons, the outer-horizon first law automatically enforces a corresponding inner-horizon law, and that the mass-independence of the horizon area product $S_+S_-$ is equivalent to $T_+S_+=T_-S_-$. Using this, the authors connect horizon thermodynamics to Kerr/CFT by identifying right/left-moving sectors with equal central charges when $S_+S_-$ is mass-independent and by extracting dimensionless CFT temperatures from thermodynamic data, which reproduce known dual pictures for 4D Kerr-Newman and 5D Myers–Perry black holes. They show mass-independence holds in 4D KN, in 4D/5D MP with certain conditions, and in BTZ, but breaks for Myers-Perry with $d\ge 6$ and Kerr-AdS with $d\ge 4$. The paper also predicts central charges and temperatures for a holographic CFT description of the 5D doubly rotating black ring, illustrating a broad applicability of horizon thermodynamics to holography and highlighting open questions in higher-dimensional and AdS settings.

Abstract

In this paper we investigate the thermodynamics of the inner horizon and its implication on the holographic description of the black hole. We focus on the black holes with two physical horizons. Under reasonable assumption, we prove that the first law of thermodynamics of the outer horizon always indicates that of the inner horizon. As a result, the fact that the area product being mass-independent is equivalent to the relation $T_+S_+=T_-S_-$, with $T_\pm$ and $S_\pm$ being the Hawking temperatures and the entropies of the outer and inner horizon respectively. We find that the mass-independence of area product breaks down in general Myers-Perry black holes with spacetime dimension $d\geq6$ and Kerr-AdS black holes with $d\geq4$. Moreover we discuss the implication of the first laws of the outer and inner horizons on the thermodynamics of the right- and left-moving sectors of dual CFT in Kerr/CFT correspondence. We show that once the relation $T_+S_+=T_-S_-$ is satisfied, the central charges of two sectors must be same. Furthermore from the thermodynamics relations, we read the dimensionless temperatures of microscopic CFT, which are in exact agreement with the ones obtained from hidden conformal symmetry in the low frequency scattering off the black holes, and then determine the central charges. This method works well in well-known cases in Kerr/CFT correspondence, and reproduce successfully the holographic pictures for 4D Kerr-Newman and 5D Kerr black holes. We go on to predict the central charges and temperatures of a possible holographic CFT description dual to 5D doubly rotating black ring.