Thermodynamics of Kerr-AdS black holes in the restricted phase space

TL;DR

This paper explores the thermodynamics of Kerr-AdS black holes using a new restricted phase space approach that includes the central charge and chemical potential, revealing phase transitions and similarities to RN-AdS black holes.

Contribution

It introduces a restricted phase space formalism for Kerr-AdS black holes, explicitly analyzing thermodynamic processes and phase transitions without pressure and volume variables.

Findings

Identification of a first order supercritical phase transition at fixed angular momentum.

Discovery of a non-equilibrium transition from small to large black holes.

Observation of a Hawking-Page phase transition in the chemical potential-central charge plane.

Abstract

The thermodynamics for Kerr-AdS black hole in four dimensions is revisited using the recently proposed restricted phase space formalism, which includes the central charge $C$ of the dual CFT and the chemical potential $\mu$, but excludes the pressure and the conjugate volume, as thermodynamic variables. The Euler relation holds automatically, and the first order homogeneity of the mass and the zeroth order homogeneity of the intensive variables are made explicit. Thermodynamic processes involving each pair of conjugate variables are studied in some detail, with emphasis on the scaling properties of the equations of states. It turns out that the thermodynamic behavior of the Kerr-AdS black hole is very similar to that of the RN-AdS black hole studied earlier. In particular, it is found that, there is a first order supercritical phase equilibrium in the $T-S$ processes at fixed nonvanishing angular momentum, while at vanishing angular momentum or at fixed angular velocities, there is always a non-equilibrium transition from a small unstable black hole state to a large stable black hole state. Moreover, there is a Hawking-Page phase transition in the $\mu-C$ processes. Due to the complicatedness of the Kerr metric, the exact critical point and the Hawking-Page temperature are worked out explicitly only in the slow rotating limit, however the characteristic thermodynamic properties do not rely on the slow rotating approximation.