First Law, Counterterms and Kerr-AdS_5 Black Holes

TL;DR

This paper applies the counterterm subtraction method to compute thermodynamic quantities of five-dimensional Kerr-AdS black holes with different boundary metrics, ensuring the first law of thermodynamics is satisfied.

Contribution

It demonstrates that choosing an appropriate boundary metric with arbitrary angular velocity resolves previous violations of the first law in Kerr-AdS_5 calculations.

Findings

Thermodynamic quantities satisfy the first law with proper boundary conditions.

The choice of boundary metric affects the validity of thermodynamic laws.

A new coordinate system ensures consistency with the first law.

Abstract

We apply the counterterm subtraction technique to calculate the action and other quantities for the Kerr--AdS black hole in five dimensions using two boundary metrics; the Einstein universe and rotating Einstein universe with arbitrary angular velocity. In both cases, the resulting thermodynamic quantities satisfy the first law of thermodynamics. We point out that the reason for the violation of the first law in previous calculations is that the rotating Einstein universe, used as a boundary metric, was rotating with an angular velocity that depends on the black hole rotation parameter. Using a new coordinate system with a boundary metric that has an arbitrary angular velocity, one can show that the resulting physical quantities satisfy the first law.