The First Law of Thermodynamics for Kerr-Anti-de Sitter Black Holes

TL;DR

This paper derives consistent expressions for the mass and angular momenta of Kerr-AdS black holes across multiple dimensions, confirming the first law of thermodynamics and clarifying their thermodynamic properties within the AdS/CFT framework.

Contribution

It provides new, consistent formulas for black hole mass and angular momentum in AdS backgrounds that satisfy the first law in higher dimensions, differing from previous literature.

Findings

Expressions satisfy the first law of thermodynamics in all dimensions.

Mass matches the conformal conserved charge by Ashtekar, Magnon, and Das.

Thermodynamic potential equals background-subtracted Euclidean action times temperature.

Abstract

We obtain expressions for the mass and angular momenta of rotating black holes in anti-de Sitter backgrounds in four, five and higher dimensions. We verify explicitly that our expressions satisfy the first law of thermodynamics, thus allowing an unambiguous identification of the entropy of these black holes with $\ft14$ of the area. We find that the associated thermodynamic potential equals the background-subtracted Euclidean action multiplied by the temperature. Our expressions differ from many given in the literature. We find that in more than four dimensions, only our expressions satisfy the first law of thermodynamics. Moreover, in all dimensions we show that our expression for the mass coincides with that given by the conformal conserved charge introduced by Ashtekar, Magnon and Das. We indicate the relevance of these results to the AdS/CFT correspondence.