First Law for Kerr Taub-NUT AdS Black Holes

TL;DR

This paper establishes the first law of thermodynamics for Kerr Taub-NUT AdS black holes, incorporating variations in multiple charges and demonstrating consistency between different charge definitions.

Contribution

It introduces a geometric and scaling approach to derive the first law for these black holes, including NUT charges and angular momenta variations.

Findings

Derived the first law including NUT and angular momentum variations.

Established equivalence of Komar and Brown-York charge definitions.

Highlighted the importance of the Killing vector choice for thermodynamic mass.

Abstract

The first law of black hole mechanics, which relates the change of energy to the change of entropy and other conserved charges, has been the main motivation for probing the thermodynamic properties of black holes. In this work, we investigate the thermodynamics of Kerr Taub-NUT AdS black holes. We present geometric Komar definitions for the black hole charges, that by construction satisfy the Smarr formula. Further, by a scaling argument based on Euler's theorem, we establish the first law for the Kerr Taub-NUT AdS black holes. The corresponding first law includes variations in the cosmological constant, NUT charges and angular momenta. The key new ingredient in the construction are the independent variations of both angular momenta, the black hole and Misner string angular momenta. Employing the Brown-York quasilocal charge definitions we show that our expression for the mass and spin coincide with our generalized Komar expressions. We indicate the relevance of these results to the thermodynamics of rotating AdS black holes, including the proper choice of time-like Killing vector to produce the correct thermodynamic mass.