General Smarr relation and first law of Nutty dyonic black hole

TL;DR

This paper explores the thermodynamics of charged NUT black holes with Misner strings, deriving generalized Smarr relations and first laws that depend on gauge choices, and establishing conditions for their consistency.

Contribution

It introduces a geometric method to derive Smarr relations and first laws for NUT black holes, emphasizing the role of gauge choices in their thermodynamic descriptions.

Findings

Misner charge obtained via Komar integration.

Thermodynamic laws depend on gauge choices for potentials.

Conditions for consistent thermodynamic laws derived.

Abstract

In this paper, we investigate the thermodynamics of the charged NUT black hole with the Misner strings present in the Einstein-Maxwell gravity. We show that the Misner charge $N$ can be obtained by performing Komar integration over the Misner strings in the presence of electric and magnetic charges and the corresponding Smarr relation and first law can also be naturally established based on this geometric perspective. Besides, we demonstrate that the electric and magnetic charges appearing in the first law and Smarr relation are contingent on the special choice of gauge freedom for the electric and magnetic potentials. We re-derive two different versions of thermodynamics according to distinct choices of gauge and furthermore, we try to formulate more general thermodynamic laws under other arbitrary choices of gauge and deduce the conditions necessary to satisfy the Smarr relations and first laws simultaneously.