The Disjointed Thermodynamics of Rotating Black Holes With a NUT Twist

TL;DR

This paper investigates Euclidean Kerr-bolt solutions with NUT charge, revealing a quantization condition that affects the validity and interpretation of the black hole thermodynamics first law.

Contribution

It demonstrates that regularity conditions impose a quantization constraint, fundamentally altering the applicability of the first law of thermodynamics for these solutions.

Findings

Most parameter combinations are excluded due to singularity removal.

Continuous transitions between solution manifolds are impossible.

The quantization condition affects the interpretation of thermodynamic relations.

Abstract

We study the solutions of the Euclidean-signature Einstein equations (gravitational instantons) whose line-element takes the Kerr-bolt form, characterized by three real parameters: a size, a NUT charge, and a spin rate. The exclusion of singularities eliminates most combinations of these parameters, leaving only separated solution-manifolds between which continuous transitions are impossible (The angular velocity divided by the temperature is forced to be a rational multiple of $2\pi$). This ``quantization'' prevents the free variations presupposed by an equation like $T dS=dM + \Omega dJ$, and thereby renders the first law true, false, meaningless, or tautological, depending on how one approaches it.