# Misner Gravitational Charges and Variable String Strengths

## TL;DR

This paper provides a geometric interpretation of Misner charges in Taub-NUT spacetimes, linking them to surface gravity and Komar integrals, and extends the thermodynamic first law to include variable string strengths.

## Contribution

It introduces a geometrical framework for Misner charges using surface gravity and Komar integrals, and generalizes the thermodynamic first law for Taub-NUT solutions.

## Key findings

- Misner potential identified with surface gravity of the string
- Misner charge obtained via Komar-type integration
- First law extended to include variable string strengths

## Abstract

As shown recently (arXiv:1903.08668), consistent thermodynamics of the Lorentzian Taub-NUT solutions with Misner strings present can be formulated provided a new pair of conjugate quantities (related to the NUT parameter) $\psi-N$ is introduced. In (arXiv:1903.08668) this pair was calculated from the Euclidean action but no geometrical interpretation for the new quantities was provided. In this paper we propose that the potential $\psi$ should be identified with the surface gravity of the Misner string and the conjugate Misner charge $N$ can be obtained by a Komar-type integration over the tubes surrounding the string singularities. We show that similar tube contributions also modify the Komar formula for the thermodynamic volume. To render the integrals finite we employ the method of Killing co-potentials. By construction the new charges then satisfy the Smarr relation. Equipped with these geometrical notions, we generalize the first law for the (possibly charged) Taub-NUT spacetimes to account for asymmetric distributions of Misner strings and their potential variable strengths.