Misner Gravitational Charges and Variable String Strengths

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.