On the Smarr formulas for electrovac spacetimes with line singularities

TL;DR

This paper derives new Smarr mass formulas for stationary axisymmetric Einstein-Maxwell solutions with line singularities, clarifying the roles of NUT parameters, Misner strings, and the rod structure in these spacetimes.

Contribution

It introduces revised Smarr formulas incorporating NUT parameters and Misner strings, correcting previous inaccuracies and providing a unified framework for such spacetimes.

Findings

Derived new horizon and global Smarr formulas with NUT parameter.

Identified the necessity of symmetric tuning to prevent divergence of angular momentum.

Clarified the contribution of Misner strings to total mass as angular momentum, electric charge, and length terms.

Abstract

Using the revised Komar-Tomimatsu approach, we derive Smarr mass formulas for stationary axisymmetric solutions of the Einstein-Maxwell equations containing line singularities (defects) on the polar axis. In terms of the rod structure associated with Weyl representation of the metric, the horizons and the defects are formally similar up to differences due to their timelike/spacelike character. We derive (previously unknown or incorrect) horizon and global Smarr formulas in presence of a Newman-Unti-Tamburino (NUT) parameter. To avoid the divergence of the Komar angular momentum of semi-infinite Dirac and Misner strings, it is necessary to use a symmetric tuning. We also note that the horizon mass Smarr formula does not include either magnetic charge, or NUT parameter, correcting some statements in the literature. The contribution of each Misner string to the total mass consists in an angular momentum term, an electric charge term, and a length term, which can also be presented as the product of the spacelike analogue of surface gravity and the area of the string.