The first law for stationary axisymmetric multi-black hole systems

TL;DR

This paper derives a first law of mechanics for stationary axisymmetric multi-black hole systems in Einstein-Maxwell theory, accounting for black holes and Misner strings with electric, magnetic, and gravimagnetic fluxes.

Contribution

It introduces a new first law that includes contributions from Misner-Dirac strings and their fluxes, extending previous black hole mechanics to complex multi-black hole configurations.

Findings

First law incorporates areas, angular momenta, and electric charges of horizons.

When gravimagnetic fluxes vanish, the law simplifies to include string tensions and magnetic fluxes.

The form of the law is not invariant under electric-magnetic duality in general.

Abstract

In the framework of Einstein-Maxwell theory, we consider collinear arrays of rotating electrically charged black holes connected by Misner-Dirac strings carrying gravimagnetic and magnetic fluxes. The first law of mechanics for these systems is derived. It involves as dynamical variables the areas, angular momenta and electric charges of the various Killing horizons -- black holes and Misner strings. When the gravimagnetic fluxes all vanish, the first law reduces to a form where the dynamical variables associated with the strings are the string tensions and magnetic fluxes. This form is not generically invariant under electric-magnetic duality.