On the Smarr formula for rotating dyonic black holes

TL;DR

This paper confirms that the classic Smarr formula remains valid for rotating dyonic black holes, correcting previous claims and clarifying the relation between horizon and asymptotic quantities.

Contribution

It demonstrates that the standard Smarr relation holds for dyonic black holes, correcting earlier derivations and extending understanding of mass and angular momentum in these solutions.

Findings

The usual Smarr formula applies to dyonic Kerr-Newman black holes.

The difference between asymptotic and horizon masses equals the sum of Dirac string masses.

Results are relevant for dyonic dihole solutions.

Abstract

We revisit the derivation by Tomimatsu of the generalized Komar integrals giving the mass and angular momentum of rotating Einstein-Maxwell black holes. We show that, contrary to Tomimatsu's claim, the usual Smarr formula relating the horizon mass and angular momentum still holds in the presence of both electric and magnetic charges. The simplest case is that of dyonic Kerr-Newman black holes, for which we recover the modified Smarr formula relating the asymptotic mass and angular momentum, the difference between asymptotic and horizon masses being equal to the sum of the two Dirac string masses. Our results apply in particular to the case of dyonic dihole solutions which have been investigated recently.