Dyonic black holes: The theory of two electromagnetic potentials. II

TL;DR

This paper extends the theory of dyonic black holes with two electromagnetic potentials to stationary axially symmetric cases, refining the Ernst formalism, analyzing the Kerr-Newman solution, and clarifying the evaluation of the black hole mass.

Contribution

It introduces a symmetrical formalism for electric and magnetic charges in stationary black holes, confirming the validity of the Tomimatsu mass formula and clarifying misconceptions about magnetic charge potentials.

Findings

The extended Ernst formalism includes a magnetic potential similar to the electric one.

The Komar mass must be correctly evaluated in dyonic black hole models.

The standard Tomimatsu mass formula remains valid for dyonic black holes.

Abstract

The results obtained in our previous paper are now extended to the case of stationary axially symmetric dyonic black boles within the theory of two electromagnetic potentials. We slightly enlarge the classical Ernst formalism by introducing, with the aid of the $t$- and $\varphi$-components of the dual potential $B_\mu$, the magnetic potential $\Phi_m$ which, similar to the known electric potential $\Phi_e$, also takes constant value on the black hole horizon. We analyze in detail the case of the dyonic Kerr-Newman black hole and show how the Komar mass must be evaluated correctly in this stationary dyonic model. In particular, we rigorously prove the validity of the standard Tomimatsu mass formula and point out that attempts to "improve" it made in recent years are explained by misunderstanding of the auxiliary role that singular potentials play in the description of magnetic charges. Our approach is symmetrical with respect to electric and magnetic charges and, like in the static case considered earlier, Dirac strings of all kind are excluded from the physical picture of the stationary black hole dyonic spacetimes.