Smarr formula for black holes endowed with both electric and magnetic charges

TL;DR

This paper refines the Smarr formula for black holes with both electric and magnetic charges by incorporating magnetic contributions using Tomimatsu's Komar integrals, demonstrating a unified complex charge approach in dyonic solutions.

Contribution

It introduces a consistent method to include magnetic charges in the Smarr formula using complex charges and potentials, clarifying previous ambiguities.

Findings

The electromagnetic terms in Smarr's formula can be expressed as a complex charge times an extended potential.

The approach is validated through three dyonic black hole solutions.

The method unifies electric and magnetic contributions in the mass formula.

Abstract

The present paper clarifies how to consistently take into account the contribution of magnetic charges in the generalized Smarr mass formula by using properly the Tomimatsu's representation of Komar integrals. It is shown in particular that in all three examples of the dyonic solutions considered by us, the sum of the two electromagnetic terms in Smarr's formula can be cast into the form $\bar{\cal Q}\Phi^H_{ext}$, where ${\cal Q}$ is the complex charge and $\Phi^H_{ext}$ the complex extension of the electric potential.