Smarr's formula for black holes with non-linear electrodynamics

TL;DR

This paper explores how Smarr's formula for black hole mass is modified in the context of nonlinear electrodynamics, specifically for Bardeen and Ayon-Beato-Garcia black holes, highlighting the conditions under which it remains valid.

Contribution

It derives the modified Smarr's formula for specific nonlinear electrodynamics black hole solutions, clarifying its applicability and limitations.

Findings

Smarr's formula holds for Bardeen black holes with magnetic monopoles.

The formula does not hold when electric charge variations are involved.

The work clarifies the conditions for the validity of Smarr's relation in nonlinear electrodynamics.

Abstract

It is known that for nonlinear electrodynamics the First Law of Black Hole Mechanics holds, however the Smarr's formula for the total mass does not. In this contribution we discuss the point and determine the corresponding expressions for the Bardeen black hole solution that represents a nonlinear magnetic monopole. The same is done for the regular black hole solution derived by Ayon-Beato and Garcia, showing that in the case that variations of the electric charge are involved, the Smarr's formula does not longer is valid.