Smarr’s formula for black holes with non-linear 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 Ayón–Beato and García [1], showing that in the case that variations of the electric charge are involved, the Smarr’s formula is no longer valid.