A Smarr formula for charged black holes in nonlinear electrodynamics

TL;DR

This paper derives a new Smarr formula for charged black holes in nonlinear electrodynamics, accounting for the non-vanishing trace of the energy-momentum tensor, and confirms its consistency with existing solutions.

Contribution

It introduces a generalized Smarr formula for nonlinear electrodynamics black holes, extending the classical relation to cases with non-zero trace of the energy-momentum tensor.

Findings

Derived a new Smarr-type formula for nonlinear electrodynamics black holes.

Confirmed the formula's consistency with known solutions.

Highlights the role of the energy-momentum tensor trace in black hole thermodynamics.

Abstract

It is well known that the Smarr formula does not hold for black holes in non-linear electrodynamics. The main reason for this is the fact that the trace of the energy momentum tensor for nonlinear electrodynamics does not vanish as it is for Maxwell's electrodynamics. Starting from the Komar integral, we derived a new Smarr-type formula for spherically symmetric static electrically charged black hole solutions in nonlinear electrodynamics. We show that this general formula is in agreement with some that are obtained for black hole solutions with nonlinear electrodynamics.