Derivation of the Smarr formula from the Komar charge in Einstein-nonlinear electrodynamics theories and applications to regular black holes

TL;DR

This paper derives a generalized Smarr formula for black holes in Einstein-nonlinear electrodynamics theories using a novel Komar charge approach, including the role of coupling constants, and applies it to analyze regular black holes.

Contribution

It introduces a method to incorporate coupling constants into the Smarr formula via a dynamical field, extending previous homogeneity-based approaches.

Findings

Derived a generalized Smarr formula including coupling constants.

Validated the approach on various Einstein--NLED theories.

Analyzed the thermodynamics and regularity of the Bardeen black hole.

Abstract

We construct the generalized Komar charge of generic, non-linear theories of electrodynamics (NLED) in 4 dimensions coupled to Einstein gravity. The contribution of the dimensionful coupling constant present in all these theories is obtained by promoting it to a dynamical field which is forced to be constant on-shell by a Lagrange multiplier. We use this charge to derive a Smarr formula for asymptotically-flat black-hole and soliton solutions of these theories that includes the contribution of the coupling constant. Previously, this contribution had been found using homogeneity arguments. We test our results on a broad class of Einstein--NLED theories and analyze in detail the thermodynamics of the regular Bardeen black hole using the conservation of the generalized Komar charge to understand the regularity of regular black holes inside the event horizon.