Generalizations of the Smarr formula for black holes with nonlinear electromagnetic fields

TL;DR

This paper derives a generalized Smarr formula for stationary axially symmetric black holes with nonlinear electromagnetic fields, unifying previous forms and including quantum corrections from the Euler-Heisenberg Lagrangian.

Contribution

It provides a geometric derivation of the generalized Smarr formula for black holes with nonlinear electromagnetic fields, extending previous spherically symmetric results and incorporating quantum effects.

Findings

Derived a geometric form of the generalized Smarr formula.

Unified previous spherically symmetric results within a broader framework.

Included quantum corrections from the Euler-Heisenberg Lagrangian.

Abstract

We present a direct, geometric derivation of the generalized Smarr formula for the stationary axially symmetric black holes with nonlinear electromagnetic fields. The additional term is proven to be proportional to the integral of the trace of the electromagnetic energy-momentum tensor and can be written as a product of two conjugate variables. From the novel relation we can deduce all previously proposed forms of the generalized Smarr formula, which were derived only for the spherically symmetric black holes, and provide the lowest order quantum correction to the classical relation from the Euler-Heisenberg Lagrangian.