Non-Linear Electrodynamics: Zeroth and First Laws of Black Hole Mechanics

TL;DR

This paper derives the Zeroth and First Laws of Black Hole Mechanics within non-linear electrodynamics coupled to gravity, extending classical results to more general matter fields and specific theories like Born-Infeld.

Contribution

It provides a general derivation of black hole laws in non-linear electrodynamics, including a corrected mass variation formula and extensions to scalar fields.

Findings

Zeroth Law holds even if the Dominant Energy Condition is violated.

The First Law is validated for non-linear electrodynamics, including Born-Infeld theory.

A comprehensive mass variation formula is derived, accounting for previously omitted terms.

Abstract

The Zeroth and First Laws of Black Hole Mechanics are derived in the context of non-linear electrodynamics coupled to gravity. The Zeroth Law is shown to hold quite generally even if the Dominant Energy Condition is violated. The derivation of the First Law is discussed in detail for general matter fields coupled to gravity. The general mass variation formula obtained includes a term previously omitted in some of the literature. This is then applied to the case of non-linear electrodynamics and the usual First Law is found to hold true. As an example, Born-Infeld theory is discussed. The results are extended to include scalar fields in a very general way, including additional terms arising from the variation of the asymptotic values of the scalars.