Note on the First Law with p-form potentials

TL;DR

This paper defines conserved charges for p-form gauge fields in gravity, compares different methods, and applies the formalism to black rings, plane waves, and horizon laws, advancing understanding of gauge charge properties.

Contribution

It introduces a Lagrangian-based formalism for conserved charges in p-form gauge fields and demonstrates its consistency with covariant phase space methods, including applications to black rings and plane waves.

Findings

Surface charges are explicitly defined and compared with previous methods.

The first law of black hole mechanics is validated for regular gauge fields on horizons.

The formalism provides a new way to define energy in plane wave backgrounds.

Abstract

The conserved charges for $p$-form gauge fields coupled to gravity are defined using Lagrangian methods. Our expression for the surface charges is compared with an earlier expression derived using covariant phase space methods. Additional properties of the surfaces charges are discussed. The proof of the first law for gauge fields that are regular when pulled-back on the future horizon is detailed and is shown to be valid on the bifurcation surface as well. The formalism is applied to black rings with dipole charges and is also used to provide a definition of energy in plane wave backgrounds.