The first law of black hole mechanics in the Einstein-Maxwell theory revisited

TL;DR

This paper re-derives the first law of black hole mechanics within Einstein-Maxwell theory using gauge-invariant methods, introducing momentum maps that generalize thermodynamical potentials and applying these ideas to higher-dimensional charged black holes.

Contribution

It introduces a gauge-invariant derivation of the first law using momentum maps, extending the framework to more complex scenarios beyond previous approaches.

Findings

Momentum maps serve as generalized thermodynamical potentials.

The approach satisfies generalized zeroth laws.

Application to higher-dimensional Reissner-Nordström-Tangherlini black holes confirms the validity.

Abstract

We re-derive the first law of black hole mechanics in the context of the Einstein-Maxwell theory in a gauge-invariant way introducing "momentum maps" associated to field strengths and the vectors that generate their symmetries. These objects play the role of generalized thermodynamical potentials in the first law and satisfy generalized zeroth laws, as first observed in the context of principal gauge bundles by Prabhu, but they can be generalized to more complex situations. We test our ideas on the $d$-dimensional Reissner-Nordstr\"om-Tangherlini black hole.