Dynamical entropy of charged black objects

TL;DR

This paper develops a covariant phase space framework for the first law of black hole mechanics, incorporating electric and magnetic charges in various gravity theories with gauge fields, applicable to dynamical perturbations.

Contribution

It introduces a gauge-invariant method to include electric and magnetic charges in the first law for dynamical black objects in arbitrary dimensions.

Findings

Derived comparison and physical process versions of the first law.

Extended the construction to include magnetic charges with a gauge-invariant prescription.

Applied the framework to dyonic AdS black holes, black rings, and charged black branes.

Abstract

We develop a general framework for electromagnetic potential-charge contributions to the first law of black hole mechanics, applicable to dynamical first-order perturbations of stationary black objects with possibly non-compact bifurcate Killing horizons. Working in the covariant phase space formalism, we derive both comparison and physical process versions of the first law. We consider generic diffeomorphism-invariant theories of gravity in $D$ spacetime dimensions, containing non-minimally coupled abelian $p$-form gauge fields. The pullback of the gauge field to the horizon is allowed to diverge while its field strength remains smooth, yielding gauge-invariant electric potential-charge pairs in the first law. We further extend the construction to include magnetic charges by developing a bundle-covariant, gauge-invariant prescription that fixes the Jacobson-Kang-Myers ambiguity in the improved Noether charge. Electric and magnetic charges are, respectively, associated with non-trivial $(D - p - 1)$- and $(p + 1)$-cycles of the horizon cross-section, whose homology classes determine the number of independent potential-charge pairs through the Betti numbers $b_{D - p - 1}$ and $b_{p + 1}$. Further, the dynamical gravitational entropy entering the first law is identified with the gauge-invariant part of the improved Noether charge, giving a gauge-invariant extension of the recent proposal by Hollands, Wald and Zhang. We illustrate our framework with dyonic AdS black holes, dipole black rings, and charged black branes.