First law of black hole mechanics in Einstein-Maxwell and Einstein-Yang-Mills theories

TL;DR

The paper generalizes the first law of black hole mechanics to stationary black holes in Einstein–Maxwell and Einstein–Yang–Mills theories using the Iyer–Wald covariant phase space formalism, avoiding reliance on a bifurcation surface. It shows how horizon data and boundary/flux terms reproduce the canonical energy and angular momentum variations, yielding a horizon-based expression that includes a charge-related term for EM and a horizon surface term for YM. The EM result recovers the familiar relation between area, mass, angular momentum, and charge, while the YM case highlights non-Abelian horizon contributions and the role of gauge choices in simplifying the law. The approach applies to extremal black holes and extends Wald’s framework to non-Abelian gauge fields, clarifying the thermodynamic content of horizon charges.

Abstract

The first law of black hole mechanics is derived from the Einstein-Maxwell (EM) Lagrangian by comparing two infinitesimally nearby stationary black holes. With similar arguments, the first law of black hole mechanics in Einstein-Yang-Mills (EYM) theory is also derived.