Surface Charges for Gravity and Electromagnetism in the First Order Formalism

TL;DR

This work develops a covariant, quasilocal framework for defining surface charges in 3+1 gravity coupled to electromagnetism using the tetrad-connection (first-order) formalism and covariant symplectic methods. It provides explicit charge expressions that are coordinate-, gauge-, and background-independent, and demonstrates their conservation under exact symmetries; the approach reproduces a quasilocal first law for Kerr-Newman-(A)dS black holes and extends naturally to Lovelock gravity in arbitrary dimensions. The analysis shows the two covariant-symplectic routes yield equivalent charges and that boundary or topological terms do not affect the charges, thereby offering a robust platform for horizon dynamics and asymptotic symmetry studies without reliance on asymptotic structure. The results unify gravity and electromagnetism in a quasilocal charge framework and set the stage for exploring higher-dimensional and horizon-focused thermodynamics in broad gravity theories.

Abstract

A new derivation of surface charges for 3+1 gravity coupled to Electromagnetism is obtained. Gravity theory is written in the tetrad-connection variables. The general derivation starts from the Lagrangian and uses the covariant symplectic formalism in the language of forms. For gauge theories surface charges disentangle physical from gauge symmetries through the use of Noether identities and the exactness symmetry condition. The surface charges are quasilocal, explicitly coordinate independent, gauge invariant, and background independent. For a black hole family solution the surface charge conservation implies the first law of black hole mechanics. As a check we show the first law for black hole electrically charged, rotating, and with an asymptotically constant curvature (the Kerr-Newman (anti-)de Sitter family). The charges, including the would-be mass term appearing in the first law, are quasilocal. It is not required a reference to the asymptotic structure of the spacetime nor boundary conditions, and therefore topological terms do not play a rôle. Finally, surface charges formulae for Lovelock gravity coupled to Electromagnetism are exhibited. It generalizes the one derived in a recent work by G. Barnich, P. Mao, and R. Ruzziconi. The two different symplectic methods to define surface charges are compared and shown equivalent.