# Surface Charges for Gravity and Electromagnetism in the First Order Formalism

## TL;DR

This paper derives coordinate-independent, gauge-invariant surface charges in 3+1 gravity coupled with electromagnetism using covariant symplectic formalism, applicable to black holes and Lovelock gravity, without relying on asymptotic conditions.

## Contribution

It introduces a new covariant derivation of quasilocal surface charges in gravity-electromagnetism, extending to Lovelock gravity, and compares different symplectic methods for their calculation.

## Key findings

- Surface charges are coordinate independent and gauge invariant.
- First law of black hole mechanics is recovered for various black hole solutions.
- The formalism applies to Lovelock gravity coupled with electromagnetism.

## 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 r\^ole. 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.