Surface Charges for Gravity and Electromagnetism in the First Order Formalism

A new derivation of surface charges for 3+1 gravity coupled to Electromagnetism is obtained. The derivation uses the symplectic formalism and the language of forms. Gravity theory is written in the tetrad-connection variables. As a gauge theory, physical symmetry transformations are disentangled from gauge at the level of the presymplectic structure density, through the use of N\"other identities and the exactness symmetry condition. The resulting surface charges are explicitly coordinate independent, gauge invariant, and background independent. For a black hole family solution the surface charges conservation implies the first law of black hole mechanics. As a preliminary check we show it for the family of black hole solutions electrically charged, rotating, and with an asymptotically constant curvature (the Kerr-Newman (anti-)de Sitter family). The computations, including the would-be mass term appearing in the first law, can be performed in a quasilocal way. It is not required a reference to the asymptotic structure of the spacetime nor boundary conditions. Finally, surface charges formulae for a gravity theory coupled to Electromagnetism in an arbitrary dimension are exhibited. It generalizes the one derived in a recent work by G. Barnich, P. Mao, and R. Ruzziconi. A comparison of the two different symplectic prescriptions are discussed and shown to be equivalent.