1 The First Law of Isolated Horizons via Noether Theorem

A general recipe proposed elsewhere to define, via Noether theorem, the variation of energy for a natural field theory is applied to Einstein-Maxwell theory. The electromagnetic field is analysed in the geometric framework of natural bundles. Einstein-Maxwell theory turns then out to be natural rather than gauge-natural. As a consequence of this assumption a correction termà la Regge-Teitelboim is needed to define the variation of energy, also for the pure electromagnetic part of the Einstein-Maxwell Lagrangian. Integrability conditions for the variational equation which defines the variation of energy are analysed in relation with boundary conditions on physical data. As an application the first law of thermodynamics for rigidly rotating horizons is obtained. 1 Introduction In a previous paper [28] a recipe to define, via Noether theorem, the variation of Noether conserved quantities in natural field theories was proposed. The main result of the theory there developed was to describe, for vacuum General Relativity, the quasilocal stress-energy content of a spatially bounded region of spacetime with non-orthogonal boundaries. The purpose of the present paper is to generalise the recipe of [28] to Einstein-Maxwell theory in order to calculate the Noether quasilocal conserved quantities, still remaining in the general case of spacetime regions with non-orthogonal boundaries. As a remarkable application we define a first principle of thermodynamics for spacetimes admitting an isolated horizon (in the sense of [2]). Using a geometric approach to field theories in the framework of natural theories, it is possible to associate to each infinitesimal symmetry ξ on the base manifold M (spacetime of dimension m) a conserved quantity Q(ξ). This Noether charge is obtained by integrating the Noether current on a compact (m − 1) submanifold Σ of M with boundary ∂Σ and it can be decomposed into a bulk term Q bulk (ξ) and a surface term Q surf ace (ξ). The bulk term is obtained by integrating on Σ the reduced Noether current, which is a (m − 1)-form vanishing on shell (i.e. along solutions). The surface term is the integral over ∂Σ of the superpotential related to the symmetry generated by ξ which in turn, is a (m − 2)-form algorithmically calculated starting from the Lagrangian of the theory [22]. To obtain physically expected values, in analogy with the original prescription suggested by Regge-Teitelboim in [43] (when dealing with the analysis of Hamiltonian boundary *