Energy in Einstein–Maxwell theory and the first law of isolated horizons via the Noether theorem

A general recipe proposed elsewhere to define, via the 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. The Einstein–Maxwell theory then turns out to be natural rather than gauge-natural. As a consequence of this assumption, a correction term like that used by Regge and Teitelboim is needed to define the variation of energy, as well as 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 to boundary conditions on physical data. As an application the first law of thermodynamics for rigidly rotating horizons is obtained.