Coupled gravitational and electromagnetic perturbations of Reissner-Nordstr\"om spacetime in a polarized setting

We derive a system of equations governing the coupled gravitational and electromagnetic perturbations of Reissner-Nordstr"om spacetime. The equations are derived in the context of global non-linear stability of Reissner-Nordstr"om under axially symmetric polarized perturbations, as a generalization of the recent work on non-linear stability of Schwarzschild spacetime of Klainerman-Szeftel. The main result consists in deriving, through a Chandrasekhar-type transformation, a gauge invariant quantity associated to the electromagnetic tensor that verifies a Regge-Wheeler equation. In this paper, we present the derivation of the main equations, postponing the estimates of the non-linear system and the construction of the dynamical spacetime to a later time.