Quantum Field Theory of Black Hole Perturbations with Backreaction IV. Spherically symmetric 2nd order Einstein-Maxwell sector in generalised gauges

In previous papers of this series we analysed the reduced phase space approach to perturbations of Einstein-Maxwell theory to second order around spherically symmetric backgrounds in the Gullstrand Painlevé Gauge and confirmed consistency with previous approaches. In this paper we generalize this result and show that the analysis can be performed in gauges for the background variables compatible with the Gullstrand Painlevé gauge. We obtain the same structure for the reduced Hamiltonian that contains the well known Regge-Wheeler and Zerilli potentials. Possible applications of this generalization are discussed.