In this paper we extend reduced phase space approach to black hole perturbation theory to Maxwell matter. We expand the resulting reduced Hamiltonian to second order in the graviton and photon perturbations and find that the corresponding equations of motion match the ones derived in the literature. Accordingly our approach reproduces previous results at second order. Its real virtue lies in the fact that it extends to any order in perturbation theory in a manifestly gauge invariant fashion.
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Neuser et al. (2024) studied this question.