  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.
