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We consider the gravitational and electromagnetic fields produced by a charged (or uncharged) test particle moving in a Reissner-Nordstr\"om geometry as perturbations on the background Reissner-Nordstr\"om geometry and its associated electric field, respectively. The gravitational perturbations are expanded in tensor harmonics in the manner of Regge and Wheeler, while the electromagnetic field is expanded in vector harmonics. Following a previously proposed convention, we find that in the Einstein-Maxwell system of equations, electric gravitational multipoles couple only to electric (TM) electromagnetic multipoles and similarly for magnetic multipoles. It is possible to reduce the entire Einstein-Maxwell system for each type of multipole to two second-order Schr\"odinger-type equations.
Frank J. Zerilli (Fri,) studied this question.
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