Gravitational and Electromagnetic Perturbations of a Charged Black Hole in a General Gauge Condition

We derive a set of coupled equations for the gravitational and electromagnetic perturbation in the Reissner-Nordstrom geometry using the Newman-Penrose formalism. We show that the information of the physical gravitational signal is contained in the Weyl scalar function Psi(4), as is well known, but for the electromagnetic signal, the information is encoded in the function chi, which relates the perturbations of the radiative Maxwell scalars phi(2) and the Weyl scalar Psi(3). In deriving the perturbation equations, we do not impose any gauge condition and as a limiting case, our analysis contains previously obtained results, for instance, those from Chandrashekhar's book. In our analysis, we also include the sources for the perturbations and focus on a dust-like charged fluid distribution falling radially into the black hole. Finally, by writing the functions on the basis of spin-weighted spherical harmonics and the Reissner-Nordstrom spacetime in Kerr-Schild type coordinates, a hyperbolic system of coupled partial differential equations is presented and numerically solved. In this way, we completely solve a system that generates a gravitational signal as well as an electromagnetic/gravitational one, which sets the basis to find correlations between them and thus facilitates gravitational wave detection via electromagnetic signals.