Imbedding a Reissner-Nordstr\"om charged mass into cosmology

We present the extension of the method of imbedding a mass into cosmology proposed by Gautreau, for the case of a Reissner-Nordstr\"om charged mass. We work in curvature coordinates $(R,T)$ where the coordinate time $T$ is measured by clocks at fixed points $R =$ const., and geodesic coordinates $(R,\tau)$ where $\tau$ is the time recorded by clocks moving along the radial geodesics. The Einstein equations are solved for a energy-momentum tensor which is composed of a cosmological fluid and an electrostatic field outside a radius $R_{b}$. Inside $R_{b}$ we have a part of cosmological fluid plus a Reissner-Nordstr\"om charged mass. We offer metrics for a Reissner-Nordstr\"om charged mass imbedded into different cosmological scenarios with zero curvature. An important consequence of our results, is that orbits will spiral when a charged mass is imbedded into a cosmological background. We found the generalized equation for the change of orbital radius, by using geodesic coordinates. Some criticism to the Newtonian calculation done by Gautreau for the orbital spiralling is discussed.