Self-interaction and massive vector field in the Schwarzschild space-time

Using a multipole expansion, we determine formally the massive vector field generated by a point source held fixed in the Schwarzschild space-time. We prove that its limit, when the mass of the Proca field goes to zero, does not tend to the corresponding massless vector field. In this limit we evaluate the expression of the self-force acting on the particle. The result is in accordance with that of Vilenkin, without the assumption that the point source is at a large distance from the horizon, that we extend to the case of a Reissner-Nordström space-time. We also investigate a further case: a point source within a spherical shell of matter for which the Proca field tends to the corresponding massless vector field.