Self-force on a scalar particle in spherically-symmetric spacetime via mode-sum regularization: radial trajectories

Recently, we proposed a method for calculating the ``radiation reaction'' self-force exerted on a charged particle moving in a strong field orbit in a black hole spacetime. In this approach, one first calculates the contribution to the ``tail'' part of the self force due to each multipole mode of the particle's self field. A certain analytic procedure is then applied to regularize the (otherwise divergent) sum over modes. This involves the derivation of certain regularization parameters using local analysis of the (retarded) Green's function. In the present paper we present a detailed formulation of this mode-sum regularization scheme for a scalar charge on a class of static spherically-symmetric backgrounds (including, e.g., the Schwarzschild, Reissner-Nordstr\"{o}m, and Schwarzschild-de Sitter spacetimes). We fully implement the regularization scheme for an arbitrary radial trajectory (not necessarily geodesic) by explicitly calculating all necessary regularization parameters in this case.