Covariant self-force regularization of a particle orbiting a Schwarzschild black hole - Mode decomposition regularization

The covariant structure of the self-force of a particle in a general curved background has been elucidated in the cases of scalar, electromagnetic, and gravitational charges. Specifically, what we need is the part of the self-field that is non-vanishing off and within the past light cone of the particle's location, the so-called tail. The radiation reaction force in the absence of external fields is entirely contained in the tail. In this paper, we develop the mathematical tools necessary for the regularization and propose a practical method to calculate the self-force of a particle orbiting a Schwarzschild black hole.