Regularization parameters for the self force in Schwarzschild spacetime: I. scalar case

TL;DR

This paper derives explicit regularization parameters for calculating the self-force on a scalar particle in a Schwarzschild spacetime, providing detailed formulas essential for mode-sum regularization in self-force computations.

Contribution

It provides the complete derivation of regularization parameters for the scalar case, advancing the mode-sum method for self-force calculations in curved spacetime.

Findings

Explicit regularization parameters derived for scalar self-force.

Methodology applicable to arbitrary geodesic orbits.

Foundation for future electromagnetic and gravitational self-force studies.

Abstract

We derive the explicit values of all regularization parameters (RP) for a scalar particle in an arbitrary geodesic orbit around a Schwarzschild black hole. These RP are required within the previously introduced mode-sum method, for calculating the local self-force acting on the particle. In this method one first calculates the (finite) contribution to the self-force due to each individual multipole mode of the particle's field, and then applies a certain regularization procedure to the mode sum, involving the RP. The explicit values of the RP were presented in a recent Letter [Phys. Rev. Lett. {\bf 88}, 091101 (2002)]. Here we give full details of the RP derivation in the scalar case. The calculation of the RP in the electromagnetic and gravitational cases will be discussed in an accompanying paper.