High-order expansions of the Detweiler-Whiting singular field in Schwarzschild spacetime

TL;DR

This paper develops high-order expansions of the singular field in Schwarzschild spacetime for scalar, electromagnetic, and gravitational cases, improving the accuracy of self-force calculations in general relativity.

Contribution

It provides new high-order regularization parameters and expressions for the effective source approach, enhancing precision in self-force computations for generic geodesic orbits.

Findings

Agreement between coordinate and covariant approaches confirmed

New regularization parameters computed for mode-sum regularization

High-order expressions for effective source approach derived

Abstract

The self field of a charged particle has a component that diverges at the particle. We use both coordinate and covariant approaches to compute an expansion of this singular field for generic geodesic orbits in Schwarzschild spacetime for scalar, electromagnetic and graviational cases. We check agreement of both approaches and give, as an application, the calculation of previously unknown regularisation parameters. In this so-called "mode-sum regularization" approach, each mode of the field is finite, while their sum diverges. The sum may be rendered finite and convergent by the subtraction of "regularization parameters". Higher order parameters lead to faster convergence in the mode-sum. As a second example application, we compute high order expressions for the effective source approach to self-force calculations.