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

Abstract

The self-field of a charged particle has a singular component which diverges at the particle. We use both coordinate and covariant approaches to compute an expansion of this singular field for particles in generic geodesic orbits about a Schwarzschild black hole for scalar, electromagnetic and gravitational cases. We check that both approaches yield identical results and give, as an application, the calculation of previously unknown mode-sum regularization parameters. In the so-called mode-sum regularization approach to self-force calculations, each mode of the retarded field is finite, while their sum diverges. The sum may be rendered finite and convergent by the subtraction of appropriate regularization parameters. Higher-order parameters lead to faster convergence in the mode sum. To demonstrate the significant benefit which they yield, we use our newly derived parameters to calculate a highly accurate value of ${F}_{r}=0.000013784482575667959(3)$ for the self-force on a scalar particle in a circular orbit around a Schwarzschild black hole. Finally, as a second example application of our high-order expansions, we compute high-order expressions for use in the effective source approach to self-force calculations.