Self-Force and Green Function in Schwarzschild spacetime via Quasinormal Modes and Branch Cut

TL;DR

This paper presents a semi-analytic method to compute the self-force on a scalar charge in Schwarzschild spacetime by combining local Hadamard form regularization with spectral decomposition into quasinormal modes and branch cut integrals, providing insights relevant for gravitational wave modeling.

Contribution

It introduces a novel approach that combines local Green function regularization with spectral decomposition for accurate self-force calculations in Schwarzschild spacetime.

Findings

Successful computation of the self-force using matched Green function expansions.

Validation of the overlap region where both expansions agree.

Illustration of the method's potential for gravitational self-force problems.

Abstract

The motion of a small compact object in a curved background spacetime deviates from a geodesic due to the action of its own field, giving rise to a self-force. This self-force may be calculated by integrating the Green function for the wave equation over the past worldline of the small object. We compute the self-force in this way for the case of a scalar charge in Schwarzschild spacetime, making use of the semi-analytic method of matched Green function expansions. Inside a local neighbourhood of the compact object, this method uses the Hadamard form for the Green function in order to render regularization trivial. Outside this local neighbourhood, we calculate the Green function using a spectral decomposition into poles (quasinormal modes) and a branch cut integral in the complex-frequency plane. We show that both expansions overlap in a sufficiently large matching region for an accurate calculation of the self-force to be possible. The scalar case studied here is a useful and illustrative toy-model for the gravitational case, which serves to model astrophysical binary systems in the extreme mass-ratio limit.