Self-force on a scalar charge in radial infall from rest using the Hadamard-WKB expansion

TL;DR

This paper introduces an analytic Hadamard-WKB expansion method to compute the self-force on a scalar charge in radial infall within Schwarzschild spacetime, providing a useful check for numerical and analytical approaches.

Contribution

It develops an analytic approach based on Hadamard-WKB expansion for short-duration radial infall self-force calculations in Schwarzschild spacetime.

Findings

Analytic expression for self-force during short infall periods.

Method expresses Green's function integral in terms of two manageable integrals.

Useful for validating numerical and approximate self-force calculations.

Abstract

We present an analytic method based on the Hadamard-WKB expansion to calculate the self-force for a particle with scalar charge that undergoes radial infall in a Schwarzschild spacetime after being held at rest until a time t = 0. Our result is valid in the case of short duration from the start. It is possible to use the Hadamard-WKB expansion in this case because the value of the integral of the retarded Green's function over the particle's entire past trajectory can be expressed in terms of two integrals over the time period that the particle has been falling. This analytic result is expected to be useful as a check for numerical prescriptions including those involving mode sum regularization and for any other analytical approximations to self-force calculations.