Radiation Reaction in Schwarzschild Spacetime: Retarded Green's Function via Hadamard-WKB Expansion

TL;DR

This paper presents an analytic method using Hadamard-WKB expansion to derive the tail term of the retarded Green's function in Schwarzschild spacetime, aiding understanding of radiation reaction forces in curved spacetime.

Contribution

It introduces a WKB-based expansion method for the retarded Green's function's tail term applicable to static spherically symmetric spacetimes, including Schwarzschild.

Findings

Derived explicit expansion terms for Schwarzschild spacetime

Established equivalence with DeWitt-Schwinger expansion

Provided a practical approach for tail term calculations

Abstract

An analytic method is given for deriving the part of the retarded Green's function v(x,x') that contributes to the tail term in the radiation reaction force felt by a particle coupled to a massless minimally coupled scalar field. The method gives an expansion of v(x,x') for small separations of the points x, x' valid for an arbitrary static spherically symmetric spacetime. It is obtained by using a WKB approximation for the Euclidean Green's function for the massless minimally coupled scalar field and is equivalent to the DeWitt-Schwinger expansion for v(x,x'). The first few terms in this expansion are displayed here for the case of Schwarzschild spacetime.