Radiation reaction in Schwarzschild spacetime: Retarded Green’s function via Hadamard-WKB expansion

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.