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 ${v(x,x}^{\prime })$ 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 ${v(x,x}^{\prime })$ for small separations of the points ${x,x}^{\prime }$ 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 ${v(x,x}^{\prime }).\mathrm{}$ The first few terms in this expansion are displayed here for the case of Schwarzschild spacetime.