A new method for solving covariant wave equations by means of higher-order fundamental solutions: proof of an algorithm

Abstract

Proceeding from the classical fundamental solution (Green's function), a new method for solving covariant inhomogeneous wave equations on a causal domain of curved spacetimes is considered. A simple recurrent algorithm for calculating the solutions of the wave equation by means of higher-order fundamental solutions is presented and the corresponding theorems are proved. The method can be applied both for obtaining exact multipole solutions of the wave equation on curved spacetimes, if the exact form of the classical fundamental solution and multipole expansion of the source term are known, as well as for obtaining approximate ones. The efficiency of the proposed method is demonstrated by way of giving a short and elegant proof of a result known earlier, namely, the universality of the form of the tail correction term in the Schwarzschild spacetime.