Complete algorithms for calculating the higher-order fundamental solutions of wave equations

Abstract

Complete recurrent algorithms for calculating the higher-order fundamental solutions of covariant linear wave equations for scalar and tensor wave fields on an arbitrary curved spacetime are derived. The higher-order fundamental solutions are the distributions that satisfy the wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. Like the classical Green's function for a scalar wave equation, the higher-order fundamental solutions contain the terms which have support on, and only on, the lightcone as well as tail terms which have support inside the lightcone. With the help of the higher-order fundamental solutions found it is possible to compute the exact multipole solutions of wave equations in a form convenient for practical computations. As applications we consider the exact field of a dipole source of variable strength travelling in an arbitrary curved spacetime and the tail term of scalar multipole waves in the Friedman dust-dominated universe for the case of minimal coupling.