Higher-order fundamental solutions of second-order wave equations on a curved spacetime

Abstract

Higher-order fundamental solutions are defined as the distributions that satisfy the wave equations with inhomogeneous terms which are point distributions of the corresponding order. Starting from the Hadamard fundamental solution, the construction of the local higherorder fundamental solutions of the covariant scalar wave equation on a causal domain is considered. A simple recurrent algorithm for calculating such solutions is found.

On the asymptotic character of electromagnetic waves in a Friedmann?Robertson?Walker universe

“…Expansions to generalized spherical harmonics have also been presented by Mankin et al [13][14][15]26], and by Laas et al [27] to construct exact and approximate solutions to electromagnetic wave equations in a curved spacetime using higher-order Green's function method.…”

New approach to electromagnetic wave tails on a curved spacetime

“…, µ + 1. This statement was first proposed by us without proof in [27], its proof can be found in [24]. Eqs.…”

“…Finally, after quite simple but voluminous calculations we obtain from Eqs. (27)(28)(29) the seeked expression for the first-order tail term…”

“…V. SUMMARY AND DISCUSSION (i) On the basis of our earlier works [27,24,25], we developed simple recurrent formulas, i.e. Eq.…”

“…We have recently developed a new method [24][25][26] for calculating the exact solutions of scalar and tensor wave equations whose source terms are arbitrary-order multipoles on a curved spacetime. The developed method is based on the higher-order fundamental solutions (Green's functions) for wave equations which are defined in our paper [27] as the distributions that satisfy wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. Provided that the classical Green's function and the multipole expansion of the source term are given, there is no need for a small expansion parameter within the framework of our formalism and in certain cases we can even find the exact multipole solutions for a strong field.…”