Retarded potentials and radiation in odd dimensions
Abstract
Free massless fields of any spin in flat D-dimensional spacetime propagate at the speed of light. But the retarded fields produced by the corresponding point-like moving sources share this property only for even D. Since the Green's functions of the d'Alembert equation are localized on the light cone in even-dimensional spacetime, but not in odd dimensions, extraction of the emitted part of the retarded field in odd D requires some care. We consider the wave equations for spins 0, 1, and 2 in five-dimensional spacetime and analyze the fall-off conditions for the retarded fields at large distances. It is shown that the farthest part of the field contains a component propagating at the speed of light, while the non-derivative terms propagate with all velocities up to that of light. The generated radiation will contain a radiation tail corresponding to the complete prehistory of the source's motion preceding the retarded moment of time. We also demonstrate that dividing the Green's function into a part localized on the light cone and another part that is not zero inside the light cone gives separately the divergent terms in the Coulomb field of a point source. Their sum, however, is finite and corresponds to the usual power-law behaviour.
Cite
@article{arxiv.2109.10586,
title = {Retarded potentials and radiation in odd dimensions},
author = {D. V. Gal'tsov and M. Khlopunov},
journal= {arXiv preprint arXiv:2109.10586},
year = {2021}
}
Comments
14 pages, to be submitted to the Proceedings of the 16th Marcel Grossmann meeting