Response to "Comment on `Method of handling the divergences in the radiation theory of sources that move faster than their own waves'" [J. Math. Phys. 40, 4331 (1999)]

TL;DR

This paper clarifies the fundamental differences between the retarded potential and field solutions for superluminal sources, showing the boundary term's significance and explaining the non-spherical decay of radiation.

Contribution

It demonstrates that for superluminal sources, the boundary term in the wave equation's solution dominates, leading to a 1/R^(1/2) decay, challenging previous assumptions of spherical decay.

Findings

Boundary term in wave equation is larger than source term for superluminal sources.

Radiation field from superluminal sources decays as 1/R^(1/2).

Neglecting boundary terms leads to incorrect decay predictions.

Abstract

There is a fundamental difference between the classical expression for the retarded electromagnetic potential and the corresponding retarded solution of the wave equation that governs the electromagnetic field. While the boundary contribution to the retarded solution for the potential can always be rendered equal to zero by means of a gauge transformation that preserves the Lorenz condition, the boundary contribution to the retarded solution of the wave equation governing the field may be neglected only if it diminishes with distance faster than the contribution of the source density in the far zone. In the case of a source whose distribution pattern rotates superluminally (i.e., faster than the speed of light in vacuo), the boundary term in the retarded solution governing the field is by a factor of the order of R^(1/2) larger than the source term of this solution in the limit where the distance R of the boundary from the source tends to infinity. This result is consistent with the prediction of the retarded potential that the radiation field generated by a rotating superluminal source decays as 1/R^(1/2), instead of 1/R. It also explains why an argument based on the solution of the wave equation governing the field in which the boundary term is neglected, such as Hannay presents in his Comment, misses the nonspherical decay of the field.