# Inadequacies in the conventional treatment of the radiation field of moving sources

## TL;DR

This paper reveals fundamental differences between the classical retarded potential and the wave equation solution for electromagnetic fields of moving sources, especially for superluminal patterns, explaining why conventional methods overlook certain radiation decay behaviors.

## Contribution

It demonstrates that neglecting boundary terms in the wave equation solution leads to incorrect predictions for superluminal source radiation fields.

## Key findings

- Boundary term in wave equation is larger than source term for superluminal sources.
- Radiation from rotating superluminal sources decays as R^{-1/2}, not R^{-1}.
- Experimental confirmation of the R^{-1/2} decay prediction.

## 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 {\em 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 {\em 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 both rotates and travels faster than light {\em in vacuo}, as realized in recent experiments, the boundary term in the retarded solution governing the field is by a factor of the order of $R^{1/2}$ {\em larger} than the source term of this solution in the limit that the distance $R$ of the boundary from the source tends to infinity. This result is consistent with the prediction of the retarded potential that part of the radiation field generated by a rotating superluminal source decays as $R^{-1/2}$, instead of $R^{-1}$, a prediction that is confirmed experimentally. More importantly, it pinpoints the reason why an argument based on a solution of the wave equation governing the field in which the boundary term is neglected (such as appears in the published literature) misses the nonspherical decay of the field.