On the use of projection operators in electrodynamics

TL;DR

This paper examines the differences between proper and simplified transverse projection operators in electrodynamics, demonstrating that using the simplified operator can lead to significant errors in certain scenarios involving moving charges.

Contribution

It provides a detailed comparison of proper and simplified projection operators, showing that the common approximation can cause substantial inaccuracies in the vector potential calculations.

Findings

Simplified projection can produce errors comparable to the vector potential itself.

Errors are significant when the source has unbounded spatial motion.

Proper projection is necessary for accurate results in such cases.

Abstract

In classical electrodynamics all the measurable quantities can be derived from the gauge invariant Faraday tensor $F_{\alpha\beta}$. Nevertheless, it is often advantageous to work with gauge dependent variables. In [4],[2] and [8], and in the present note too, the transformation of the vector potential in Lorenz gauge to that in Coulomb gauge is considered. This transformation can be done by applying a projection operator that extracts the transverse part of spatial vectors. In many circumstances the proper projection operator is replaced by a simplified transverse one. It is widely held that such a replacement does not affect the result in the radiation zone. In this paper the action of the proper and simplified transverse projections will be compared by making use of specific examples of a moving point charge. It will be demonstrated that whenever the interminable spatial motion of the source is unbounded with respect to the reference frame of the observer the replacement of the proper projection operator by the simplified transverse one yields, even in the radiation zone, an erroneous result with error which is of the same order as the proper Coulomb gauge vector potential itself.