What gauges can be used in applied electromagnetic calculations?

TL;DR

This work investigates gauge choices in classical electrodynamics and argues that the Coulomb gauge ($v\to\infty$) and the velocity gauge ($v=c$, with the Lorenz gauge as the limiting case) can produce unphysical, superluminal predictions for the electric field. By deriving an explicit expression for $E_{C,x}$ in the Coulomb gauge, $E_{C,x} = \frac{q}{(D-ut)^2}\left(1 - \frac{u^2}{c^2}\ln\left[1 - \frac{c^2 t^2}{D^2}\right]\right)$, the paper shows that a signal may appear for $D>ct$, indicating propagation faster than $c$ due to nonlocal source terms and incomplete cancellation with the vector-potential contribution. It further critiques proposed alternative representations of ${\bf A}^{(v)}$ (e.g., Yang 1976 and BC approaches), arguing they rely on unjustified wave equations or improper Green-function constructions that fail to resolve the issue. The central conclusion is that only the Lorenz gauge provides a consistent framework for general charged systems, avoiding the pathological superluminal propagation, and thus should be used in applied electromagnetic calculations.

Abstract

In the classical electrodynamics, different gauges, i.e. connections between the electromagnetic potentials, are used. Some of these are quite specific and intended for calculations in special systems (absence of free charges, etc.). All of these specific gauges are reductions of the Lorenz gauge. However, in addition to this gauge, two more, i.e., the Coulomb and velocity gauges, can be used to describe systems of charges and currents without any restrictions. It is commonly accepted opinion that these three gauges are equivalent, meaning that the expressions for electromagnetic fields obtained from the potentials defined in these gauges are identical. However, it can be shown that the Coulomb and velocity gauges yield solutions corresponding to `superluminal propagation' of the electric field. Since such a propagation of the electric field has not been observed experimentally and, moreover, is forbidden by special relativity, it can be concluded that calculations in these gauges may yield incorrect results. Therefore, these gauges cannot be used in applied electromagnetic calculations.

Figures (2)

• Figure 1: A system where calculations of the electric field in the Coulomb gauge predict superluminal propagation.
• Figure 2: A system where calculations of the electric field in the velocity gauge predict superluminal propagation.