A low order extension the Lienard-Wiechert retardation equations to include the Thomas precession

TL;DR

This paper extends the Lienard-Wiechert equations to include Thomas precession effects, deriving solutions that relate to Maxwell and Proca equations, and discusses the mathematical structure of these solutions in terms of tensor irreducibility.

Contribution

It introduces a low order extension of the Lienard-Wiechert equations to incorporate Thomas precession and analyzes their relation to Maxwell and Proca equations.

Findings

Thomas precession vanishes when acceleration and velocity are parallel

Solutions for dipole antennas match Lienard-Wiechert solutions and satisfy Maxwell equations

Current loop antenna solutions do not satisfy Maxwell equations

Abstract

In a calculation that directly parallels the derivation of the Thomas precession, the first time derivative of the retarded potentials is derived. The solutions have to be integrated in time to obtain the potential solution. The Thomas precession vanishes when the acceleration and velocity vectors are parallel, causing the solution for the dipole antenna to be the same as for the Lienard-Wiechert solution, and those solutions are in turn always solutions to the Maxwell equations. The solution for the current loop antenna is not a solution to the Maxwell equations. Field equations are obtained by restructuring the Proca equations that are commensurate with the low order retardation solutions. The solutions are not in the Lorentz gauge and they are not solutions to the unmodified Proca equations. The high order terms are not solutions to the equations. In representing angular relationships, an argument is developed that derivatives beyond the first will be required for more complete solutions. The calculations are not in tensor form, but the tensors represent angular relationships, and the inference is based on the tensor irreducibility theorem. In being linear equations expressing angular relationships, the theorem implies that exact retardation equations do not exist unless the contravariant tensor of rank n+1 is reducible.