Complete Lorentz transformation of a charge-current density

TL;DR

This paper clarifies that a complete Lorentz transformation of a charge-current density does not induce a charge distribution or electric dipole moment in a moving current loop, correcting a common misconception.

Contribution

It demonstrates that using the full Lorentz transformation, including the transformed time variable, eliminates the predicted charge density and electric dipole moment.

Findings

No induced charge density in a moving current loop.

No electric dipole moment arises from Lorentz transformation.

Corrects a common assumption in electromagnetic theory.

Abstract

It is generally assumed in the literature that a Lorentz transformation on a neutral current loop results in a moving current loop with a nonvanishing charge distribution and an electric dipole moment. We show in this paper that this is not, in fact, correct. The derivation that leads to the charge distribution was based on an incomplete Lorentz transformation, which transforms the charge-current four-vector $j^\mu=[\rho({\bf r},t),{\bf j(r},t)]$, but not the space-time four-vector $x^\mu=(t,{\bf r})$. We show that completing the Lorentz transformation by using the variable $t'$ in the moving frame, rather than keeping the rest frame time variable $t$, results in there being no induced charge density and no resulting electric dipole moment.