Spurious causality hiding an action at a distance

TL;DR

The paper reveals that the Coulomb law's instantaneous action at a distance is hidden within the causal formulation of the electric field, with only the transverse component being truly causal.

Contribution

It demonstrates that the longitudinal electric field component remains instantaneous despite being expressed through causal wave equations.

Findings

Longitudinal electric field $E_ ext{parallel}$ is instantaneous.

Causal wave equations only describe the transverse electric field.

Spurious causality hides the action at a distance.

Abstract

The Coulomb/Gauss law given in the Maxwell equations describes a spatial relation between the electric field component $E_\parallel({\bf r}, t)$ and its source $\rho({\bf r}, t)$ that is instantaneous, occurring at the same time $t$. This instantaneous action at a distance can be hidden by writing it formally as the sum of two causal terms. I show here that the causal expression of the total electric field ${\bf E}$ suffers from such a spurious causality hiding the action at a distance that is $E_\parallel$. In fact, $E_\parallel$ remains instantaneous in time even when it is part of a wave equation. Only the ${\bf E}_\perp$ part is really causal.