Gustav Mie's longitudinal electrical field lines and superluminal signals and longitudinal waves

Abstract

In his textbook, Gustav Mie states, that the radiation field consists solely of transverse electrical field lines, which are closed. The longitudinal field lines goes without interruption from the positive to the negative charges; they cannot be expelled. This raises the question about the speed of propagation along them. For this, a mathematical description of Mie’s field lines is developed (the Helmholtz-transverse and longitudinal components of $\overrightarrow{\mathit{E}}$ are non-local auxiliary quantities and thus not suitable for that). On the other hand, Mie also states, that the propagation of electromagnetic fields is governed by Maxwell’s equations and Poynting’s theorem. Here, however, there is a single wave equation for all vector components of the electrical field. In this contribution, a minimum modification of Maxwell’s equations is proposed, which yields both, an instantaneous Coulomb field and longitudinal electrical radiation.