Scalar-Longitudinal Radiation in Extended Electrodynamics with Multipole Theory and a Compensated Source Model

Abstract

Extended electrodynamics (EED) leaves the Lorenz gauge condition unimposed and treats the scalar combination $C=\nabla\cdot\mathbf{A}+c^{-2}\partialΦ/\partial t$ as a dynamical field. For a conserved source with no scalar initial field, $C=0$ and the theory reduces to classical electrodynamics. A source with a nonzero local continuity anomaly has no Maxwell solution, but EED remains well posed and can support a scalar-longitudinal sector. For each radiating frequency of a localized source, the far field separates into the usual transverse Maxwell channel and a scalar-longitudinal channel with a longitudinal electric field, no magnetic field of its own, and a co-propagating $C$ field. Under the field-only energy balance used here, the time-averaged fluxes add without interference. The scalar channel depends only on $Λ=\partialρ/\partial t+\nabla\cdot\mathbf{J}$ and radiates when its moments at $k=ω/c$ are nonzero. An all-orders multipole formula is derived for this flux. Bound polarization and magnetization sources conserve charge identically and cannot excite the scalar channel. A compensated polarized carrier with a globally neutral anomalous surface layer isolates the channel and gives its dipole flux in closed form. The connection between the adopted flux and a physical stress-energy tensor remains unresolved. These results are conditional predictions of EED and do not imply a failure of charge conservation in classical electrodynamics.