Quasi-Lacunae of Maxwell's Equations

Abstract

Classical lacunae in the solutions of hyperbolic differential equations and systems (in the spaces of odd dimension) are a manifestation of the Huygens' principle. If the source terms are compactly supported in space and time, then, at any finite location in space, the solution becomes identically zero after a finite interval of time. In other words, the propagating waves have sharp aft fronts. For Maxwell's equations though, even if the currents that drive the field are compactly supported in time, they may still lead to the accumulation of charges. In that case, the solution won't have the lacunae per se. We show, however, that the notion of classical lacunae can be generalized, and that even when the steady-state charges are present, the waves still have sharp aft fronts. Yet behind those aft fronts, there is a nonzero electrostatic solution rather than one identically zero.