Electromagnetic Charge Density with a Causal Current Density Vector and Infinite Speed of Light

In past work we have obtained necessary and sufficient conditions for the existence of an infinite speed of light c for reference frames in relative motion. These frames could house general linear or specific nonlinear systems and the characteristic for this result was a temporal discontinuity in the relative velocity of the motion of the frames. The search for the same infinite c effect in an electromagnetic system was affirmative provided that the current density vector in motion possessed a temporal discontinuity. The failure of the Lorenz condition was utilized to devise an approach that yielded this result. This property meant the failure of Maxwell’s equations. For the Lorenz condition and the wave equations for the potentials are equivalent to Maxwell’s equations. Also obtained was the side result of emergence of magnetic sources. The purpose of the present work is to show that in an electromagnetic system an infinite c is possible along with the advent of magnetic sources when the current density vector is simply causal, be it with or without a temporal discontinuity provided that the acceleration of the charge density is nonzero at the inception instant of the motion. We thus extend the previously determined infinite c result for the motion of a charge density with infinite initial acceleration to also include now the case of a motion with a nonzero but finite initial acceleration. With finite and infinite accelerations, we respectively have continuous and discontinuous current density vectors assuming the charge density function is not zero at t = 0. For the continuous case magnetic sources emerge unless the charge density function and the Green’s functions for the scalar and vector potentials satisfy a specific condition. For the abruptly starting motion case there exists no restriction. However, for a stationary charge density as in a Hertz dipole (and hence for an arbitrary antenna) where a velocity function is not defined, the causal current density vector continuous at t = 0 will not support an infinite c. An exemplary computation is provided for when the current density vector is continuous using the Green’s function for an infinite half space.