A Stationary Charge Density and Radiation at Infinite Speed of Light

In previous work by the author feasibility of an infinite speed of light $c$ was predicted for reference frames which undergo a motion with a time discontinuity in the relative velocity. Among such systems for which an infinite light speed can be attained is an electric charge density function that in the time coordinate abruptly starts an arbitrary motion. In the present paper we extend the environment where an infinite light speed can be found to the case of an electric charge density function whose space coordinate variables are independent of time, which is a time-varying function for $t > 0$, is time invariant for $t < 0$ and is continuous at $t=0$. Moreover, the corresponding current density vector must be non-zero at $=0$. In this case, just as in the case of a charge density with an abruptly starting motion, the Lorenz condition for the potentials of the problem fails even with an infinite $c$ stipulation which is necessary, but not sufficient to enforce the Lorenz condition for all $t\geq 0$. Then to have Maxwell's equations satisfied in the present environment too magnetic sources have to be introduced into Maxwell's equations. It is noted that these sources share the same support as the electric charge and current density functions. The solutions of Maxwell's equations incorporating magnetic sources in terms of potentials are indicated to be of the same form as in the case of a density function in motion. An example is included which consists in an electric dipole which in the limit form is a Hertz dipole. It is shown that this structure also supports an infinite $c$ and yet magnetic sources are needed to satisfy Maxwell's equations.