Motion of a Charge Density and the Speed of Light in Vacuum

In a series of articles, it has been claimed by the author that speed of light in vacuum c can be infinite. This result was reached by considering electromagnetic problems or more general processes and the Lorentz transformation or the differential Lorentz transformation of space-time coordinates was assumed as the basis for the development in each work. However, in the present work for the motion of a charge density, an infinite speed of light in vacuum result is obtained without recourse to the Lorentz transformation. The discussion is on the scalar and vector potentials of a charge density that abruptly starts an arbitrary motion. First it is shown that for a point charge with an abruptly starting uniform rectilinear motion, the potential formulas obtained using the Lorenz condition do not satisfy the Lorenz condition unless c is set equal to infinity. Then for a general charge distribution it is found that the Lorenz condition upon which the formulas for the scalar and vector potentials are based, is satisfied if and only if the $\displaystyle \frac{\partial}{\partial t}$ and $\nabla$. operators respectively can be interchanged with the volume integrals needed to compute these potentials. Next for the potentials of a discontinuous arbitrary motion of a charge density, it is proved that these differentiations under the integral are permitted only if c is infinite. These results give, without resorting to the Lorentz transformation, the fact that for an arbitrary charge density set into abrupt general motion, the potentials satisfy the Lorenz condition only if c is infinite or that the potentials are no longer retarded.