On non-equivalence of Lorentz and Coulomb gauges within classical electrodynamics

Abstract

It is shown that the well-known procedure for proving the equivalence of the expressions for the electric field calculated using the Lorentz and Coulomb gauges is incorrect. The difference between the two gauges is due to the difference in the speed of propagation of a disturbance of the scalar potential. As an auxiliary result, it is proven that the solution for the electric field cannot be obtained directly from the Maxwell equations, i.e. without introducing the scalar and vector potentials.