Maxwell Equations in Cartesian Coordinate Systems

We would like to make the laws of Electrodynamics in the vacuum to be invariant under the Galilean transformations. For this purpose we refer to the analogy with the Maxwell equations in a medium. It is well known that the classical Maxwell equations in a medium have the form of \( {\left\{ \begin{array}{ll} {\text {curl}}_{\textbf{x}} \textbf{H}= \frac{4\pi }{c}\textbf{j}+\frac{1}{c}\frac{\partial \textbf{D}}{\partial t} ,\\ {{\,\textrm{div}\,}}_{\textbf{x}} \textbf{D}= 4\pi \rho ,\\ {\text {curl}}_{\textbf{x}} \textbf{E}+\frac{1}{c}\frac{\partial \textbf{B}}{\partial t}= 0 ,\\ {{\,\textrm{div}\,}}_{\textbf{x}} \textbf{B}= 0 . \end{array}\right. }\) Here \(\textbf{E}\) is the electric field, \(\textbf{B}\) is the magnetic field, \(\textbf{D}\) is the electric displacement field, \(\textbf{H}\) is the \(\textbf{H}\) -magnetic field, \(\rho \) is the charge density, \(\textbf{j}\) is the current density and c is the universal constant, called speed of light. It is assumed in the Classical Electrodynamics that for the vacuum we always have \(\textbf{D}= \textbf{E}\) and \(\textbf{H}= \textbf{B}\) .