Comments on the expanded Maxwell's equations for moving charged media system

Abstract: In the recent work~\cite{Wang:2021p2}, the author proposed the expanded Maxwell's equations for moving charged media system, which seems subtle. Considering a very short time, we can approximately define the inertial frame of reference. If we assume all the physical quantities are defined in the same reference frame by default, Maxwell's equations for the static media system and moving media system are definitely the traditional Maxwell's equations, which are covariant and consistent with the two fundamental postulates of special relativity. We even prove the covariance of Maxwell's equations explicitly by considering the Lorentz transformation under the ${\cal O} (v)$ order approximation and the Galileo approximation, respectively. Therefore, it seems to us that the fields in the expanded Maxwell's equations cannot be in the same reference frame. Defining the fields in the lab and co-moving frames explicitly, we derive the expanded Maxwell's equations for moving media system. Furthermore, we discuss another possible variant of Maxwell's equations, which has an additional coefficient $\alpha$ related to the media. However, it is still subtle from theoretical point of view.