The Constitutive Relations and the Magnetoelectric Effect for Moving Media

In this paper the constitutive relations for moving media with homogeneous and isotropic electric and magnetic properties are presented as the connections between the generalized magnetization-polarization bivector $$ and the electromagnetic field F. Using the decompositions of F and $\mathcal{M}$, it is shown how the polarization vector P(x) and the magnetization vector M(x) depend on E, B and two different velocity vectors, u - the bulk velocity vector of the medium, and v - the velocity vector of the observers who measure E and B fields. These constitutive relations with four-dimensional geometric quantities, which correctly transform under the Lorentz transformations (LT), are compared with Minkowski's constitutive relations with the 3-vectors and several essential differences are pointed out. They are caused by the fact that, contrary to the general opinion, the usual transformations of the 3-vectors $$, $\mathbf{B}$, $\mathbf{P}$, $\mathbf{M}$, etc. are not the LT. The physical explanation is presented for the existence of the magnetoelectric effect in moving media that essentially differs from the traditional one.