Snell's Law for the Poynting Vector in a Semi-infinite Dielectric Moving Perpendicularly to the Surface

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Abstract
Snell's law for the Poynting vector is obtained for the case of a semi-infinite, non-dispersive (in its proper frame), isotropic and linear dielectric that is moving in a direction perpendicular to the interface vacuum-dielectric. Maxwell's equations and Minkowski's constitutive equations in covariant form, as well as in three-dimensional form, are used to obtain a wave equation for the transmitted four-potential A α t = (ϕ/ t /c, A t). By using ϕ/t and A t in the form of plane waves, the transmitted fields E t, H t and the transmitted Poynting vectors, S t = E t × H t (instantaneous) and 〈S t〉 = ½ Re (E t × H∗t) (time-average), are obtained. Then, with the help of the relativistic transformations of the propagation vector k t and the frequency ω t, the transmitted angle of the Poynting vector is expressed in terms of the incident one.