On the invariance in the inhomogeneous Lorentz group SO(1,3) in the context of optical vortex description

Abstract

Recent studies of optical vortexes have elicited the need for a new kind of transformations, comprising a variety of Lorentz transformations that would embody not only translational but rotational quantities as well. The model described in this paper was motivated by a conviction in the need for a uniform approach that would enable exposition of translational and rotational motion one equal grounds—one that would have to be reflected in the very foundations of theoretical physics. This paper shows that transformations can be grounded in a generalized invariance of translational and rotational motion instead of ordinary linear invariance known from the general theory of relativity. We have proven that the geodesic principle holds in a Lorentz group parameter space for free translational and a rotational motion of a solid body. It has been demonstrated that the new transformation group would naturally lead to extending the special theory of relativity with the principles describing rotational motion.