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.

中文翻译：

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光学涡旋描述中非齐次洛伦兹群SO(1,3)的不变性

最近对光学涡旋的研究引发了对一种新变换的需求，包括各种洛伦兹变换，这些变换不仅体现了平移量，还体现了旋转量。本文中描述的模型源于一种信念，即需要一种统一的方法，该方法将使平移和旋转运动能够在平等的基础上进行阐述——这一点必须反映在理论物理学的基础中。本文表明，变换可以基于平移和旋转运动的广义不变性，而不是广义相对论中已知的普通线性不变性。我们已经证明了测地线原理在洛伦兹群参数空间中适用于固体的自由平移和旋转运动。