Hyperbolic Rotation with Euclidean Angle Illuminates Space Time Spinors

Abstract

In this article it will be shown that by introducing a hyperbolic rotation as function of a Euclidean rotation angle, all possible Lorentz group spacetime rotations can be performed by Euclidean rotation parameters. This unification in Lorentz group rotations parameters (all Euclidean) allows for the calculation of the volume of a spacetime region bounded by the light cone of a past event, i.e., a causality-volume with the shape of a spacetime three-sphere. As will be calculated, the surface area of the spacetime three-sphere is the volume of our 3D space. Additionally, the unification in Lorentz group rotation parameters allows for the derivation of a Dirac spinor purely from the spacetime three-sphere symmetries. This derivation from the spacetime symmetries of a geometrical object yields the same result as solving the Dirac equation with quantum mechanical eigenvalue eigenvector complex matrix calculations.