On the Relationship between Rotations and Lorentz Transformations in Two, Three, and Four Dimensions

Abstract

Rotations and Lorentz transformations are compared geometrically and algebraically for two-, three-, and four-dimensional space and space-time. The close analogy between the two types of transformation is revealed by means of orthogonal and pseudo-orthogonal matrices, complex numbers and double numbers, and quaternions and tetrons. The most natural and elegant forms are shown to be the complex and hypercomplex number representations. The relationship is essentially based on the direct correspondence between the circular functions sinθ and cosθ, and the hyperbolic functions sinhφ and coshφ.