Parametric Realization of the Lorentz Transformation Group in Pseudo-Euclidean Spaces

Abstract

The Lorentz transformation group ${\rm SO}(m,n)$, $m,n\in \mathbb{N}$, is a group of Lorentz transformations of order $(m,n)$, that is, a group of special linear transformations in a pseudo-Euclidean space $\mathbb{R}^{m,n}$ of signature $(m,n)$ that leave the pseudo-Euclidean inner product invariant. A parametrization of ${\rm SO}(m,n)$ is presented, giving rise to the composition law of Lorentz transformations of order $(m,n)$ in terms of parameter composition. The parameter composition, in turn, gives rise to a novel group-like structure that $\mathbb{R}^{m,n}$ possesses, called a bi-gyrogroup. Bi-gyrogroups form a natural generalization of gyrogroups where the latter form a natural generalization of groups. Like the abstract gyrogroup, the abstract bi-gyrogroup can play a universal computational role which extends far beyond the domain of pseudo-Euclidean spaces.