Journal of Geometry and Symmetry in Physics

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

Abraham A. Ungar

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.

Article information

Source
J. Geom. Symmetry Phys., Volume 38 (2015), 39-108.

Dates
First available in Project Euclid: 27 May 2017

https://projecteuclid.org/euclid.jgsp/1495850536

Digital Object Identifier
doi:10.7546/jgsp-38-2015-39-108

Mathematical Reviews number (MathSciNet)
MR3380218

Zentralblatt MATH identifier
1352.20043

Citation

Ungar, Abraham A. Parametric Realization of the Lorentz Transformation Group in Pseudo-Euclidean Spaces. J. Geom. Symmetry Phys. 38 (2015), 39--108. doi:10.7546/jgsp-38-2015-39-108. https://projecteuclid.org/euclid.jgsp/1495850536