From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups

Document Type: Special Issue: International Conference on Architecture and Mathematics

Author

North Dakota State University

Abstract

‎The Lorentz transformation of order $(m=1,n)$‎, ‎$n\in\Nb$‎, ‎is the well-known ‎Lorentz transformation of special relativity theory‎. ‎It is a transformation of time-space coordinates of the ‎pseudo-Euclidean space $\Rb^{m=1,n}$ of one time dimension and ‎$n$ space dimensions ($n=3$ in physical applications)‎. ‎A Lorentz transformation without rotations is called a {\it boost}‎. ‎Commonly‎, ‎the special relativistic boost is ‎parametrized by a relativistically admissible velocity parameter $\vb$‎, ‎$\vb\in\Rcn$‎, ‎whose domain is the $c$-ball $\Rcn$ of all ‎relativistically admissible velocities‎, ‎$\Rcn=\{\vb\in\Rn:\|\vb\|<c\}$‎, ‎where the ambient space $\Rn$ is the ‎Euclidean $n$-space‎, ‎and $c>0$ is an arbitrarily fixed ‎positive constant that represents the vacuum speed of light‎. ‎The study of the Lorentz transformation composition law in terms of ‎parameter composition reveals that the group structure of the ‎Lorentz transformation of order $(m=1,n)$ induces a gyrogroup and ‎a gyrovector space structure that regulate ‎the parameter space $\Rcn$‎. ‎The gyrogroup and gyrovector space structure ‎of the ball $\Rcn$‎, ‎in turn‎, ‎form the algebraic setting for the Beltrami-Klein ball model ‎of hyperbolic geometry‎, ‎which underlies the ball $\Rcn$‎. ‎The aim of this article is to extend the study of the ‎Lorentz transformation of order $(m,n)$ from $m=1$ and $n\ge1$ to ‎all $m,n\in\Nb$‎, ‎obtaining algebraic structures called ‎a {\it bi-gyrogroup} and a {\it bi-gyrovector space}‎. ‎A bi-gyrogroup is ‎a gyrogroup each gyration of which is a pair of ‎a left gyration and a right gyration‎. ‎A bi-gyrovector space is constructed from a bi-gyrocommutative bi-gyrogroup ‎that admits a scalar multiplication‎.

Keywords

Main Subjects


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