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

Abstract

‎The Lorentz transformation of order $(m=1,n)$‎, ‎$ninNb$‎, ‎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 $vb$‎, ‎$vbinRcn$‎, ‎whose domain is the $c$-ball $Rcn$ of all ‎relativistically admissible velocities‎, ‎$Rcn={vbinRn:|vb|0$ 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 $nge1$ to ‎all $m,ninNb$‎, ‎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.