Mathematics Interdisciplinary Research (Jan 2016)
From the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups
Abstract
The Lorentz transformation of order (m=1,n), n∈N, is the well-known Lorentz transformation of special relativity theory. It is a transformation of time-space coordinates of the pseudo-Euclidean space Rm=1,n of one time dimension and n space dimensions (n=3 in physical applications). A Lorentz transformation without rotations is called a boost. Commonly, the special relativistic boost is parametrized by a relativistically admissible velocity parameter v, v ∈ Rcn, whose domain is the c-ball Rcn of all relativistically admissible velocities, Rcn={v ∈ Rn: ||v||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≥1 to all m,n∈N, obtaining algebraic structures called a bi-gyrogroup and a 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.
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