On the Geometry Induced by Lorentz Transformations in Pseudo-Euclidean Spaces

Abstract

The Lorentz transformations of order $(m,n)$ in pseudo-Euclidean spaces with indefinite inner product of signature $(m,n)$ are extended in this work from $m=1$ and $n\ge1$ to all $m,n\ge1$. A parametric realization of the Lorentz transformation group of any order $(m,n)$ is presented, giving rise to generalized gyrogroups and gyrovector spaces called bi-gyrogroups and bi-gyrovector spaces. The latter, in turn, form the setting for generalized analytic hyperbolic geometry that underlies generalized balls called eigenballs.