Hyperboloid preservation implies the Lorentz and Poincar\'e groups without dilations

TL;DR

This paper presents a hyperboloid preservation theorem that characterizes the Poincaré group without dilations, extending the Alexandrov-Zeeman theorem to hyperbolic geometries and one-dimensional cases.

Contribution

It introduces a hyperboloid-based analogue of the Alexandrov-Zeeman theorem, uniquely characterizing the Poincaré group without dilations, including in one-dimensional space.

Findings

Hyperboloid preservation characterizes the Poincaré group.

The theorem applies to one-dimensional space.

An orthochronous version based on forward hyperboloid shells is established.

Abstract

An analogue of the Alexandrov-Zeeman theorem, based on hyperboloid preservation, as opposed to light cone preservation, is provided. This characterizes exactly the Poincar\'e group, as opposed to the Poincar\'e group extended by dilations. The hyperbolic analogue, as opposed to the cone-based Alexandrov-Zeeman theorem, is also valid in the case of a single space dimension. An orthochronous version also holds, based on the preservation of forward hyperboloid shells.