Geometries for Possible Kinematics

TL;DR

This paper classifies and relates all possible Lorentzian and Euclidean kinematic geometries with isotropy, revealing their contraction relations and identifying nine fundamental geometries with specific signatures and isotropy properties.

Contribution

It provides a comprehensive classification of kinematic geometries via contraction methods, revealing pairings and a key $t \leftrightarrow 1/( u^2 t)$ correspondence among them.

Findings

Almost all geometries form pairs with specific correspondences.

Identified only 9 fundamental geometries with right signatures and isotropy.

Classified geometries into relativistic, absolute-time, and absolute-space categories.

Abstract

The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are revealed. Almost all geometries fall into pairs. There exists $t \leftrightarrow 1/(\nu^2t)$ correspondence in each pair. In the viewpoint of differential geometry, there are only 9 geometries, which have right signature and geometrical spatial isotropy. They are 3 relativistic geometries, 3 absolute-time geometries, and 3 absolute-space geometries.