Principle of Relativity, 24 possible kinematical algebras and new geometries with Poincar\'e symmetry

TL;DR

This paper explores 24 possible kinematical algebras derived from the principle of relativity with two invariants, revealing new geometries with Poincaré symmetry and discussing free particle motion in these novel space-times.

Contribution

It identifies and classifies 24 kinematical algebras, including a new Poincaré algebra, and introduces associated geometries with potential physical implications.

Findings

Discovery of 24 kinematical algebras from relativity principles.

Introduction of new geometries with Poincaré symmetry.

Analysis of free particle motion in these new space-times.

Abstract

From the principle of relativity with two universal invariant parameters $c$ and $l$, 24 possible kinematical (including geometrical and static) algebras can be obtained. Each algebra is of 10 dimensional, generating the symmetry of a 4 dimensional homogeneous space-time or a pure space. In addition to the ordinary Poincar\'e algebra, there is another Poincar\'e algebra among the 24 algebras. New 4d geometries with the new Poincar\'e symmetry are presented. The motion of free particles on one of the new space-times is discussed.