Clifford Fibrations and Possible Kinematics

TL;DR

This paper introduces a novel geometric approach using Clifford fibrations to construct and analyze 2D spacetimes based on various kinematical algebras, providing a unified algebraic framework.

Contribution

It develops a new geometric construction based on Clifford fibrations to study kinematical algebras and their associated spacetimes, differing from traditional Hopf fibration approaches.

Findings

Algebraic properties of Clifford fibrations describe geometrical features of kinematical groups.

Constructs homogeneous spaces for all but one kinematic algebra.

Provides a unified algebraic framework for 2D kinematic spacetimes.

Abstract

Following Herranz and Santander [Herranz F.J., Santander M., Mem. Real Acad. Cienc. Exact. Fis. Natur. Madrid 32 (1998), 59-84, physics/9702030] we will construct homogeneous spaces based on possible kinematical algebras and groups [Bacry H., Levy-Leblond J.-M., J. Math. Phys. 9 (1967), 1605-1614] and their contractions for 2-dimensional spacetimes. Our construction is different in that it is based on a generalized Clifford fibration: Following Penrose [Penrose R., Alfred A. Knopf, Inc., New York, 2005] we will call our fibration a Clifford fibration and not a Hopf fibration, as our fibration is a geometrical construction. The simple algebraic properties of the fibration describe the geometrical properties of the kinematical algebras and groups as well as the spacetimes that are derived from them. We develop an algebraic framework that handles all possible kinematic algebras save one, the static algebra.