Cromlech, menhirs and celestial sphere: an unusual representation of the Lorentz group

TL;DR

This paper introduces a novel geometric and algebraic representation of the Lorentz group using sphere reversions, complex matrices, quaternions, and Clifford algebras, leading to a simple formula for relativistic velocity addition.

Contribution

It presents a new geometric and algebraic framework for the Lorentz group, connecting sphere reversions with algebraic structures like quaternions and Clifford algebras.

Findings

Derived a simple formula for relativistic velocity composition

Connected Lorentz boosts with celestial sphere diffeomorphisms

Proposed a unified geometric and algebraic Lorentz group representation

Abstract

We present a novel representation of the Lorentz group, the geometric version of which uses "reversions" of a sphere while the algebraic version uses pseudounitary 2x2 matrices over complex numbers and quaternions, and Clifford algebras in general. A remarkably simple formula for relativistic composition of velocities and an accompanying geometric construction follows. The method is derived from the diffeomorphisms of the celestial sphere induced by Lorentz boost.