Thomas-Wigner rotation via Clifford algebras

TL;DR

The paper provides a self-contained, undergraduate-accessible derivation of the Thomas–Wigner rotation’s angle and axis using Clifford-algebra techniques. It frames boosts and rotations as Clifford conjugations, derives a compact expression for the three-boost composition, and obtains the Macfarlane formula for the Thomas–Wigner angle by analyzing the conjugating element in the appropriate spacelike plane. The approach emphasizes a transparent geometric and algebraic pathway, linking reflections, boosts, and rotations within a unified Clifford framework. This yields a pedagogically clear route to Macfarlane’s result and clarifies how boost compositions induce rotations in the orthogonal spacelike plane.

Abstract

We derive Macfarlane's formula for the Thomas-Wigner angle of rotation using Clifford-algebra methods. The presentation is pedagogical and elementary, suitable for students with some basic knowledge of special relativity; no prior knowledge of Clifford algebras is required.