Visualization of Thomas–Wigner Rotations

Georg Beyerle

doi:10.3390/sym9120292

Abstract

It is well known that a sequence of two non-collinear Lorentz boosts (pure Lorentz transformations) does not correspond to a Lorentz boost, but involves a spatial rotation, the Wigner or Thomas–Wigner rotation. We visualize the interrelation between this rotation and the relativity of distant simultaneity by moving a Born-rigid object on a closed trajectory in several steps of uniform proper acceleration. Born-rigidity implies that the stern of the boosted object accelerates faster than its bow. It is shown that at least five boost steps are required to return the object’s center to its starting position, if in each step the center is assumed to accelerate uniformly and for the same proper time duration. With these assumptions, the Thomas–Wigner rotation angle depends on a single parameter only. Furthermore, it is illustrated that accelerated motion implies the formation of a “frame boundary”. The boundaries associated with the five boosts constitute a natural barrier and ensure the object’s finite size.

Highlights

This spatial rotation is known as Wigner rotation or Thomas–Wigner rotation, the corresponding rotation angle is the Thomas–Wigner angle
The Thomas–Wigner rotation angle is calculated from the known boost angles ζ 1,2 (γ) and ζ 2,3 (γ) (Equation (35))
It is well known that pure Lorentz transformations do not form a group in the mathematical sense, since the composition of two transformations in general is not a pure Lorentz transformation again, but involves the Thomas–Wigner spatial rotation

Summary

Introduction

Thomas’s analysis [2,3] explains the observed deviations in terms of a special relativistic [4] effect. He recognized that a sequence of two non-collinear pure Lorentz transformations (boosts) cannot be expressed as one single boost. Two non-collinear boosts correspond to a pure Lorentz transformation combined with a spatial rotation. This spatial rotation is known as Wigner rotation or Thomas–Wigner rotation, the corresponding rotation angle is the Thomas–Wigner angle This spatial rotation is known as Wigner rotation or Thomas–Wigner rotation, the corresponding rotation angle is the Thomas–Wigner angle (see e.g., ref. [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21], and references therein)

Methods

Discussion

Conclusion