Rotations associated with Lorentz boosts

TL;DR

This paper explores the relationship between two different rotation angles associated with successive Lorentz boosts, revealing a fundamental connection and implications across various physics domains including optics.

Contribution

It demonstrates that two distinct rotation angles linked to Lorentz boosts are different but their sum equals the angle between initial and final boosts, with applications across physics.

Findings

The sum of the two rotation angles equals the boost angle.

The two angles differ but are related by a simple sum.

Applications extend to classical ray optics and other physics branches.

Abstract

It is possible to associate two angles with two successive non-collinear Lorentz boosts. If one boost is applied after the initial boost, the result is the final boost preceded by a rotation called the Wigner rotation. The other rotation is associated with Wigner's O(3)-like little group. These two angles are shown to be different. However, it is shown that the sum of these two rotation angles is equal to the angle between the initial and final boosts. This relation is studied for both low-speed and high-speed limits. Furthermore, it is noted that the two-by-two matrices which are under the responsibility of other branches of physics can be interpreted in terms of the transformations of the Lorentz group, or vice versa. Classical ray optics is mentioned as a case in point.