Ordered addition of two Lorentz boosts through spatial and space-time rotations

TL;DR

This paper introduces a novel method for adding two Lorentz boosts through a sequence of spatial rotations and one-dimensional Lorentz transformations, extending from 2D to 3D space using matrix operations and invariants.

Contribution

The paper presents a new approach to combine Lorentz boosts via rotations and Lorentz transformations, providing a practical matrix-based method that extends to three-dimensional space.

Findings

The method works for two-dimensional space and extends to three dimensions.

Two rotations with a boost cannot always be simplified to a single boost with rotations.

The approach uses only matrix multiplication and invariant quantities.

Abstract

The ordered addition of two Lorentz boosts is normally shown to result in a boost by utilizing concepts from group theory and non-Euclidian geometry. We present a method for achieving this addition by performing a sequence of spatial rotations and uni-dimensional Lorentz transformations. The method is first developed for two-dimensional space and it is then extended to three-dimensional space by utilizing the commutative property of the rotation of the y-z plane and a boost along the x-axis. The method employs only matrix multiplication and certain invariant quantities that are natural consequences of spatial rotations and Lorentz transformations. The combining of two boosts in different directions into a single boost cannot be expected a priori because we show that the converse of this statement is not true. That is, two rotations interspersed with a boost cannot always be reduced to a single rotation preceded and followed by boosts.