Lorentz transformations with arbitrary line of motion

TL;DR

This paper develops Lorentz transformations for observers with non-collinear lines of motion, revealing non-orthogonality of axes in different frames and demonstrating their application through example problems.

Contribution

It introduces a matrix algebra method to derive Lorentz transformations with arbitrary lines of motion, showing axes become non-orthogonal in different frames.

Findings

Axes are observed to be non-orthogonal in different frames.

The method extends Lorentz transformations to arbitrary directions of motion.

Application to the rod-slot problem demonstrates practical usefulness.

Abstract

Sometimes it becomes a matter of natural choice for an observer (A) that he prefers a coordinate system of two-dimensional spatial x-y coordinates from which he observes another observer (B) who is moving at a uniform speed along a line of motion, which is not collinear with As chosen x or y axis. It becomes necessary in such cases to develop Lorentz transformations where the line of motion is not aligned with either the x or the y-axis. In this paper we develop these transformations and show that under such transformations, two orthogonal systems (in their respective frames) appear non-orthogonal to each other. We also illustrate the usefulness of the transformation by applying it to three problems including the rod-slot problem. The derivation has been done before using vector algebra. Such derivations assume that the axes of K and K-prime are parallel. Our method uses matrix algebra and shows that the axes of K and K-prime do not remain parallel, and in fact K and K-prime which are properly orthogonal are observed to be non-orthogonal by K-prime and K respectively. http://www.iop.org/EJ/abstract/0143-0807/28/2/004