Lorentz transformation from an elementary point of view

TL;DR

This paper explores the mathematical foundations of the Lorentz transformation using elementary methods, focusing on its eigen-system, exponential representation, and connections to shear maps, highlighting complex cases like hyper-singularity.

Contribution

It introduces a novel elementary approach to understanding the Lorentz transformation's eigen-system and its relation to shear maps, emphasizing the hyper-singular case.

Findings

Eigen-system characterized via exponential of G-skew symmetric matrix

Identified unconnectedness at hyper-singular case

Connected hyper-singular case to shear map through Pauli coding

Abstract

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.