Orthogonal decomposition of Lorentz transformations

TL;DR

This paper presents a method to decompose Lorentz transformations into orthogonal simple components, providing new formulas for their exponential and logarithm functions, enhancing understanding of Lorentz algebra structure.

Contribution

It introduces a canonical decomposition of Lorentz transformations into commuting simple bivectors, offering an alternative approach to exponential and logarithm calculations.

Findings

Decomposition into orthogonal simple Lorentz bivectors

New formulas for exponential and logarithm of Lorentz transformations

Enhanced understanding of Lorentz algebra structure

Abstract

The canonical decomposition of a Lorentz algebra element into a sum of orthogonal simple (decomposable) Lorentz bivectors is discussed, as well as the decomposition of a proper orthochronous Lorentz transformation into a product of commuting Lorentz transformations, each of which is the exponential of a simple bivector. As an application, we obtain an alternative method of deriving the formulas for the exponential and logarithm for Lorentz transformations.