Lorentz matrices: A review

Abstract

This is an expository review of the Lorentz transformation, which is a change of coordinates used by one “inertial observer” to those used by another one. The transformation can be represented by a four-by-four matrix, the Lorentz matrix or the Minkowski-Lorentz matrix. The most familiar, or “special,” case has thex axis of both observers parallel to their relative velocity. A more general transformation drops this constraint. But then a seeming “paradox” arises when there are three observers, and this has led to a challenge to the self-consistency of the special theory of relativity. It is shown here that this challenge is based on a misunderstanding. The properties of the more general Lorentz transformation are reviewed consistently in terms of the matrix approach, which the author believes is now the easiest approach to understand. The spectral analysis of the Lorentz matrix is also discussed. Several checks are included to “make assurance double sure.”