Unimodular Matrices Homomorphic to Lorentz Transformations in n ≥ 2 Spacelike Dimensions

Abstract

The purpose of the present paper is to establish a homomorphie correspondence between a subgroup of the unimodular matrices and the Lorentz transformations in (n +1)-dimensional space with one timelike and n ≧2 spacelike dimensions. This correspondence, which generalizes the well-known correspondence for n=3 between 2×2 unimodular matrices and 4×4 restricted Lorentz transformations, is established in Section 1. We have devoted Section 2 to two proofs, which we believe to be simple, of a theorem on the unimodular matrix associated with the transpose of an (n + 1)-dimensional Lorentz matrix for n≧2. This theorem is essentially known for the case of three spatial dimensions ^{1, 2}, the only accessible proof being, however, that in reference ².