Planar factors of proper homogeneous Lorentz transformations

Abstract

This article discusses two constructions factoring proper homogeneous Lorentz transformations H into the product of two planar transformations. A planar transformation is a proper homogeneous Lorentz transformation changing vectors in a two‐flat through the origin, called the transformation two‐flat, into new vectors in the same two‐flat and which leaves unchanged vectors in the orthogonal two‐flat, called the pointwise invariant two‐flat. The first construction provides two planar factors such that a given timelike vector lies in the transformation two‐flat of one and in the pointwise invariant two‐flat of the other; it leads to several basic conditions on the trace of H and to necessary and sufficient conditions for H to be planar. The second construction yields explicit formulas for the orthogonal factors of H when they exist and are unique, where two planar transformations are orthogonal if the transformation two‐flat of one is the pointwise invariant two‐flat of the other.