Lorentz transformations as space-time reflections

Abstract

A rank-two tensor is built out of the 4-velocities of two inertial observers, which corresponds precisely to the most general Lorentz matrix connecting the two Cartesian frames of the observers. The Lorentz tensor is then factorized as the product of two ’’complementary’’ space-time reflections. It is shown that the first tensorial factor performs the very essential tasks (i.e., FitzGerald contraction and time dilation) of the corresponding Lorentz transformation, while the second factor is just an internal reflection performed in one and the same intertial frame. Thus, in its essential features, a Lorentz transformation between two different inertial frames obtains upon performing just one space-time reflection. It is also shown that the (same) Lorentz tensor of the two inertial observers can be factorized into ’’complementary’’ reflections either by two hyperplanes with spacelike normals, or else by tow hyperplanes with timelike normals, which geometric meaning is rather simple. An application of the presented formalism to Dirac’s 4-spinor transformation law is also briefly discussed.