The abstract Lorentz transformation group

Treatments of the Lorentz transformation of special relativity at an undergraduate level usually assume that the motion of one observer is along the x axis of another observer, resulting in the (1+1)-dimensional Lorentz transformation group. The (1+n)-dimensional Lorentz group, for n=2 or n=3, is unknown to many physics students because of the simplified (1+1)-dimensional treatments found in most texts. The aim of this article is to simplify the presentation of the (homogeneous, proper, orthochronous) Lorentz group by abstraction to the point where the (1+n)-dimensional Lorentz group can readily be presented to physics students in n space dimensions where n is finite or infinite. The study of the (homogeneous, proper, orthochronous) Lorentz transformation group is simplified and generalized in this article by abstraction, thus obtaining an elegant formalism to deal with the Lorentz group. This new formalism allows one to solve in the abstract Lorentz group previously poorly understood problems in the standard, (1+3)-dimensional Lorentz group. Two such problems, studied in this article, are (i) the problem of determining the Lorentz transformation composition law in a way analogous to the well-understood determination of the Galilean transformation composition law, and (ii) the problem of determining all the Lorentz transformations linking two given points in a Minkowski space. Crucial points in the present study of the abstract Lorentz group are (i) the abstract relativistic velocity addition law; (ii) the abstract Thomas precession, called Thomas gyration; and (iii) the parametrization of the abstract Lorentz transformation by abstract velocity and abstract orientation parameters in such a way that the composition law of abstract Lorentz transformations is given by a corresponding parameter composition law. In the limit of large speed of light c, c→∞, Thomas gyration vanishes, and the Lorentz transformation composition law reduces to the Galilean transformation composition law.