The Lorentz Group and Its Abstraction

Abstract

In 1966 Ehlers, Rindler and Robinson [ERR66] proposed a new formalism for dealing with the Lorentz group. Their formalism, however, did not find its way to the mainstream literature. Therefore, thirty three years later, two of them suggested considering the “notorious Thomas precession formula” (in their words, p. 431 in [RR99]) as an indicator of the quality of a formalism for dealing with the Lorentz group. The idea of Rindler and Robinson to use the “notorious Thomas precession formula” as an indicator works fine for gyrogroup formalism, where the ugly duckling of special relativity, the “notorious Thomas precession formula”, becomes the beautiful swan of gyrogroup and gyrovector space theory, the Thomas gyration formula in Theorem 2.21, p. 49. Indeed, we will see in this Chapter that the formalism of gyrogroup and gyrovector space theory is well suited for the study of Lorentz groups.