A categorical approach for relativity theory

Abstract

We give a mathematical interpretation for a previously investigated model, which is based on a set of axioms incorporating fundamental aspects from both Galilei and special relativity (SR). The model considers two time variables, one of them identified with absolute time [characteristic of Galilei relativity (GR)] and the other identified with the local time of SR. Its main characteristics rely on two classes of transformations. One is what we call generalized Lorentz transformation, which includes the standard Lorentz transformations as a particular case. The other is the h-map, which essentially relates absolute and local times. The h-map also determines the basic kinematics of SR from the corresponding kinematics of GR. This allows us to express a Lorentz transformation in terms of a Galilei transformation. It suggests us to see the h-map as a natural transformation between two functors \(\overline{G}\) and \(\overline{L}\), representing the notions of Galilei and Lorentz transformations in functorial language. The categorical framework we develop not only elucidates the structure of Galilei and Lorentz transformations but also provides a unified model for Galilei and special relativity, where the former acquires a distinguished role in itself, not being a mere low speed limit of the latter.