On the geometry of the kinematic space in special relativity

TL;DR

This paper explores the hyperbolic geometric structure of the space of inertial frames in special relativity, revealing deep connections between hyperbolic geometry and relativistic physics.

Contribution

It provides a geometric framework for special relativity by characterizing its kinematic space as hyperbolic, highlighting differences from classical mechanics through geometric perspectives.

Findings

The space of inertial frames is naturally hyperbolic.

Hyperbolic geometry underpins key relativistic phenomena.

Differences from classical mechanics are geometrically interpreted.

Abstract

The classifying space of inertial reference frames in special relativity is naturally hyperbolic. There is a remarkable interplay between central elements of hyperbolic geometry and those of special relativity -- which, to a certain extent, have already been observed in the past -- that we present and further discuss in the paper. We aim at a geometrization of special relativity at the level of kinematic space by giving to physical concepts/phenomena purely geometric definitions/descriptions. In this way, the differences between special relativity and classical mechanics can be seen as a manifestation of the distinct geometric natures of their kinematic spaces.