GYROVECTOR SPACES AND THEIR DIFFERENTIAL GEOMETRY

Abstract

This article adds physical appeal to Einstein addition, the Einstein velocity addition law of relativistically admissible velocities. Einstein addition turns out to be isomorphic to MÄobius addition in the sense of isomorphisms between gyrovector spaces. Gyrovector spaces, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. A remarkable link between the gyrovector spaces that we study in this article and hyperbolic geometry is provided by differential geometry.