Affine Geometry and Relativity

Abstract

We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincar\'e transformations of a plane. In particular, a 1-parametric family of natural deformations of the Poincar\'e group is described. We also visualized the underlying groups of Galilean, Euclidean, and pseudo-Euclidean rotations within the special linear group.