Continuous space-time transformations

TL;DR

This paper characterizes continuous transformations in four-dimensional Minkowski space that preserve light cones in one direction, showing they are either Poincaré similarities or degenerate forms, using homotopy theory.

Contribution

It provides a complete classification of continuous light cone-preserving maps in Minkowski space, extending understanding of spacetime symmetries.

Findings

Transformations are either Poincaré similarities or degenerate forms.

Homotopy theory of spheres is used as a key proof tool.

Results apply to continuous maps preserving light cones in one direction.

Abstract

We prove that every continuous map acting on the four-dimensional Minkowski space and preserving light cones in one direction only is either a Poincar\'e similarity, that is, a product of a Lorentz transformation and a dilation, or it is of a very special degenerate form. In the presence of the continuity assumption the main tool in the proof is a basic result from the homotopy theory of spheres.