Optimal version of the fundamental theorem of chronogeometry

TL;DR

This paper characterizes lightlikeness preserving maps in 4D Minkowski spacetime, showing they are either Lorentz transformations with scalings and translations, or map almost all points into a light cone or lightlike line, without regularity assumptions.

Contribution

It provides the optimal classification of lightlikeness preserving maps in Minkowski spacetime without assuming continuity, surjectivity, or injectivity.

Findings

Maps are either Lorentz transformations with scalings and translations.

Alternatively, maps send almost all points into a single light cone or lightlike line.

Results extend to subsets and compactifications of Minkowski spacetime.

Abstract

We study lightlikeness preserving mappings from the $4$-dimensional Minkowski spacetime $\mathcal{M}_4$ to itself under no additional regularity assumptions like continuity, surjectivity, or injectivity. We prove that such a mapping $\phi$ satisfies one of the following three conditions. (1) The mapping $\phi$ can be written as a composition of a Lorentz transformation, a multiplication by a positive scalar, and a translation. (2) There is an event $r\in \mathcal{M}_4$ such that $\phi(\mathcal{M}_4\setminus\{r\})$ is contained in one light cone. (3) There is a lightlike line $\ell$ such that $\phi(\mathcal{M}_4\setminus \ell)$ is contained in another lightlike line. Here, a line that is contained in some light cone in $\mathcal{M}_4$ is called a lightlike line. We also give several similar results on mappings defined on a certain subset of $\mathcal{M}_4$ or the compactification of $\mathcal{M}_4$.