A supplement to the Alexandrov–Lester Theorem

Abstract

Let $V$ be the 4-dimensional Minkowski-space of special relativity over the reals with quadratic form Q. Consider a mapping $\psi:V→ V$ such that $Q(x-y)=0\Leftrightarrow Q(x\psi-y\psi)=0$ for all $x,y \in V$. Under the assumption that $\psi$ is a bijection Alexandrov's theorem states that $\psi$ is a linear bijection followed by a translation. Our results imply (as a special case) that the assumption of $\psi$ being a bijection an be dropped.