Optimal version of the fundamental theorem of chronogeometry

We study lightlikeness preserving mappings from the 4-dimensional Minkowski spacetime $M_4$ to itself under no additional regularity assumptions like continuity, surjectivity, or injectivity. We prove that such a mapping ϕ satisfies one of the following three conditions.

• (1)
  The mapping ϕ can be written as a composition of a Lorentz transformation, a multiplication by a positive scalar, and a translation.
• (2)
  There is an event $r\in M_4$ such that $\varphi (M_4\setminus \{r\left\}\right)$ is contained in one light cone.
• (3)
  There is a lightlike line ℓ such that $\varphi (M_4\setminus \ell )$ is contained in another lightlike line.

Here, a line that is contained in some light cone in $M_4$ is called a lightlike line. We also give several similar results on mappings defined on a certain subset of $M_4$ or the compactification of $M_4$.