On Minkowski space and finite geometry

TL;DR

This paper characterizes maps on finite Minkowski space that preserve light-like pairs, revealing automatic bijectivity in many cases and linking the problem to a central unsolved issue in finite geometry involving ovoids.

Contribution

It provides new insights into the structure of light-like preserving maps on finite Minkowski spaces and connects the problem to the existence of ovoids in orthogonal polar spaces.

Findings

Bijectivity and light-likeness preservation are automatic in many cases.

The existence of non-bijective light-like maps relates to an open problem in finite geometry.

Results extend previous work on affine polar graphs and polar spaces.

Abstract

The main aim of this interdisciplinary paper is to characterize all maps on finite Minkowski space of arbitrary dimension $n$ that map pairs of distinct light-like events into pairs of distinct light-like events. Neither bijectivity of maps nor preservation of light-likeness in the opposite direction, i.e. from codomain to domain, is assumed. We succeed in in many cases, which include the one with $n$ divisible by 4 and the one with $n$ odd and $\geq 9$, by showing that both bijectivity of maps and the preservation of light-likeness in the opposite direction is obtained automatically. In general, the problem of whether there exist non-bijective mappings that map pairs od distinct light-like events into pairs of distinct light-like events is shown to be related to one of the central problems in finite geometry, namely to existence of ovoids in orthogonal polar space. This problem is still unsolved in general despite a huge amount of research done in this area in the last few decades. The proofs are based on the study of a core of an affine polar graph, which yields results that are closely related to the ones obtained previously by Cameron and Kazanidis for the point graph of a polar space.