Remarks on antichains in the causality order of space-time

TL;DR

This paper explores the properties of antichains and gradings in the causality order of space-time, revealing differences between Lorentz-invariant orders and implications for the structure of space-time.

Contribution

It provides a detailed analysis of antichain cutsets and gradings in Lorentz-invariant causality orders, including combinatorial characterizations and proofs related to space-time structure.

Findings

Subluminal causality order lacks antichain cutsets and cannot be graded.

Maximal chains correspond to world lines of particles with or without mass.

A simple proof of the Alexandrov-Zeeman Theorem is derived from combinatorial characterizations.

Abstract

The two closely related Lorentz-invariant partial orders of space-time are distinguished with respect to the existence of antichain cutsets and the possibility of grading. World lines of particles with or without mass are the maximal chains in the causality order of space-time, and antichain cutsets are the levels of the various gradings of the causality partial order. The maximal chains of the weaker, subluminal causality order need not be connected topologically, subluminal causality has no antichain cutsets and cannot be graded. Combinatorial characterizations of optical lines and hyperplanes, separation lines, inertia planes and lines, ultimately in terms of the causality order yield a simple proof of the Alexandrov-Zeeman Theorem.