Approximately measured causality implies the Lorentz group: Alexandrov- Zeeman result made more realistic (Q1340333)

The author studies 4-dimensional Minkowski space. For two vectors \(a,b \in \mathbb{R}^ 4\) he writes \(a < b\) if and only if \(b - a\) is a future directed, causal vector. There exists a theorem by \textit{A. D. Aleksandrov} [Usp. Mat. Nauk. 5, No. 3(37), 187 (Russian) (1950)] and \textit{A. D. Aleksandrov} and \textit{V. V. Ovchinnikova} [Vestn. Leningr. Univ. 11, 95-110 (Russian) (1953)] asserting that a bijective mapping \(f : \mathbb{R}^ 4 \mapsto \mathbb{R}^ 4\) which respects this partial order is the composition of a Lorentz transformation, a shift, and a dilatation. This theorem is generalized to maps which approximately respect this order. The meaning of `approximately' is as follows: Assume that there exists a function \(h: \mathbb{R}^ + \to \mathbb{R}^ +\) with \(h(t) \to 0\) \((t \to 0)\) and let \(C \subset \mathbb{R}^ 4 \times \mathbb{R}^ 4\). \(C\) is supposed to satisfy (i) \((a,b) \in C \Rightarrow \exists b' \in \mathbb{R}^ 4\) such that \(a < b'\) and \(d(b,b') < h(d(a,b)\)); (ii) \((a,b) \notin C \Rightarrow \exists b' \in \mathbb{R}^ 4\) such that \(a \nless b'\) and \(d(b,b') < h(d(a,b))\). Here \(d\) is the usual Euclidean distance. Thus \(C\) is an approximate model for \(<\). The author now proves that any homeomorphism \(f: \mathbb{R}^ 4 \to \mathbb{R}^ 4\) with \((a,b) \in C \Leftrightarrow (f(a),f(b)) \in C\) is the composition of a Lorentz transformation, a shift, and a dilatation. He also gives some physical motivation for his theorem.