Causality explains why spatial and temporal translations commute: A remark
The author defines a spacetime as a pair \((M,\leq)\) with \(M\) a set and \(\leq \) an order. (The exact properties of \(\leq\) are not specified but it is to be regarded as a ``causality'' ordering. Also a ``non-trivial'' restriction which the reviewer does not understand is imposed on \(M\).) A 1-1 map \(g : M \to M\) is called a symmetry if for \(a,b \in M\), \(a \leq b \Leftrightarrow g(a) \leq g(b)\). A map \(x : M \to \mathbb{R}\) is called a coordinate and it is an inertial coordinate if for each \(q \in G\) there exists a number \(S_x(g)\) such that \(x(g(a)) = x(a) + S_x(g)\) for all \(a \in M\). A coordinate \(t\) is called temporal if \(a \leq b \Rightarrow t(a) \leq t(b)\). Finally, the pair \((M,G)\) is called natural if, given \(a,b \in M\), the condition \(t(a) \leq t(b)\) for all inertial temporal coordinates \(t\) implies \(a \leq b\). The author then proves that if \((M,G)\) is such a natural pair, \(G\) is an abelian group.
scientific article; zbMATH DE number 889807