Axiomatics of the general-relativistic and the Finsler space-time by means of causality (Q1824229)

A. D. Aleksandrov (1982) proposed to develop the axiomatics of the general theory of relativity in the spirit of chronogeometry. In chronogeometry it is established that the space-time of special relativity can be regarded as a lattice \((M,<,{\mathfrak G})\) where M is a set, \(<\) is an order relation representing causality and \({\mathfrak G}\) is a sufficiently rich group of automorphisms g: (M,\(<)\to (M,<)\). Since the general-relativistic pseudo-Riemannian space \(^{n-1}V_ n\), generally speaking, does not have a rich set of automorphisms, this approach is not directly applicable. In this paper the author shows that \(^{n-1}V_ n\) can be presented in the form of a lattice as above is \({\mathfrak G}\) is conceived as automorphisms \(R^ M\to R^ M\) on the set of order preserving functions f: (M,\(<)\to (M,<)\). Another difficulty appears in connection with the order relation \(<\). That is why the author uses his relation \(\prec\) named the relation of local succession [introduced in his book Kinematic spaces (1970; Zbl 0203.283)]. The present paper relies on several previous papers of the same author listed in the references at the end of the article.