Theory of ordered spaces. II: The local differential structure
This paper continues the investigation of the implications of Einstein causality for the local structure of space-time which was begun in part I [\textit{H.-J. Borchers} and \textit{R. N. Sen}, Commun. Math. Phys. 132, No. 3, 593-611 (1990; Zbl 0725.54026)]. In that paper the basic framework, namely the axiomatization of Einstein causality as a partial order, was set up and its local topological consequences explored. This axiomatization did not use any number system apart from the natural numbers. One sees from an example of the present paper that these axioms are not strong enough to force the emergence of the real number system. In this paper the authors investigate the conditions under which the ordered spaces defined in the previous paper are locally diffeomorphic to \(\mathbb R\). The authors introduce this structure via the new connectedness axiom, and work out some of its immediate consequences. The (somewhat involved) construction of timelike curves in a \(D\)-set is given, which are used to prove a nested interval theorem for ordered spaces, the contractibility of order intervals in \(D\)-sets, and that order intervals in \(D\)-sets are star-shaped. The notion of \(D\)-countability (meaning that a \(D\)-set has a countable base in the subspace topology) is introduced. The Urysohn lemma shows that a \(D\)-countable ordered space is locally metrizable. If this space is also locally compact, then it has finite topological dimension \(N\). The local differential structure now follows from known results: the embedding of such spaces in \(\mathbb R^{2N+1},\) and the result that an open star-shaped region in \(\mathbb R^n\) is diffeomorphic to \(\mathbb R^n\). The interpretation of Einstein causality as a partial order and the investigation of the local structure of these spaces reveal the following: There exist ordered spaces in which the connectedness axiom does not hold, and those in which \(D\)-countability or local compactness fail. A \(D\)-countable and locally compact ordered space satisfying the connectedness axiom is locally smoothly diffeomorphic to \(\mathbb R^n\) for some \(n\).
scientific article; zbMATH DE number 1379925