Theory of ordered spaces. II: The local differential structure
type: Item
instance of: scholarly article
title: Theory of ordered spaces. II: The local differential structure
zbMATH Open document ID: 0968.83029
published in: Communications in Mathematical Physics
publication date: 1999-12-16T00:00:00Z
review text: 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 loc…
Mathematics Subject Classification ID: 83C75
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 51G05
zbMATH DE Number: 1379925
zbMATH Keywords: partial order
zbMATH Keywords: light rays
zbMATH Keywords: axiomatization of Einstein causality
zbMATH Keywords: connectedness axiom
zbMATH Keywords: contractibility
zbMATH Keywords: \(D\)-sets
zbMATH Keywords: \(D\)-countability
zbMATH Keywords: locally metrizable space
zbMATH Keywords: timelike curves
zbMATH Keywords: parametrization of the double
author: H. J. Borchers
author: R. Nath. Sen
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: s002200050654
OpenAlex ID: W2073410452
DOI: 10.1007/S002200050654
description: scientific article; zbMATH DE number 1379925