Zeeman topologies on space-times of general relativity theory

Abstract

In 1964 Zeeman published a paper showing [independently of Alexandrov (1953)] that the causal structure of the light cones on Minkowski space M determines the linear structure of M. This initiated the question whether a topology (more physically than the ordinary one) on M, related to the light cones also implies the linear structure of M. In 1967 Zeeman defined such a new topology -here called Zeeman-topology 3o ~ o n Minkowski space and solved this question for M. In that paper he asked whether it is possible to generalize this program to general relativity. Two of his main questions were: (a) What is the structure of the group G(S) of all homeomorphisms of a spacetime S with respect to the general relativistic analogue of 3o (defined in § 3)? (b) What are the world lines (defined in § 1) with respect to 3o? Without any restrictions on the space-time S we will give the answers: (a) G(S) is the group of all homothetic transformations on S (for an explicite discussion of this result we refer to § 5). (b) World lines are broken geodesies. Including external fields (like Maxwell fields and deviations of3 0 )the answer (b) can be generalized in different physical directions; cf. § 3. Ein Fernrohr wird gezeigt, womit man seinen eίgenen Rϋcken sieht. Es fύhrt durchs Weltall deinen Blick im Kreis zurύck auf dein Genick. Zwar braucht es so geraume Frist, daβ du schon lάngsί verstorben bist, dock wird ein Standbild dir geweiht, empfάngt es ihn zu seiner Zeit. Christian Morgenstern, 9i Bόhmischer Jahrmarkt" Contents § 1. Introduction and Discussion 290 § 2. Definition and Notations 292 § 3. Zeeman Topology on a General Relativistic Space-Time 293 § 4. Further Properties of the Zeeman Topologies on Space-Times 297 §5. The Group of All Homeomorphisms of a Space-Time 298 §6. References 306Note Added in Proof. These topologies will be considered in two forthcoming papers by Hawking, King, McCarthy [42] and Gobel [43]. In [43] the results of [23] will be corrected.

Infinity in string cosmology: A review through open problems

“…• Zeeman also conjectured in [93] that an extension to the curved spacetimes of general relativity should be possible, that is for a general spacetime the homothetic group must be isomorphic to the homeomorphism group of the Zeeman topology, a conjecture that was shown to be correct by Göbel in [94].…”

Signal spaces—an axiomatic approach to space-time

“…Zeeman type topologies are defined by the topologies they induce on certain subsets, for example, the interval topology on straight timelike lines. This idea can, with modifications, be extended to space-times in general [2,5]. This type of topology does not generally reproduce the manifold topology in space-times (usually it is too fine), but has interesting properties in its own right.…”

Invariance of domain in Minkowski space with path topology

“…However, it is none of normal, regular, paracompact, locally compact, second countable or simply connected [1,2,9,10]. Generalizations of the non-Euclidean fine and s-topologies to the spacetime of general relativity have been studied by Göbel [5,6] and Domiaty [3,4].…”