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Zeeman topologies on space-times of general relativity theory

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Abstract

In 1964 Zeeman published a paper showing [independently of Alexandrov (1953)] that the causal structure of the light cones on Minkowski spaceM determines the linear structure ofM. This initiated the question whether a topology (more physically than the ordinary one) onM, related to the light cones also implies the linear structure ofM. In 1967 Zeeman defined such a new topology — here called Zeeman-topology ℨ0 — on Minkowski space and solved this question forM. 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 groupG(S) of all homeomorphisms of a space-timeS with respect to the general relativistic analogue of ℨ0 (defined in § 3)? (b) What are the world lines (defined in § 1) with respect to ℨ0? Without any restrictions on the space-timeS we will give the answers: (a)G(S) is the group of all homothetic transformations onS (for an explicite discussion of this result we refer to § 5). (b) World lines are broken geodesics. Including external fields (like Maxwell fields and deviations of ℨ0) the answer (b) can be generalized in different physical directions; cf. § 3.

Ein Fernrohr wird gezeigt, womit man seinen eigenen Rücken sieht. Es führt durchs Weltall deinen Blick im Kreis zurück auf dein Genick. Zwar braucht es so geraume Frist, daß du schon längst verstorben bist, doch wird ein Standbild dir geweiht, empfängt es ihn zu seiner Zeit.

Christian Morgenstern, „Böhmischer Jahrmarkt“

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  1. Fachbereich 6, Mathematik, Universität Essen, GHS, D-4300, Essen 1, Federal Republic of Germany

    Rüdiger Göbel

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Communicated by J. Ehlers

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Göbel, R. Zeeman topologies on space-times of general relativity theory. Commun.Math. Phys. 46, 289–307 (1976). https://doi.org/10.1007/BF01609125

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