On the causal structure of Lorentzian Lie groups. A globality theorem for Lorentzian cones in certain solvable Lie algebras
type: Item
instance of: scholarly article
title: On the causal structure of Lorentzian Lie groups. A globality theorem for Lorentzian cones in certain solvable Lie algebras
zbMATH Open document ID: 0854.22007
DOI: 10.1007/BF01444225
author: Dirk Mittenhuber
published in: Mathematische Annalen
publication date: 1996-10-07T00:00:00Z
full work available at URL: https://eudml.org/doc/182991
review text: Let \(G\) be a Lie group equipped with a Lorentzian manifold structure, then the causality properties of this manifold are closely related to the properties of a subsemigroup \(S\) of \(G\) which is generated by a Lorentzian cone \(C\) in the Lie algebra. Roughly speaking, the Lorentzian manifold has reasonable causality properties if and only if the semigroup \(S\) is pointed, i.e., its subgroup of units is trivial: \(S \cap S^{-1} = \{\mathbf{1}\}\). In this case the cone \(C\) is called global. The main theorem states that if \(\mathfrak g\) is a Lie algebra of the form \({\mathfrak g} = {\mathfrak h} \rtimes \mathbb{R}\) with \(\mathfrak h\) nilpotent of length 2 and if the Lorentzian cone \(C\) lies in the halfspace \({\mathfrak h} \times \mathbb{R}^+\), then \(C\) is global. As a direct application it is proven that a Lorentzian cone in a nilpotent Lie algebra of length 2 is either global or controllable, i.e., the semigroup generated by \(C\) is either pointed or the whole group. This means that the corresponding spacetime is either past and future distinguishing or totally vicious.
reviewed by: Dirk Mittenhuber
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 22E15
zbMATH DE Number: 931961
zbMATH Keywords: Lie group
zbMATH Keywords: Lorentzian cone
zbMATH Keywords: Lorentzian manifold
zbMATH Keywords: nilpotent Lie algebra
zbMATH Keywords: spacetime
MaRDI profile type: Publication
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Wikidata QID: Q115393544
OpenAlex ID: W2912265616
description: scientific article; zbMATH DE number 931961