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On the causal structure of Lorentzian Lie groups. A globality theorem for Lorentzian cones in certain solvable Lie algebras. (English) Zbl 0854.22007

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.

MSC:

22E25 Nilpotent and solvable Lie groups
53C30 Differential geometry of homogeneous manifolds
43A85 Harmonic analysis on homogeneous spaces
22E60 Lie algebras of Lie groups
22E15 General properties and structure of real Lie groups
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References:

[1] Beem, J.K., Ehrlich, P.E.: Global Lorentzian Geometry. New York, Basel: Marcel Dekker 1981. · Zbl 0462.53001
[2] Hilgert, J.: The halfspace method for causal structures on homogeneous manifolds. In: Hofmann, K.H., Lawson, J.D., Vinberg, E.B. (eds.) Semigroups in Algebra, Geometry and Analysis, pp. 33-55. Berlin: Walter de Gruyter 1995 · Zbl 0848.22006
[3] Hilgert, J., Hofmann, K.H.: On the Causal Structure of Homogenous Manifolds. Mathematica Scandinavica67, 119-144 (1990) · Zbl 0739.53041
[4] Hilgert, J., Hofmann, K.H., Lawson, J.D.: Lie Groups, Convex Cones and Semigroups. Oxford University Press 1989. · Zbl 0701.22001
[5] Hilgert, J., Neeb, K.H.: Lie Semigroups and their applications (Lect. Notes in Math., vol. 1552) Berlin Heidelberg New York Tokyo: Springer 1993 · Zbl 0807.22001
[6] Hofmann, K.H.: Hyperplane-subalgebras of real Lie-algebras. Geometriae Dedicata36, 207-224 (1990) · Zbl 0718.17006 · doi:10.1007/BF00150789
[7] Levichev, A.: On Causal Structure of Homogeneous Lorentzian Manifolds. General Relativity and Gravitation21, 1027-1045 (1989) · Zbl 0697.53057 · doi:10.1007/BF00774087
[8] Levichev, A., Levicheva, V.: Distinguishability Condition and the Future Subsemigroup. Seminar Sophus Lie2, 205-212 (1992) · Zbl 0791.53060
[9] Mittenhuber, D.: A globality theorem for Lie wedges that are bounded by a hyperplane-ideal. Seminar Sophus Lie2, 213-221 (1992) · Zbl 0788.22010
[10] Neeb, K.-H.: The Duality between Subsemigroups of Lie Groups and Monotone Functions. Transactions of the AMS329, 653-677 (1992) · Zbl 0747.22006 · doi:10.2307/2153957
[11] Penrose, R.: Techniques of Differential Topology in Relativity. SIAM Regional Conference Series in Applied Mathematics, (1972) · Zbl 0321.53001
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