On the causal structure of Lorentzian Lie groups. A globality theorem for Lorentzian cones in certain solvable Lie algebras (Q1923234)

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.