Sub-Lorentzian Extremals Defined by an Antinorm

Abstract

We consider a left-invariant sub-Lorentzian structure on a Lie group. This structure is assumed to be defined by a closed convex salient cone in the corresponding Lie algebra and a continuous antinorm associated with this cone. We derive the Hamiltonian system for sub-Lorentzian extremals and give conditions under which normal extremal trajectories keep their causal type. Tangent vectors of abnormal extremal trajectories are either lightlike or are tangent vectors of sub-Riemannian abnormal extremal trajectories for the sub-Riemannian distribution spanned by the cone.