Sub-Lorentzian Extremals Defined by an Antinorm

We consider a left-invariant sub-Lorentzian structure on a Lie group. This structure isassumed to be defined by a closed convex salient cone in the corresponding Lie algebra and acontinuous antinorm associated with this cone. We derive the Hamiltonian system forsub-Lorentzian extremals and give conditions under which normal extremal trajectories keep theircausal type. Tangent vectors of abnormal extremal trajectories are either lightlike or are tangentvectors of sub-Riemannian abnormal extremal trajectories for the sub-Riemannian distributionspanned by the cone.