Lorentzian geometry in the Lobachevsky plane

We consider left-invariant Lorentzian problems on the group of proper affine functions on the line. These problems have constant sectional curvature, thus are locally isometric to standard constant curvature Lorentzian manifolds (Minkowski space, de Sitter space, and anti-de Sitter space).
For these problems, the attainability set is described, existence of optimal trajectories is studied, a parameterization of Lorentzian length maximizers is obtained, and Lorentzian distance and spheres are described.
For zero curvature problem a global isometry into a half-plane of Minkowski plane is constructed.