One-parametric series of SO(1,1)-symmetric (sub-)Lorentzian structures on the universal covering of SL(2,R)

We consider a one-parametric series of left-invariant Lorentzian structures on the universal covering of the Lie group SL(2,R). These structures have SO(1,1)-symmetry and are deformations of the anti-de Sitter Lorentzian manifold. We study the global optimality of geodesics, i.e., we describe the longest arcs. The sub-Lorentzian structure appears as a limit case of the considered series of Lorentzian structures. We study how the several properties of the Lorentzian structures deform to the properties of the sub-Lorentzian structure.