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

TL;DR

This paper explores a one-parameter family of SO(1,1)-symmetric Lorentzian structures on the universal cover of SL(2,R), analyzing their extremal trajectories and the transition to sub-Lorentzian structures.

Contribution

It introduces a new series of Lorentzian structures with SO(1,1) symmetry and examines their deformation into sub-Lorentzian structures, including optimality of trajectories.

Findings

Describes the longest extremal trajectories in the Lorentzian structures.

Shows how properties deform from Lorentzian to sub-Lorentzian structures.

Identifies the sub-Lorentzian structure as a limit case of the series.

Abstract

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 they are deformations of the anti-de Sitter Lorentzian manifold. We study the global optimality of extremal trajectories, 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.