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

A. V. Podobryaev

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 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.

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

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 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.
Paper Structure (11 sections, 25 theorems, 104 equations, 12 figures)

This paper contains 11 sections, 25 theorems, 104 equations, 12 figures.

Table of Contents

  1. Optimal control problem statement
  2. Geodesic equations
  3. The oblate case
  4. The attainable set
  5. Conjugate points
  6. Optimality of geodesics
  7. The sub-Lorentzian case
  8. The prolate case
  9. The attainable set
  10. The conjugate points
  11. The Maxwell set

Key Result

Proposition 1

(1) The extremals are solutions of the following Hamiltonian system corresponding to the maximized Hamiltonian $H = \frac{1}{2}\left( -\frac{h_1^2}{I_1} + \frac{h_2^2}{I_1} + \frac{h_3^2}{I_3} \right)$ (2) The initial covectors of normal extremals (corresponding to $\nu = 1$) are located on the half of the hyperboloid $H = -\frac{1}{2}$, $h_1 < 0$. These normal extremals are time-like and have uni

Figures (12)

  • Figure 1: The phase portraits for the Hamiltonian system for covectors. The normal case corresponding to time-like geodesics (left) and the abnormal case corresponding to light-like geodesics (right).
  • Figure 2: Initial covectors for different types of geodesics. The section $h_2 = 0$. Solid lines are corresponding to the hyperboloid $H(h) = -\frac{1}{2}$ and the cone $H(h) = 0$. Dashed lines are corresponding to the hyperboloids $|h| = 1$, or, equivalently, $\mathop{\mathrm{Kil}}\nolimits{(h)} = \pm 1$.
  • Figure 3: The location of the first positive root of the equation \ref{['eq-tgth']}.
  • Figure 4: Orbits of the $\mathrm{SO}_{1,1}$-action at the group $G$ for different sections $\mathop{\mathrm{Im}}\nolimits{w} = \mathrm{const}$.
  • Figure 5: The Maxwell time corresponding to the reflection with respect to the plane $h_3 = 0$ is greater or equal to the Maxwell time corresponding to the hyperbolic rotations.
  • ...and 7 more figures

Theorems & Definitions (61)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • Remark 3
  • Proposition 3
  • proof
  • Lemma 1
  • ...and 51 more