Sub-Lorentzian geodesics on ${\mathrm{GL}}^{+}(2,\mathbb{C})$ with the generating space of Hermitian matrices in the Lie algebra ${\mathfrak{gl}}^{+}(2,\mathbb{C})$

Abstract

The Lie subgroup ${\mathrm{GL}}^{+}(2,\mathbb{C})$ of all matrices in the Lie group $\mathrm{GL}(2,\mathbb{C})$ with positive real determinant is equipped with a left-invariant sub-Lorentzian (anti)metric defined by the natural structure of the 4-dimensional Minkowski space-time on the subspace of Hermitian matrices in its Lie algebra. In base of the corresponding time-anti-optimal control problem, formulated in the paper, and Pontryagin minimum principle for it, using geodesics and shortest arcs of the corresponding left-invariant sub-Riemannian metric on the Lie subgroup $\mathrm{SL}(2,\mathbb{C})$, the authors found sub-Lorentzian nonspacelike geodesics and longest arcs.

27 pages