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

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