Lorentzian distance on the Lobachevsky plane
The problem considered is that of admissible geodesics of a Lorentzian structure; here admissible means time-like, in relativistic terms. The problem is formulated as an optimal control problem, with the time-like constraint then formulated as a constraint on the controls. Many ideas are initially presented in a general manner (with references to their more complete presentation), but the new results in the paper are those pertaining to Lorentzian structures on the space of orientation-preserving affine mappings of \(\mathbb{R}\), this space being a Lie group diffeomorphic to the upper half-plane. The Lorentzian structures considered are left-invariant.\N\NThe author uses the Pontryagin maximum principle to study the problem of the existence and character of length minimizing time-like curves. The emphasis is on explicit characterization of geometric properties like curvature, geodesic connectedness, completeness, distance, spheres, isometries, etc. The paper closes with a particularly detailed analysis of a few particular left-invariant Lorentzian structures on the upper half-plane.\N\NFor a reader who is familiar with the Pontryagin maximum principle, the paper is quite easy to follow. The techniques are all ``elementary'' (not intended in a pejorative sense), and the paper is very well written.
scientific article; zbMATH DE number 7896366
scientific article