Properties of reachable sets in the sub-Lorentzian geometry

The aim of this paper is to develop local theory of future timelike, nonspacelike and null reachable sets from a given point _q0$q_0$ in the sub-Lorentzian geometry. In particular, we prove that if U$U$ is a normal neighbourhood of _q0$q_0$ then the three reachable sets, computed relative to U$U$, have identical interiors and boundaries with respect to U$U$. Further, among other things, we show that for Lorentzian metrics on contact distributions on R²ⁿ⁺¹${\mathbb{R}}^{2n+1}$, n≥1$n\ge 1$, the boundary of reachable sets from _q0$q_0$ is, in a neighbourhood of _q0$q_0$, made up of null future directed curves starting from _q0$q_0$. Every such curve has only a finite number of non-smooth points; smooth pieces of every such curve are Hamiltonian geodesics. For general sub-Lorentzian structures, contrary to the Lorentzian case, timelike curves may appear on the boundary. It turns out that such curves are always Goh curves. We also generalize a classical result on null Lorentzian geodesics: every null future directed Hamiltonian sub-Lorentzian geodesic initiating at _q0$q_0$ is contained, at least to a certain moment of time, in the boundary of the reachable set from _q0$q_0$.