Billiard in the space with a time machine

We study a system of an elastic ball moving in the non-relativistic spacetime with a nontrivial causal structure produced by a wormhole-based time machine. For such a system it is possible to formulate a simple model of the so-called grandfather paradox': for certain paradoxical’ initial conditions the standard straight trajectory of the ball would self-collide inconsistently. We analyze globally consistent solutions of local equations of motion, namely, we find all trajectories with one self-collision. It is demonstrated that all standard initial conditions have a consistent evolution, including those `paradoxical’ ones, for which the inconsistent collision-free trajectory is superseded by a special consistent self-colliding trajectory. Moreover, it is shown that for a wide class of initial conditions more than one globally consistent evolution exist. The nontrivial causal structure thus breaks the uniqueness of the classical theory even for locally deterministic physical laws.