Billiard in the space with a time machine

TL;DR

This paper investigates the behavior of an elastic ball in a spacetime with a wormhole-based time machine, demonstrating that multiple consistent solutions can exist for paradoxical initial conditions, challenging classical determinism.

Contribution

It provides a detailed analysis of self-colliding trajectories in a nontrivial causal structure, showing the existence of multiple solutions and resolving the grandfather paradox.

Findings

All paradoxical initial conditions have at least one consistent solution.

Multiple globally consistent evolutions can exist for the same initial data.

The causal structure breaks the uniqueness of classical solutions.

Abstract

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a consistent evolution, including those `paradoxical' ones, for which the inconsistent collision-free trajectory is superseded by a special consistent self-colliding trajectory.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.