Three-Dimensional Billiards with Time Machine

TL;DR

This paper explores the dynamics of a classical point particle in a spacetime with closed time-like curves, analyzing self-collisions and their equivalence to collisions with identical particles, using a three-dimensional billiard model with a wormhole time machine.

Contribution

It introduces a detailed analysis of self-collisions in a 3D time-machine spacetime and demonstrates the existence of equivalent configurations involving identical particles.

Findings

Self-collision configurations are explicitly constructed.

Multiple wormhole traversals are analyzed.

Equivalent scenarios with identical particles are identified.

Abstract

Self-collision of a non-relativistic classical point-like body, or particle, in the spacetime containing closed time-like curves (time-machine spacetime) is considered. A point-like body (particle) is an idealization of a small ideal elastic billiard ball. The known model of a time machine is used containing a wormhole leading to the past. If the body enters one of the mouths of the wormhole, it emerges from another mouth in an earlier time so that both the particle and its "incarnation" coexist during some time and may collide. Such self-collisions are considered in the case when the size of the body is much less than the radius of the mouth, and the latter is much less than the distance between the mouths. Three-dimensional configurations of trajectories with a self-collision are presented. Their dynamics is investigated in detail. Configurations corresponding to multiple wormhole traversals are discussed. It is shown that, for each world line describing self-collision of a particle, dynamically equivalent configurations exist in which the particle collides not with itself but with an identical particle having a closed trajectory (Jinnee of Time Machine).