Billiards and toy gravitons

TL;DR

This paper explores the dynamics of one-dimensional elastic collisions involving positive and negative mass particles, revealing their equivalence to billiard systems and uncovering unique behaviors like attraction and continuous acceleration.

Contribution

It introduces a novel framework linking negative mass collisions to billiard systems and characterizes conditions for finite collisions and continuous acceleration effects.

Findings

Negative mass particles induce attraction between positive mass particles.

Finitely many collisions occur under specific conditions.

Negative mass leads to continuous acceleration with a $U(r)=-k/r^2$ potential.

Abstract

In this article we study the one-dimensional dynamics of elastic collisions of particles with positive and negative mass. We show that such systems are equivalent to billiards induced by an inner product of possibly indefinite signature, we characterize the systems with finitely many collisions and we prove that a small particle of negative mass between two particles of positive mass acts like an attracting particle with discrete acceleration (at the collisions) provided that the total kinetic energy is negative. In the limit of the negative mass going to zero, with fixed negative kinetic energy, we obtain a continuous acceleration with potential energy of the form $U(r)=-k/r^2$.