Nonconservation of Energy and Loss of Determinism I. Infinitely Many Balls

TL;DR

This paper demonstrates that in an infinite system of elastically colliding balls, energy and momentum may not be conserved globally and the process can be indeterministic, challenging classical assumptions.

Contribution

It reveals that infinite elastic collisions can lead to nonconservation of energy and momentum and introduces indeterminism in classical and relativistic systems.

Findings

Energy and momentum are not necessarily conserved globally.

The system exhibits indeterminism with arbitrary energy injection.

Examples show all balls can move with the same velocity after collisions.

Abstract

An infinite number of elastically colliding balls is considered in a classical, and then in a relativistic setting. Energy and momentum are not necessarily conserved globally, even though each collision does separately conserve them. This result holds in particular when the total mass of all the balls is finite, and even when the spatial extent and temporal duration of the process are also finite. Further, the process is shown to be indeterministic: there is an arbitrary parameter in the general solution that corresponds to the injection of an arbitrary amount of energy (classically), or energy-momentum (relativistically), into the system at the point of accumulation of the locations of the balls. Specific examples are given that illustrate these counter-intuitive results, including one in which all the balls move with the same velocity after every collision has taken place.