Point particle motion in topologically nontrivial space-times

TL;DR

This paper investigates how different topologies of space, like tori and Klein bottles, influence point-particle motion and field interactions, revealing effects like velocity-dependent forces and stable points due to topological symmetry breaking.

Contribution

It provides a detailed analysis of particle dynamics in nontrivial topologies, highlighting the emergence of forces and potentials caused by topological effects, which were not previously well understood.

Findings

Particles can move at constant velocity on a torus despite broken Lorentz invariance.

Acceleration induces non-local, velocity-dependent forces in toroidal topologies.

Topologies like the Klein bottle generate effective potentials that prevent uniform motion.

Abstract

It is well known that compactifying a space can break symmetries that are present in the covering space. In this paper we study the effects of such topological symmetry breaking on point-particle motion when the particle is coupled to a massless field on the space. For a torus topology where Lorentz invariance is broken but translation invariance is maintained, particles can move at a constant velocity through the space; however, non-local, velocity-dependent forces arise whenever the particle is accelerated. For a topology where translation invariance is broken, such as the Klein bottle, interactions with the massless field generate an effective potential as a function of position. The potential creates special stable points in the space, and prevents constant velocity motion. This latter would appear to be the generic case. This class of effects may be applicable whenever a localized object moves through a compactified bulk, such as in brane-world cosmology, or some condensed matter systems.