Classical confinement of test particles in higher-dimensional models: stability criteria and a new energy condition

TL;DR

This paper investigates conditions for stable confinement of test particles in higher-dimensional spacetimes, deriving criteria based on geometric and energy considerations, with implications for braneworld models.

Contribution

It introduces a new stability criterion for particle confinement in higher-dimensional models and defines a confinement energy condition applicable to various geometries.

Findings

Confined paths are stable if gravitational stress-energy density on the embedding space exceeds that on the submanifold.

Explicit confirmation of the stability condition in Einstein spaces and 5D vacuum solutions.

Introduction of a confinement energy condition for classifying geometries with totally geodesic submanifolds.

Abstract

We review the circumstances under which test particles can be localized around a spacetime section \Sigma_0 smoothly contained within a codimension-1 embedding space M. If such a confinement is possible, \Sigma_0 is said to be totally geodesic. Using three different methods, we derive a stability condition for trapped test particles in terms of intrinsic geometrical quantities on \Sigma_0 and M; namely, confined paths are stable against perturbations if the gravitational stress-energy density on M is larger than that on \Sigma_0, as measured by an observed travelling along the unperturbed trajectory. We confirm our general result explicitly in two different cases: the warped-product metric ansatz for (n+1)-dimensional Einstein spaces, and a known solution of the 5-dimensional vacuum field equation embedding certain 4-dimensional cosmologies. We conclude by defining a confinement energy condition that can be used to classify geometries incorporating totally geodesic submanifolds, such as those found in thick braneworld and other 5-dimensional scenarios.