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

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 $\mathrm{}(n+1\mathrm{})\mathrm{}$-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.