Classical confinement of test particles in higher-dimensional models: stability criteria and a new energy condition
Abstract
We review the circumstances under which test particles can be localized around a spacetime section 0 smoothly contained within a codimension-1 embedding space M. If such a confinement is possible, 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 0 and M; namely, confined paths are stable against perturbations if the gravitational stress-energy density on M is larger than that on 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.
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