Geodesic motion in the neighbourhood of submanifolds embedded in warped product spaces

TL;DR

This paper analyzes the geodesic motion of particles and photons in higher-dimensional warped spaces, identifying conditions for confinement to four-dimensional spacetime and examining stability, with implications for models involving extra dimensions.

Contribution

It provides a general analysis of geodesic confinement and stability conditions in warped product spaces, extending previous results to higher codimensions and arbitrary extrinsic curvature.

Findings

Conditions for particle and photon confinement to 4D spacetime.

Motion of photons is unaffected by extrinsic curvature.

Established stability criteria for confined geodesics.

Abstract

We study the classical geodesic motions of nonzero rest mass test particles and photons in (3+1+n)- dimensional warped product spaces. An important feature of these spaces is that they allow a natural decoupling between the motions in the (3+1)-dimensional spacetime and those in the extra n dimensions. Using this decoupling and employing phase space analysis we investigate the conditions for confinement of particles and photons to the (3+1)- spacetime submanifold. In addition to providing information regarding the motion of photons, we also show that these motions are not constrained by the value of the extrinsic curvature. We obtain the general conditions for the confinement of geodesics in the case of pseudo-Riemannian manifolds as well as establishing the conditions for the stability of such confinement. These results also generalise a recent result of the authors concerning the embeddings of hypersurfaces with codimension one.