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

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.

arXiv comment8 pages, 1 figure. To appear in General Relativity and Gravitation as a contributed paper to Mashhoon Festschrift

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• Mach's Principle and Higher-Dimensional Dynamics
• Particle motion in the field of a five-dimensional charged black hole
• Five-dimensional warped product space-time with time-dependent warp factor and cosmology of the four-dimensional universe
• On the classical confinement of test particles to a thin 3-brane in the absence of non-gravitational forces
• Metamaterial branes