Five-Dimensional Warped Product Space-Time with Time-Dependent Warping and a Scalar Field in the Bulk

Abstract

We consider gravity in a five-dimensional warped product space-time, with a time-dependent warp factor and a time-dependent extra dimension. The five-dimensional field equations are derived for a spatially flat FRW brane, and the energy conditions and the nature of bulk geometry are examined. It is found that the expansion of the four-dimensional universe depends on its location along the extra dimension and is different at different locations in the bulk space-time. At low energies, the trapping of fields within the brane implies a specific correlation between the warp factor and the extra-dimensional scale factor. Generally, the bulk is not conformally flat. At high energies, the bulk is assumed to be sourced by a scalar field with self-interaction. The analysis shows that the potential of the scalar field source of gravity at a given position along the fifth dimension is related to the Hubble parameter on the brane at that position in the bulk.