Brane cosmological evolution with a general bulk matter configuration

Using a fully covariant treatment for the description of the bulk geometry, we study the brane cosmological evolution in the presence of a smooth bulk matter distribution. We focus on the case of a Friedmann-Robertson-Walker (FRW) brane, invariantly characterized by the existence of a six-dimensional group of isometries acting on 3D spacelike orbits. With a FRW brane, the bulk geometry can be regarded as the 5D generalization of the \emph{inhomogeneous orthogonal family of Locally Rotationally Symmetric (LRS)} spacetimes. We show that, for \emph{any bulk matter configuration}, the expansion rate on the brane depends only on the covariantly defined \emph{comoving mass} $\mathcal{M}$ of the bulk fluid within a radius equal to the average length scale of the 3D spacelike hypersurfaces of constant curvature. This unique contribution incorporates the effects of the 5D Weyl tensor and the projected tensor related to the bulk matter, and gives a transparent physical picture that includes an effective conservation equation between the brane and the bulk matter.