Generalized Friedmann branes

We prove that for a large class of generalized Randall-Sundrum type II models the characterization of the brane-gravity sector by the effective Einstein equation, Codazzi equation and the twice-contracted Gauss equation is equivalent to the bulk Einstein equation. We give the complete set of equations in the generic case of non-${Z}_{2}$-symmetric bulk and arbitrary energy-momentum tensors both on the brane and in the bulk. Among these, the effective Einstein equation contains a varying cosmological ``constant'' and two new source terms. The first of these represents the deviation from ${Z}_{2}$ symmetry, while the second arises from the bulk energy-momentum tensor. We apply the formalism for the case of a perfect fluid on a Friedmann brane embedded in a generic bulk. The generalized Friedmann and Raychaudhuri equations are given in a form independent of both the embedding and the bulk matter. They contain two new functions obeying a first order differential system, both depending on the bulk matter and the embedding. Then we focus on Friedmann branes separating two nonidentical (inner or outer) regions of Reissner--Nordstr\"om--anti-de Sitter bulk space-times, generalizing previous non-${Z}_{2}$-symmetric treatments. Finally the analysis is repeated for the Vaidya--anti-de Sitter bulk space-time, allowing for both ingoing and outgoing radiation in each region.