Linearized gravity in the Randall-Sundrum scenario

In the literature about the Randall-Sundrum scenario, one finds, on the one hand, that there exist (small) corrections to Newton's law of gravity on the brane and, on the other, that the exact (and henceforth linearized) Einstein equations can be recovered on the brane. The explanation for these seemingly contradictory results is that the behavior of the bulk far from the brane is different in both models. We show that explicitly in this paper.