  Four dimensional gravity in the low energy limit of a higher dimensional
theory has been expected to be a (generalized) Brans-Dicke theory. A subtle
point in brane world scenarios is that the system of four dimensional effective
gravitational equations is not closed due to bulk gravitational waves and bulk
scalars. Nonetheless, weak gravity on the brane can be analyzed completely. We
revisit the theory of weak brane gravity using gauge-invariant gravitational
and scalar perturbations around a background warped geometry with a bulk scalar
between two flat branes. We obtain a simple condition for the radion
stabilization in terms of the scalar field potentials. We show that for general
potentials of the scalar field which provides radion stabilization and a
general conformal transformation to a frame in which matter on the branes are
minimally coupled to the metric, 4-dimensional Einstein gravity, not BD
gravity, is restored at low energies on either brane. In contrast, in RS brane
world scenario without a bulk scalar, low energy gravity is BD one. We
conjecture that in general brane world scenarios with more than one scalar
field, one will again encounter the situation that low energy gravity is not
described by the Einstein theory. Equipped with the weak gravity results, we
discuss the properties of 4d brane gravitational equations, in particular, the
value and sign of 4d Newton's gravitational coupling.
