  In brane world scenarios with a bulk scalar field between two branes it is
known that 4-dimensional Einstein gravity is restored at low energies on either
brane. By using a gauge-invariant gravitational and scalar perturbation
formalism we extend the theory of weak gravity in the brane world scenarios to
higher energies, or shorter distances. We argue that weak gravity on either
brane is indistinguishable from 4-dimensional higher derivative gravity,
provided that the inter-brane distance (radion) is stabilized, that the
background bulk scalar field is changing near the branes and that the
background bulk geometry near the branes is warped. This argument holds for a
general conformal transformation to a frame in which matter on the branes is
minimally coupled to the metric. In particular, Newton's constant and the
coefficients of curvature-squared terms in the 4-dimensional effective action
are determined up to an ambiguity of adding a Gauss-Bonnet topological term. In
other words, we provide the brane-world realization of the so called
$R^2$-model without utilizing a quantum theory. We discuss the appearance of
composite spin-2 and spin-0 fields in addition to the graviton on the brane and
point out a possibility that the spin-0 field may play the role of an effective
inflaton to drive brane-world inflation. Finally, we conjecture that the
sequence of higher derivative terms is an infinite series and, thus, indicates
non-locality in the brane world scenarios.
