  We discuss the non-linear gravitational interactions in the Randall-Sundrum
single brane model. If we naively write down the 4-dimensional effective action
integrating over the fifth dimension with the aid of the decomposition with
respect to eigen modes of 4-dimensional d'Alembertian, the Kaluza-Klein mode
coupling seems to be ill-defined. We carefully analyze second order
perturbations of the gravitational field induced on the 3-brane under the
assumption of the static and axial-symmetric 5-dimensional metric. It is shown
that there remains no pathological feature in the Kaluza-Klein mode coupling
after the summation over all different mass modes. Furthermore, the leading
Kaluza-Klein corrections are shown to be sufficiently suppressed in comparison
with the leading order term which is obtained by the zero mode truncation. We
confirm that the 4-dimensional Einstein gravity is approximately recovered on
the 3-brane up to second order perturbations.
