Second order perturbations in the Randall-Sundrum infinite brane-world model

Abstract

We discuss the nonlinear gravitational interactions in the Randall-Sundrum single brane model. If we naively write down the four-dimensional effective action integrating over the fifth dimension with the aid of the decomposition with respect to the eigenmodes of four-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 three-brane under the assumption of the static and axial-symmetric five-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 that is obtained by the zero mode truncation. We confirm that the four-dimensional Einstein gravity is approximately recovered on the three-brane up to second order perturbations.