Second Order Perturbations in the Randall-Sundrum Infinite Brane-World Model

TL;DR

This paper investigates second order gravitational perturbations in the Randall-Sundrum brane-world model, demonstrating the absence of pathological mode couplings and confirming the approximate recovery of 4D Einstein gravity on the brane.

Contribution

It provides a detailed analysis of second order perturbations, clarifying the behavior of Kaluza-Klein modes and their suppression, which was previously unclear.

Findings

No pathological Kaluza-Klein mode coupling after summation.

Kaluza-Klein corrections are suppressed compared to zero mode.

4D Einstein gravity is approximately recovered up to second order.

Abstract

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.