Quadrupole formula for Kaluza-Klein modes in the braneworld

TL;DR

This paper derives a quadrupole formula for Kaluza-Klein modes in the Randall-Sundrum braneworld, revealing corrections to gravitational fields and energy flux behavior of KK modes compared to standard gravitational waves.

Contribution

It introduces a new quadrupole formula for KK modes in braneworld models, providing insights into their gravitational effects and energy flux characteristics.

Findings

KK modes induce a 1/r^2 correction to gravitational fields.

Flux of KK modes vanishes at infinity for non-spherical sources.

Effective energy flux can be negative depending on source motion.

Abstract

The quadrupole formula in four-dimensional Einstein gravity is a useful tool to describe gravitational wave radiation. We derive the quadrupole formula for the Kaluza-Klein (KK) modes in the Randall-Sundrum braneworld model. The quadrupole formula provides transparent representation of the exterior weak gravitational field induced by localized sources. We find that a general isolated dynamical source gives rise to the 1/r^2 correction to the leading 1/r gravitational field. We apply the formula to an evaluation of the effective energy carried by the KK modes from the viewpoint of an observer on the brane. Contrary to the ordinary gravitational waves (zero mode), the flux of the induced KK modes by the non-spherical part of the quadrupole moment vanishes at infinity and only the spherical part contributes to the flux. Since the effect of the KK modes appears in the linear order of the metric perturbations, the effective energy flux observed on the brane is not always positive, but can become negative depending on the motion of the localized sources.