Gravity, Stability and Energy Conservation on the Randall-Sundrum Brane-World

TL;DR

This paper investigates gravitational perturbations in the Randall-Sundrum brane-world, analyzing the behavior of the Weyl tensor and curvature invariants to assess stability, energy conservation, and the impact of matter fluctuations.

Contribution

It provides a detailed analysis of gravitational perturbations using a covariant formalism, showing the asymptotic behavior of the Weyl tensor and confirming energy conservation on the brane.

Findings

Curvature invariant diverges near the Cauchy horizon but is ultimately harmless.

The Weyl tensor perturbation falls off at spatial infinity, preserving conserved quantities.

Energy conservation laws like ADM hold in asymptotically flat brane-worlds.

Abstract

We carefully investigate the gravitational perturbation of the Randall-Sundrum (RS) single brane-world solution [hep-th/9906064], based on a covariant curvature tensor formalism recently developed by us. Using this curvature formalism, it is known that the `electric' part of the 5-dimensional Weyl tensor, denoted by $E_{\mu\nu}$, gives the leading order correction to the conventional Einstein equations on the brane. We consider the general solution of the perturbation equations for the 5-dimensional Weyl tensor caused by the matter fluctuations on the brane. By analyzing its asymptotic behaviour in the direction of the 5th dimension, we find the curvature invariant diverges as we approach the Cauchy horizon. However, in the limit of asymptotic future in the vicinity of the Cauchy horizon, the curvature invariant falls off fast enough to render the divergence harmless to the brane-world. We also obtain the asymptotic behavior of $E_{\mu\nu}$ on the brane at spatial infinity, assuming the matter perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved quantities at spatial infinity. This indicates strongly that the usual conservation law, such as the ADM energy conservation, holds on the brane as far as asymptotically flat spacetimes are concerned.