Gravity, stability, and energy conservation on the Randall-Sundrum brane world - art. no. 024008

We carefully investigate the gravitational perturbation of the Randall-Sund rum (RS) single brane-world solution [L. Randell and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999)], based on a covariant curvature tensor formalism re cently developed by us. Using this curvature formalism, it is known that th e "electric" part of the five-dimensional Weyl tensor, denoted by E-mu nu, gives the leading order correction to the conventional Einstein equations o n the brane. We consider the general solution of the perturbation equations for the five-dimensional Weyl tensor caused by the matter fluctuations on the brane. By analyzing its asymptotic behavior in the direction of the fif th dimension, we find the curvature invariant diverges as we approach the C auchy horizon. However, in the limit of asymptotic future in the vicinity o f the Cauchy horizon, the curvature invariant falls off fast enough to rend er the divergence harmless to the brane world. We also obtain the asymptoti c behavior of E-mu nu on the brane at spatial infinity, assuming that the m atter perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved quantities at spatial infinity. This indicat es strongly that the usual conservation law, such as the ADM energy conserv ation, holds on the brane as far as asymptotically flat spacetimes are conc erned.