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We carefully investigate the gravitational perturbation of theRandall-Sundrum (RS) single brane-world solution [hep-th/9906064], based on acovariant curvature tensor formalism recently developed by us. Using thiscurvature formalism, it is known that the `electric' part of the 5-dimensionalWeyl tensor, denoted by $E_{\mu\nu}$, gives the leading order correction to theconventional Einstein equations on the brane. We consider the general solutionof the perturbation equations for the 5-dimensional Weyl tensor caused by thematter fluctuations on the brane. By analyzing its asymptotic behaviour in thedirection of the 5th dimension, we find the curvature invariant diverges as weapproach the Cauchy horizon. However, in the limit of asymptotic future in thevicinity of the Cauchy horizon, the curvature invariant falls off fast enoughto render the divergence harmless to the brane-world. We also obtain theasymptotic behavior of $E_{\mu\nu}$ on the brane at spatial infinity, assumingthe matter perturbation is localized. We find it falls off sufficiently fastand will not affect the conserved quantities at spatial infinity. Thisindicates strongly that the usual conservation law, such as the ADM energyconservation, holds on the brane as far as asymptotically flat spacetimes areconcerned.Comment: 10 pages, references adde

Gravity, stability, and energy conservation on the Randall-Sundrum brane world