Delta function singularities in the Weyl tensor at the brane

TL;DR

This paper identifies a delta function singularity in the Weyl tensor at a brane within higher-dimensional spacetime, clarifying its impact on the gravitational equations derived by Shiromizu, Maeda, and Sasaki.

Contribution

It explicitly evaluates a delta function singularity effect at the brane that was not discussed in prior work, confirming it does not alter the main results.

Findings

Identified a delta function singularity in the Weyl tensor at the brane.

Evaluated the effect of this singularity on the gravitational equations.

Confirmed the singularity does not modify the previously derived equations.

Abstract

In a recent paper Shiromizu, Maeda and Sasaki derived the gravitational equations of motion which would hold on a brane which is embedded in a higher dimensional bulk spacetime, showing that even when the Einstein equations are imposed in the bulk, nonetheless the embedding leads to a modification of the Einstein equations on the brane. In this comment on their work we explicitly identify and evaluate a delta function singularity effect at the brane which they do not appear to have discussed in their paper, an effect which, while actually being of interest in and of itself, nonetheless turns out not to modify their reported results.