Matching conditions for a brane of arbitrary codimension

TL;DR

This paper derives matching conditions for branes of arbitrary codimension in Lovelock gravity, showing how distributional defects and induced gravity emerge, especially for even codimension, with implications for brane-world models.

Contribution

It provides a general framework for matching conditions in Lovelock gravity for arbitrary codimension branes, including new insights into distributional defects and induced gravity effects.

Findings

Distributional defects of higher codimension exist in Lovelock theories.

Matching conditions depend on the parity of the brane codimension.

Induced Einstein tensor appears for branes with codimension greater than 2.

Abstract

We present matching conditions for distributional sources of arbitrary codimension in the context of Lovelock gravity. Then we give examples, treating maximally symmetric distributional p-branes, embedded in flat, de Sitter and anti-de Sitter spacetime. Unlike Einstein theory, distributional defects of locally smooth geometry and codimension greater than 2 are demonstrated to exist in Lovelock theories. The form of the matching conditions depends on the parity of the brane codimension. For odd codimension, the matching conditions involve discontinuities of Chern-Simons forms and are thus similar to junction conditions for hypersurfaces. For even codimension, the bulk Lovelock densities induce intrinsic Lovelock densities on the brane. In particular, this results in the appearance of the induced Einstein tensor for p>2. For the matching conditions we present, the effect of the bulk is reduced to an overall topological solid angle defect which sets the Planck scale on the brane and to extrinsic curvature terms. Moreover, for topological matching conditions and constant solid angle deficit, we find that the equations of motion are obtained from an exact p+1 dimensional action, which reduces to an induced Lovelock theory for large codimension. In essence, this signifies that the distributional part of the Lovelock bulk equations can naturally give rise to induced gravity terms on a brane of even co-dimension. We relate our findings to recent results on codimension 2 branes.