Einstein gravity on an even codimension brane

TL;DR

This paper derives the equations of motion for a self-gravitating Dirac p-brane in even codimension spacetimes governed by Lovelock gravity, revealing conditions under which induced Lovelock theories emerge on the brane.

Contribution

It provides a comprehensive derivation of brane dynamics with Lovelock gravity, including matching conditions and the role of angle defects, extending previous models to higher codimensions.

Findings

Equations of motion involve induced Lovelock densities and extrinsic curvature terms.

Extrinsic curvature terms cancel when codimension exceeds worldvolume dimension, yielding pure Lovelock theory.

A 3-brane in 8 or 10 dimensions obeys Einstein's equations with a cosmological constant.

Abstract

We give the equations of motion for a self-gravitating Dirac p-brane embedded in an even co-dimension spacetime. The dynamics of the bulk are dictated by Lovelock gravity and permit matching conditions, even when the codimension is strictly greater than 2. We show that the equations of motion involve both induced Lovelock densities on the brane and regular extrinsic curvature terms. The brane dynamics can be derived from an exact (p+1)-dimensional induced action. The Dirac charge is carried by an overall (solid) angle defect which sets the Planck scale on the brane. In particular, if the codimension is greater than the worldvolume dimension of the p-brane, we show that the extrinsic curvature terms cancel, leaving an exact induced Lovelock theory. For example, a 3-brane embedded in 8 or 10 dimensions obeys Einstein's equations with a cosmological constant.