Codimension Two Branes and Distributional Curvature

TL;DR

This paper demonstrates how to concentrate curvature and stress-energy on codimension two submanifolds in general relativity, extending the formalism beyond the well-understood codimension one case, with applications to branes and cosmology.

Contribution

It introduces a class of metrics allowing distributional curvature on codimension two surfaces, linking to self-gravitating branes with specific equations of state.

Findings

Distributional curvature exists on codimension two submanifolds.

Stress-energy corresponds to a brane with p = -ρ equation of state.

Applicable to brane world scenarios and deSitter cosmology.

Abstract

In general relativity, there is a well-developed formalism for working with the approximation that a gravitational source is concentrated on a shell, or codimension one surface. By contrast, there are obstacles to concentrating sources on surfaces that have a higher codimension, for example, a string in a spacetime with dimension greater than or equal to four. Here it is shown that, by giving up some of the generality of the codimension one case, curvature can be concentrated on submanifolds that have codimension two. A class of metrics is identified such that (1) the scalar curvature and Ricci densities exist as distributions with support on a co-dimension two submanifold, and (2) using the Einstein equation, the distributional curvature corresponds to a concentrated stress-energy with equation of state p equals minus the energy density, where p is the isotropic pressure tangent to the submanifold. This is the appropriate stress-energy to describe a self-gravitating brane that is governed by an area action, or a brane world deSitter cosmology. The possibility of having a different equation of state arise from a wider class of metrics is discussed.