Classical Stabilization of Homogeneous Extra Dimensions

TL;DR

This paper investigates the classical stability of large, homogeneous extra dimensions in general relativity, showing that only positively curved extra dimensions can be stable under small perturbations.

Contribution

It demonstrates that stable, static, homogeneous extra dimensions require positive curvature, challenging prior assumptions and clarifying conditions for classical stabilization.

Findings

Only positively curved extra dimensions are classically stable.

Static solutions with nonnegative Ricci curvature are highly nontrivial.

Intuition from non-gravitational physics can be misleading in gravitational contexts.

Abstract

If spacetime possesses extra dimensions of size and curvature radii much larger than the Planck or string scales, the dynamics of these extra dimensions should be governed by classical general relativity. We argue that in general relativity, it is highly nontrivial to obtain solutions where the extra dimensions are static and are dynamically stable to small perturbations. We also illustrate that intuition on equilibrium and stability built up from non-gravitational physics can be highly misleading. For all static, homogeneous solutions satisfying the null energy condition, we show that the Ricci curvature of space must be nonnegative in all directions. Much of our analysis focuses on a class of spacetime models where space consists of a product of homogeneous and isotropic geometries. A dimensional reduction of these models is performed, and their stability to perturbations that preserve the spatial symmetries is analyzed. We conclude that the only physically realistic examples of classically stabilized large extra dimensions are those in which the extra-dimensional manifold is positively curved.