Classical Stabilization of Homogeneous Extra Dimensions

TL;DR

The paper studies whether large extra dimensions can be stabilized in classical general relativity under the null energy condition. It shows that for static, homogeneous spacetimes, each spatial factor must have nonnegative curvature, implying that stable large extra dimensions require positive curvature and cannot be flat or negatively curved under NEC. By dimensional reduction, these models become four-dimensional theories with one or more scalar fields (radions) with self-interaction potentials; a concrete $S^2$ example with flux demonstrates a local minimum that is dynamically unstable to perturbations, leading to singular behavior, though the four-dimensional cosmology can be acceptable. Extending to multiple compact spaces, the analysis again requires positive curvature factors and indicates that barrier heights can be at TeV scales, avoiding Brustein–Steinhardt type instabilities.

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.

Figures (1)

• Figure 1: The effective potential (\ref{['potential2']}) of the radion in the model described in the text.