Pure Geometrical Evolution of the Multidimensional Universe

Abstract

An exhaustive qualitative analysis of cosmological evolution for some multidimensional universes is given. The internal space is taken to be a compact Lie group Riemannian manifold. The space is generically anisotropic; i.e., its cosmological evolution is described by its (time-dependent) volume, the dilaton, and by relative anisotropic deformation factors representing the shear of the internal dimensions during the evolution. Neither the internal space nor its subspaces need to be Einstein spaces. The total spacetime is empty, and the cosmic evolution of the external, four-dimensional world is driven by the geometric “matter” consisting of the dilaton and of the deformation factors. Since little is known about any form of matter in the extra dimensions, we do not introduce anyad hocmatter content of the Universe. We derive the four-dimensional Einstein field equations (with a cosmological term) for these geometric sources in full generality, i.e., for any compact Lie group. A detailed analysis is done for some specific internal geometries: products of 3-spheres, andSU(3) space. Asymptotic solutions exhibit power law inflation along with a process of full or partial isotropization. For theSU(3) space all the deformation factors tend to a common value, whereas in the case ofS3's each sphere isotropizes separately.