Surfaces of discontinuity in five-dimensional Kaluza-Klein cosmologies

Abstract

Based on Israel's work on surface layers in general relativity, we study a 4-dimensional Robertson-Walker space-time as a singular hypersurface of discontinuity Σ embedded in a matter-free 5-dimensional Kaluza-Klein cosmology with non-vanishing cosmological constant. As in the case of vacuum bubbles in phase transitions, the 4-dimensional hypersurface Σ has contributions from a jump discontinuity and a δ-function singularity in the cosmological constant and the 5-dimensional stress-energy tensor, respectively. Since under this model our observable 4-dimensional space-time is considered to be the surface layer Σ, this identification yields dynamic evolution of our observable spatial sector in the fifth dimension due to the singular nature of Σ. An effective radiation content is typically present in the hypersurface due to the cosmological constant in the embedding and the evolution of the internal dimension. Even for the case when the effective radiation is continuous across Σ, this hypersurface will still be singular so long as there is a δ-function singularity in the 5-dimensional stress-energy tensor there. It is possible then to rewrite the projected 5-dimensional Einstein equations on the surface layer in a way similar to the 4-dimensional Einstein equations for a Robertson-Walker cosmology with independently specifiable matter sources. The freedom to specify the equation of state on the hypersurface allows in this sense a 4-dimensional expansion behavior, as measured by an observer on Σ, independent of the behavior of the fifth dimension.