Re-Examination of Simple Kaluza-Klein Cosmologies

TL;DR

This paper re-examines simple five-dimensional Kaluza-Klein cosmological models with Ricci-flat solutions, analyzing their properties, singularities, and the potential for realistic compactification of the extra dimension.

Contribution

It provides a detailed analysis of specific Ricci-flat Kaluza-Klein models, highlighting their cosmological behaviors and conditions for stable extra dimension compactification.

Findings

Models exhibit bounce or big bang depending on curvature index

Only one model shows potential for stable extra dimension compactification

Singularities are present in some cases, with Kretschmann scalar behavior analyzed

Abstract

Simple cosmological models based upon five-dimensional Kaluza-Klein relativity are re-examined and interesting properties are indicated. These models are special cases of those obtained by Davidson et al. and Mann and Vincent, specifically, those with vanishing cosmological constant. No five-dimensional sources are present: the solutions are Ricci-flat in five dimensions. Electromagnetic degrees of freedom are assumed not to be excited, consequently the four-dimensional stress-energy tensor induced by dimensional reduction is entirely due to the scalar field, obeying the radiation equation of state. For the three choices of curvature index, the dependence of the scale factor on cosmic time corresponds to, for k = -1 either a bounce or big bang, for k = 0 a big bang, and for k = 1 a big bang followed by collapse. The Kretschmann scalar is proportional to the square of acceleration and approaches zero for k = -1 and k = 0, while revealing true singularities in some cases. Only one of the models exhibits the compactification behavior hoped for in a realistic K-K model: shrinking of the circumference of fifth dimension in the earliest times followed by an extended period of stability.