5D Kaluza-Klein gravity: singularuty and freezing of $5^{th}$ dimension

Abstract

Some reasonings are presented that the problem of a singularity in general relativity with the problem of freezing of the $5^{th}$ dimension can be connected. It is shown that some solutions in the 5D Kaluza-Klein gravity with the cross section in the Planck region have a region $(\approx l_{Pl})$ where the metric signature changes from $\{+,-,-,-,- \}$ to $\{-,-,-,-,+ \}$. The idea is discussed that such switching can not occur following some general point of view that the Planck length is the minimal length in the nature and consequently the physical quantities can not change very quickly in the course of this length. In this case the dynamic of the $G_{55}$ metric component should be frozen.