Pure quantum freezing of the 5$^{th}$ dimension

Abstract

It is shown that superthin and superlong gravitational flux tube solutions in the 5D Kaluza-Klein gravity have the region $(\approx l_{Pl})$ where the metric signature changes from $\{+,-,-,-,- \}$ to $\{-,-,-,-,+ \}$. Such change is too quickly from one of the paradigms of quantum gravity which tells 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. For avoiding such dynamic it is supposed that a pure quantum freezing of the dynamic of the $5^{th}$ dimension takes place. As the continuation of the flux tube metric in the longitudinal direction the Reissner-Nordstr\"om metric is proposed. In the consequence of such construction one can avoid the appearance of a point-like singularity in the extremal Reissner-Nordstr\"om solution.