On Signature Transition and Compactification in Kaluza-Klein Cosmology

TL;DR

This paper explores signature transitions in a 5D Kaluza-Klein universe with negative cosmological constant, showing how such transitions can lead to internal space compactification and analyzing the quantum states associated with these processes.

Contribution

It introduces a mechanism for signature transition-induced compactification in Kaluza-Klein cosmology and provides exact quantum solutions related to these transitions.

Findings

Signature transition causes metric degeneracy and internal space compactification.

The size of the universe and internal space are inversely related in the Lorentzian region.

Quantum states peak near classical solutions undergoing signature transition.

Abstract

We consider an empty (4+1) dimensional Kaluza-Klein universe with a negative cosmological constant and a Robertson-Walker type metric. It is shown that the solutions to Einstein field equations have degenerate metric and exhibit transitioins from a Euclidean to a Lorentzian domain. We then suggest a mechanism, based on signature transition which leads to compactification of the internal space in the Lorentzian region as $a \sim |\Lambda|^{1/2}$. With the assumption of a very small value for the cosmological constant we find that the size of the universe $R$ and the internal scale factor $a$ would be related according to $Ra\sim 1$ in the Lorentzian region. The corresponding Wheeler-DeWitt equation has exact solution in the mini-superspace giving rise to a quantum state which peaks in the vicinity of the classical solutions undergoing signature transition.