Non-differentiable Degrees of Freedom : Fluctuating Metric Signature

TL;DR

This paper explores the concept of metric signature as a discrete degree of freedom in a 7D vacuum universe, revealing solutions that suggest a quantum origin of a Lorentzian universe with frozen extra dimensions.

Contribution

It introduces a model where metric signature is non-differentiable, providing analytical solutions that connect higher-dimensional theories with quantum cosmology.

Findings

Found an analytical solution with a fluctuating metric signature.

Identified a Planck-scale piece that seeds the quantum birth of the universe.

Demonstrated a boundary condition leading to a Lorentzian universe with frozen extra dimensions.

Abstract

In this article we investigate the metric signature as a non-differentiable ({\it i.e.} discrete as opposed to continuous) degree of freedom. The specific model is a vacuum 7D Universe on the principal bundle with an SU(2) structural group. An analytical solution is found which to a 4D observer appears as a flat Universe with a fluctuating metric signature, and frozen extra dimensions with an SU(2) instanton gauge field. A piece of this solution with linear size of the Planck length ($\approx l_{Pl}$) can be considered as seeding the quantum birth of a regular Universe. A boundary of this piece can initiate the formation of a Lorentzian Universe filled with the gauge fields and in which the extra dimensions have been ``frozen''.