The birth of the Universe as a result of the change of the metric signature

TL;DR

This paper explores the possibility that the Universe's birth resulted from a quantum transition from Euclidean to Lorentzian signature spacetime, analyzing classical and quantum gravity frameworks to understand this process.

Contribution

It investigates the realization of metric signature change in quantum gravity, particularly within the Wheeler-DeWitt theory and extended phase space approach, linking it to the Hartle-Hawking wave function.

Findings

The idea of signature change can be mathematically realized.

The Hartle-Hawking wave function suggests possible realization.

The approach offers insights but does not fully explain the emergence of spacetime.

Abstract

In this paper, I discuss the idea that the birth of our Universe may be a result of a quantum transition from a physical continuum with the Euclidean signature to a Lorentzian spacetime. A similar idea was expressed by Andrei D. Sakharov. At the classical level, the idea was studied by George F. R. Ellis and his collaborators, who explored if solutions to the classical Einstein equation exist which admit a change of metric signature. The present paper aims at examining possible realizations of this idea at the level of quantum gravity, in the framework of the Wheeler - DeWitt theory and in the extended phase space approach to quantization of gravity. I intend to answer the questions: Does the Hartle - Hawking wave function imply such a realization? How can this idea be realized in the extended phase space approach to quantum gravity, where the change of signature is described by imposing special conditions on $g_{00}$-component of the metric in different regions of the physical continuum? The conclusion is that the idea can be realized from a formal mathematical point of view, but it can hardly help in understanding how spacetime structure and time itself appeared from a timeless continuum.