The Birth of the Universe as a Result of the Change of the Metric Signature

Tatyana P Shestakova

doi:10.3390/physics4010012

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: 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 g00-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.

Highlights

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: 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 g00 -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
We explore the idea that our Universe appeared from a region of some abstract physical continuum, where time does not exist

Summary

Introduction

The title of this Special Issue, “Light on Dark Words” has reminded me the following lines from the poem “The Night before Christmas” by the great Russian philosopher. The Khlopov’s invitation made me return to the papers of Sakharov and look at them from the viewpoint of the extended phase space approach to quantization of gravity, which I have been developing for many years It has resulted in my paper [2], where I commented on Sakharov’s work “Cosmological transitions with changes in the signature of the metric” [3]. Can one say that the Hartle–Hawking wave function describes the physical continuum consisting of the Euclidean and Lorentzian regions, while their boundary corresponding the change of the signature and the origin of the observed Universe?. Given the fact that the “no boundary” proposal was discussed mainly in the framework of the Wheeler–DeWitt quantum geometrodynamics (see, for example, [14,15,16]), the proposal, as well as the signature change, can be considered from the viewpoint of other approaches to quantization of gravity. In any case, introducing a new field requires additional assumptions [22], so that in the present paper we shall not go into this question

The Wheeler–DeWitt Equation and Its Solution

Discontinuous Signature Change

Continuous Signature Change

Discussion