On A. D. Sakharov’s Hypothesis of Cosmological Transitions with Changes in the Signature of the Metric

Tatyana Shestakova

doi:10.3390/universe7050151

Abstract

The paper discusses possible consequences of A. D. Sakharov’s hypothesis of cosmological transitions with changes in the signature of the metric, based on the path integral approach. This hypothesis raises a number of mathematical and philosophical questions. Mathematical questions concern the definition of the path integral to include integration over spacetime regions with different signatures of the metric. One possible way to describe the changes in the signature is to admit time and space coordinates to be purely imaginary. It may look like a generalization of what we have in the case of pseudo-Riemannian manifolds with a non-trivial topology. The signature in these regions can be fixed by special gauge conditions on components of the metric tensor. The problem is what boundary conditions should be imposed on the boundaries of these regions and how they should be taken into account in the definition of the path integral. The philosophical question is what distinguishes the time coordinate among other coordinates but the sign of the corresponding principal value of the metric tensor. In particular, there is an attempt in speculating how the existence of the regions with different signature can affect the evolution of the Universe.

Highlights

The problem is what boundary conditions should be imposed on the boundaries of these regions and how they should be taken into account in the definition of the path integral
In his paper “Cosmological transitions with changes in the signature of the metric” [2] published in 1984, he wrote: “It is conjectured that there exist states of the physical continuum which include regions with different signatures of the metric. . . ”
The signature in different regions of the continuum can be fixed by special gauge conditions on components of the metric tensor. It may be the mentioned above conditions, otherwise some condition may be imposed on the determinant of the metric tensor

Summary

Introduction

The signature change implies that a temporal coordinate x0 becomes a spatial one, or vice versa It can be formally reached if one admits complex-valued transformations like t → −iy, that causes a g00component of the metric tensor to change its sign. In the class of transformations discussed above, the signature in different regions can be fixed by special gauge conditions It leads to the question if the requirement of gauge invariance of the path integral is applicable in this case. It will be Universe 2021, 7, 151 compared with the situation when the physical continuum includes regions with different signatures of the metric tensor.

The Definition of the Path Integral over Spatial Regions

Non-Trivial Topology and Gauge Invariance

Time Coordinates and Evolution in Time

Discussion