On A. D. Sakharov's hypothesis of cosmological transitions with changes in the signature of the metric

TL;DR

This paper explores the theoretical implications of Sakharov's hypothesis that the universe experienced cosmological transitions involving changes in the metric signature, discussing mathematical formulations and philosophical questions about time and spacetime regions.

Contribution

It analyzes the mathematical and philosophical aspects of including signature-changing regions in the path integral approach to cosmology, proposing possible boundary conditions and interpretations.

Findings

Signature changes can be modeled by imaginary coordinates.

Boundary conditions on metric regions are crucial for path integral formulation.

Regions with different signatures may influence universe evolution.

Abstract

I discuss 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, I try to speculate how the existence of the regions with different signature can affect the evolution of the Universe.