The Geometry of classical change of signature

TL;DR

This paper investigates the classical change of spacetime signature in general relativity, analyzing conditions under which such a change can occur without requiring the second fundamental form to vanish.

Contribution

It provides a new analysis of classical signature change, considering more general continuity assumptions and demonstrating that the change can occur without the second fundamental form vanishing.

Findings

Change of signature can occur with non-vanishing second fundamental form.

Various continuity assumptions at the signature change surface are possible.

The analysis extends previous studies by relaxing certain conditions.

Abstract

The proposal of the possibility of change of signature in quantum cosmology has led to the study of this phenomenon in classical general relativity theory, where there has been some controversy about what is and is not possible. We here present a new analysis of such a change of signature, based on previous studies of the initial value problem in general relativity. We emphasize that there are various continuity suppositions one can make at a classical change of signature, and consider more general assumptions than made up to now. We confirm that in general such a change can take place even when the second fundamental form of the surface of change does not vanish.