Signature Changing Space-times and the New Generalised Functions

TL;DR

This paper investigates whether Colombeau algebra of generalized functions can resolve disputes over the nature of signature change and junction conditions in signature changing spacetimes, concluding it cannot.

Contribution

It applies a covariant Colombeau algebra framework to analyze signature change, revealing its limitations in settling junction condition disputes.

Findings

Colombeau algebra does not exclude continuous or discontinuous signature change

It cannot resolve the debate over junction conditions

The framework's properties are extended covariantly

Abstract

A signature changing spacetime is one where an initially Riemannian manifold with Euclidean signature evolves into the Lorentzian universe we see today. This concept is motivated by problems in causality implied by the isotropy and homogeneity of the universe. As initially time and space are indistinguishable in signature change, these problems are removed. There has been some dispute as to the nature of the junction conditions across the signature change, and in particular, whether or not the metric is continuous there. We determine to what extent the Colombeau algebra of new generalised functions resolves this dispute by analysing both types of signature change within its framework. A covariant formulation of the Colombeau algebra is used, in which the usual properties of the new generalised functions are extended. We find that the Colombeau algebra is insufficient to preclude either continuous or discontinuous signature change, and is also unable to settle the dispute over the nature of the junction conditions.