Junction Conditions for Signature Change

TL;DR

This paper explores the mathematical conditions for changing the signature of a metric, correcting existing methods for Klein-Gordon fields and cosmologies to ensure consistent physical equations across signature transitions.

Contribution

It provides corrected approaches for signature change in Klein-Gordon fields and cosmologies, ensuring the validity of fundamental equations at the junction.

Findings

Corrected the approach of Dray et al. for Klein-Gordon fields on signature-changing backgrounds.

Adjusted Ellis et al.'s method for isotropic cosmologies to satisfy Einstein-Klein-Gordon equations.

Established that the standard junction condition requires vanishing second fundamental form for a well-defined Ricci tensor.

Abstract

The change of signature of a metric is explained using simple examples and methods. The Klein-Gordon field on a signature-changing background is discussed, and it is shown how the approach of Dray et al.\ can be corrected to ensure that the Klein-Gordon equation holds. Isotropic cosmologies are discussed, and it is shown how the approach of Ellis et al.\ can be corrected to ensure that the Einstein-Klein-Gordon equations hold. A straightforward calculation shows that a well defined Ricci tensor requires the standard junction condition, namely vanishing of the second fundamental form of the junction surface.