Einstein's Equations in the Presence of Signature Change

TL;DR

This paper explores Einstein's equations with signature change, deriving a generalized relation between stress tensor discontinuities and extrinsic curvature, showing no surface layer exists when curvature is continuous.

Contribution

It generalizes the Lanczos equation for signature-changing spacetimes and clarifies conditions for the absence of surface layers in Einstein's equations.

Findings

No distributional stress tensor term when extrinsic curvature is continuous.

Generalized Lanczos equation for signature-changing scenarios.

Consistency with standard results for constant signature.

Abstract

We discuss Einstein's field equations in the presence of signature change using variational methods, obtaining a generalization of the Lanczos equation relating the distributional term in the stress tensor to the discontinuity of the extrinsic curvature. In particular, there is no distributional term in the stress tensor, and hence no surface layer, precisely when the extrinsic curvature is continuous, in agreement with the standard result for constant signature.