Gravity and Signature Change

TL;DR

This paper explores how using proper time in spacetimes with Euclidean and Lorentzian regions allows for smooth frame descriptions, enabling variational and distributional treatments of Einstein's equations across signature changes.

Contribution

It introduces a method to handle signature change in spacetime using proper time, facilitating smooth orthonormal frames and generalized variational and distributional approaches.

Findings

Smooth orthonormal frames can be constructed across signature change.

Variational treatment of Einstein's equations extends to signature-changing spacetimes.

Distribution theory can be applied in the context of signature change.

Abstract

The use of proper ``time'' to describe classical ``spacetimes'' which contain both Euclidean and Lorentzian regions permits the introduction of smooth (generalized) orthonormal frames. This remarkable fact permits one to describe both a variational treatment of Einstein's equations and distribution theory using straightforward generalizations of the standard treatments for constant signature.