Comment on `Smooth and Discontinuous Signature Type Change in General Relativity'

TL;DR

This paper critiques previous boundary conditions at signature change surfaces in general relativity, proposing a weaker formulation that allows for non-vanishing extrinsic curvature and bounded energy-momentum tensors.

Contribution

It introduces a less restrictive framework for Einstein equations at signature change surfaces, challenging prior assumptions and expanding possible boundary conditions.

Findings

Weaker smoothness assumptions are sufficient for boundary conditions.

Non-vanishing extrinsic curvature is compatible with bounded energy-momentum.

Previous conditions on extrinsic curvature are not necessary.

Abstract

Kossowski and Kriele derived boundary conditions on the metric at a surface of signature change. We point out that their derivation is based not only on certain smoothness assumptions but also on a postulated form of the Einstein field equations. Since there is no canonical form of the field equations at a change of signature, their conclusions are not inescapable. We show here that a weaker formulation is possible, in which less restrictive smoothness assumptions are made, and (a slightly different form of) the Einstein field equations are satisfied. In particular, in this formulation it is possible to have a bounded energy-momentum tensor at a change of signature without satisfying their condition that the extrinsic curvature vanish.