Tensor distributions on signature-changing space-times

TL;DR

This paper explores the mathematical formalism of tensor distributions on signature-changing space-times, demonstrating that covariant derivatives can be rigorously defined in such contexts, enabling meaningful physical equations.

Contribution

It extends the theory of tensor distributions to signature-changing space-times and shows covariant derivatives are well-defined in these settings.

Findings

Covariant differentiation can be defined on tensor distributions in signature-changing space-times.

The formalism applies to both continuous and discontinuous signature changes.

This enables rigorous formulation of field equations in such space-times.

Abstract

Irregularities in the metric tensor of a signature-changing space-time suggest that field equations on such space-times might be regarded as distributional. We review the formalism of tensor distributions on differentiable manifolds, and examine to what extent rigorous meaning can be given to field equations in the presence of signature-change, in particular those involving covariant derivatives. We find that, for both continuous and discontinuous signature-change, covariant differentiation can be defined on a class of tensor distributions wide enough to be physically interesting.