On metric-connection compatibility and the signature change of space-time

TL;DR

This paper investigates conditions for the existence of metric-compatible connections in semi-pseudo-Riemannian space-times and explores the implications of signature changes on such connections.

Contribution

It establishes that metric-compatible connections exist if and only if the metric's rank and signature are constant, and analyzes the potential for space-time signature changes.

Findings

Existence of metric-compatible connections requires constant rank and signature.

Conditions for signature change in space-time are characterized.

Provides criteria for metric-connection compatibility in semi-pseudo-Riemannian geometry.

Abstract

We discuss and investigate the problem of existence of metric-compatible linear connections for a given space-time metric which is, generally, assumed to be semi-pseudo-Riemannian. We prove that under sufficiently general conditions such connections exist iff the rank and signature of the metric are constant. On this base we analyze possible changes of the space-time signature.