All metrics have curvature tensors characterised by its invariants as a limit: the \epsilon-property

TL;DR

This paper generalizes the epsilon-property, showing that in any dimension and signature, metrics not characterized by their invariants can be approximated by a background with identical invariants, through an appropriate frame choice.

Contribution

It extends the epsilon-property to all dimensions and signatures, demonstrating the approximation of metrics by background tensors with identical invariants.

Findings

Metrics not characterized by invariants can be approximated by a background with the same invariants.

A frame exists where curvature tensor components are arbitrarily close to the background.

The background is characterized solely by its curvature tensors and invariants.

Abstract

We prove a generalisation of the $\epsilon$-property, namely that for any dimension and signature, a metric which is not characterised by its polynomial scalar curvature invariants, there is a frame such that the components of the curvature tensors can be arbitrary close to a certain "background". This "background" is defined by its curvature tensors: it is characterised by its curvature tensors and has the same polynomial curvature invariants as the original metric.