Small singular regions of spacetime

TL;DR

This paper demonstrates that regions of spacetime containing incomplete half-curves can be localized to small, bounded regions in the orthonormal frame bundle, supporting the idea that singularities in general relativity are local phenomena.

Contribution

It proves that incomplete regions in spacetime can be confined to small, bounded subsets of the orthonormal frame bundle, advancing the understanding of the local nature of singularities.

Findings

Incomplete half-curves are covered by sequences of small singular regions.

Such regions are images of bounded subsets in the orthonormal frame bundle.

Results support the localizability of singular structures in classical GR.

Abstract

We prove that every open connected region of relativistic spacetime $(M,\textbf{g})$ that encloses a $b$-incomplete half-curve has an open connected subregion that encloses a $b$-incomplete half-curve and is also 'small' in the following sense: it is the image, under the bundle projection map, of some open region in the (connected) orthonormal frame bundle $O^+M$ over that spacetime which is bounded, and whose closure is Cauchy incomplete, with respect to any 'natural' distance function on $O^+M$. As a corollary, it follows that every $b$-incomplete half-curve can be covered by a sequence of singular regions which are images of a sequence of bounded subsets of $O^+M$ whose diameter, with respect to any 'natural' distance function on $O^+M$, tends to zero. We discuss to what extent these results can be interpreted in favour of the claim that singular structure in classical general relativity is 'localizable'.