'Small' singular regions of spacetime
Franciszek Cudek
TL;DR
The paper addresses how singular structure in classical general relativity, witnessed by $b$-incomplete curves, can be localized within spacetime regions. It proves that any open region containing a $b$-incomplete half-curve contains a smaller singular region that is the image, under the projection $\pi$, of a bounded, incomplete region of the orthonormal frame bundle $O^+M$. A corollary shows that such a half-curve can be covered by a sequence of shrinking singular regions whose diameters tend to zero, indicating a form of locality in the singular structure. The results hinge on relating $b$-incompleteness in $M$ to Cauchy incompleteness in $O^+M$, via lifting curves and, via Hawking and Ellis, establishing the equivalence for natural metrics on $O^+M$, and discuss the physical interpretation and limitations of these natural metrics.
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'.