Small singular regions of spacetime

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’.

💡 Research Summary

The paper addresses a long‑standing conceptual tension in classical general relativity: singularities are usually defined globally (e.g., the existence of incomplete timelike or causal geodesics), yet many physically relevant spacetimes display regions that are intuitively singular while other regions are not. To formalize the idea that singular behavior can be localized, the author adopts the weakest of the three common incompleteness notions—b‑incompleteness—and introduces a definition of a singular region: an open, connected subset (U\subset M) that contains the entire image of a b‑incomplete half‑curve (\gamma).

The technical heart of the work lies in relating b‑incompleteness on the base spacetime ((M,g)) to metric properties of the positive connected component of the orthonormal frame bundle (O^{+}M). Using the Levi‑Civita connection, one builds a family of natural Riemannian metrics on (O^{+}M) (originally due to Schmidt). Any such metric yields a distance function (d); different choices are uniformly equivalent, so the results are independent of the specific inner products used on (\mathfrak{gl}(4,\mathbb{R})) and (\mathbb{R}^{4}).

Proposition 1 states that if an open set (U\subset M) contains a curve (\gamma: