Separability of singularities and points at infinity

Establish, in general, whether pure singularities and pure points at infinity are separable in embeddings of asymptotically flat or otherwise physically relevant spacetimes, namely whether suitable envelopments admit disjoint neighborhoods of the corresponding boundary points.

Background

The paper studies the abstract-boundary classification of singularities and regions at infinity for maximally extended pseudo-Riemannian manifolds. In this framework, pure singularities and pure points at infinity are distinct types of boundary points characterized by the affine-parameter behavior of approaching curves.

The authors identify the general separability of these two classes as a fundamental unresolved issue. Establishing such separability would clarify whether singular behavior and asymptotic behavior can be topologically distinguished within suitable embeddings of spacetimes.

References

It remains an important open question, in general, as to how to separate singularities and points at infinity in embeddings?

— The relationship between spacetime singularities and regions at infinity  (2608.25317 - Liu et al., 26 Aug 2026) in Section 1, Introduction

Under physically relevant choices of curve families $\mathcal{C}$ on a smooth Lorentzian manifold $(\mathcal{M},g)$, (e.g. geodesics with affine parameter, $C1$ curves with generalised affine parameter, the causal subfamilies of either of these, etc.), points in $\mathcal{I}(\mathcal{M})$ and $\mathcal{S}_p(\mathcal{M})$ are separated from each other.

— The relationship between spacetime singularities and regions at infinity  (2608.25317 - Liu et al., 26 Aug 2026) in Conjecture A, Discussion