A mathematical analysis has identified a condition that can separate a genuine spacetime singularity from a point at infinity, two very different ways for a geometric description of the universe to end. The condition is conditional: if no pair of geodesics approaching the same endpoint is intertwined, the endpoint is classified as a pure singularity rather than a point approached only after an infinite affine journey.
The work is an arXiv preprint in gravitation and relativity. It examines maximally extended pseudo-Riemannian manifolds, the mathematical spaces used to describe spacetime, and studies incomplete geodesics through different boundary constructions called envelopments. In plain terms, an envelopment places a spacetime inside a larger mathematical space so that paths that run out of the original space can be assigned boundary endpoints.
A boundary is not always the same kind of ending
The distinction matters because a singularity and infinity can look similar when they are represented only as boundary points. The paper's classification says that, in maximally extended manifolds, points at infinity are pure, while singularities can be directional or pure. A directional singularity can cover a pure point at infinity in the framework's comparison between different envelopments, so the same abstract boundary picture can contain more than one layer of information.
The central construction uses an Endpoint Theorem, which embeds a manifold into a larger one so that a sequence with no accumulation point converges to a boundary endpoint. The authors then control a neighborhood around a central incomplete geodesic with curvature bounds and a carefully chosen neighborhood construction, preventing another geodesic from sharing that endpoint under the stated assumptions.
A re-embedding proposition takes that idea further. It constructs another envelopment in which a selected geodesic approaches a new boundary point, no other geodesic ends at that point, and the original boundary point covers the new one. Under the same non-intertwining assumptions, the selected geodesic has bounded affine parameter, the mathematical measure used to track progress along the path.
The analysis also gives a broader curve-separation result. When a pair of inextendible curves with no limit points does not intersect infinitely many times, the construction can place their endpoints at separate boundary points in different envelopments. In a maximally extended manifold, a boundary point reached only by curves with bounded parameter is then a pure singularity, and the boundary of a future g-boundary extension is classified as a pure singularity set.
What the test says about Schwarzschild spacetime
The paper applies the framework to maximally extended Schwarzschild spacetime. Its reported curvature invariant diverges as the radial coordinate r approaches zero, making every boundary point at r = 0 a curvature singularity. The geodesic analysis also finds that every geodesic terminating there has bounded affine length, so those boundary points are pure singularities in the paper's classification.
In the Penrose embedding, a compactified diagram that brings infinity and singular boundaries into a finite picture, the future and past singularity segments are both curvature singularities and pure singularities. They are approached only by geodesics with bounded parameter, while the remaining boundary points are classified as pure points at infinity. The analysis finds no intertwined geodesic pairs approaching the singularity segments.
On that basis, the authors classify the Penrose compactification of maximally extended Schwarzschild spacetime as an optimal embedding within the abstract-boundary framework. The paper also identifies region II as a connected component of the black region, hence a black hole in this construction, with its boundary at the event horizon r = 2M.
A conditional result, with a wider question still open
The findings leave the general separation problem open. The main arguments rely on maximal extension, the chosen envelopment and curve class, and non-intertwining assumptions. The authors leave open whether a directional singularity generally covers a pure singularity in another envelopment, and whether that pure singularity must be separate from a pure point at infinity.
The endpoint re-embedding proposition is stated for metrics with at least C 2,1 regularity, a technical smoothness condition. The authors say that lower regularity changes the construction of the normal neighborhood used in the proof. For the black-hole analysis, they use a modified singular-neighborhood definition restricted to pure singularity points arising from the Penrose diagram because all possible envelopments cannot be completely known.
No datasets were generated or analysed, so data sharing is not applicable. The declarations report no conflict of interest. The funding statement acknowledges Junbang Liu's ANU scholarships and support for Susan M. Scott from the Australian Research Council Centre of Excellence for Gravitational Wave Discovery, OzGrav, project CE230100016.
Paper data and sources
Original title: The relationship between spacetime singularities and regions at infinity
Authors: Junbang Liu, Ben Andrews, Susan M Scott
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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