Incomplete Curves without Singular Points

An incomplete geodesic has no final event in the spacetime: if it did, the smooth geodesic equation would normally continue it. Its finite parameter endpoint is therefore a failure of the chosen manifold and metric to contain the curve’s continuation, not a mysterious material point called “the singularity.” The same observation covers removable holes, edges produced by restricted coordinates, quasiregular defects, and genuine inextendible singular behavior; additional tests distinguish them.

Required background. Geodesics and Jacobi Fields supplies maximal affine geodesics; Curvature supplies invariant regularity tests. Helpful background. Lorentzian Metrics and Causal Cones distinguishes causal types; Causality and Global Structure supplies global endpoint language; Exact Solutions of Einstein’s Equations supplies coordinate extensions.

Why the endpoint is absent

Suppose an affinely parametrized geodesic $\gamma:[0,b)\to M$ is future inextendible with $b$ finite. If $\gamma(\lambda)$ converged to an event $p\in M$ and its tangent converged to a finite vector $v\in T_pM$, local existence and uniqueness for the geodesic equation would produce a geodesic through $(p,v)$ and extend $\gamma$ beyond $b$. Thus finite-parameter inextendibility entails a failure of this phase-space convergence in every regular chart.

The curve may leave every compact subset of $M$, its tangent may escape every compact subset of $TM$, or the metric may fail to extend with the regularity needed by the equation. None of these alternatives places a singular point inside $M$. A spacetime is, by definition, a smooth manifold carrying a nondegenerate Lorentzian metric of the declared regularity. Where that structure ceases, there is no physical tangent space or metric value unless an extension is constructed.

This is why singularity theorems conclude geodesic incompleteness rather than “there exists a singular event.” Steinbauer states the distinction before developing the theorem pattern Steinbauer 2023, §2.3, pp. 8–9. The conclusion is deliberately weak enough to cover several mechanisms.

Removable holes and coordinate edges

Delete the origin from two-dimensional Minkowski spacetime. The null line

$$\gamma(\lambda)=(\lambda,\lambda), \qquad -1\lt\lambda\lt0,$$

has finite affine endpoint as $\lambda\to0$, but that endpoint is not in the punctured manifold. Every curvature component vanishes along the curve. Restoring the origin gives a smooth, flat extension and continues the geodesic. The incompleteness is real for the original spacetime, while its cause is a removable omission.

A coordinate chart can create a similar appearance without making the underlying spacetime incomplete. Rindler coordinates end at a null surface, and Schwarzschild static coordinates become singular at the horizon. If the manifold already contains regular events beyond that chart, a coordinate curve reaching its edge has not reached a spacetime endpoint at all; it has merely left the chart. One must distinguish:

1. the endpoint of a coordinate range;
2. the endpoint of a curve within a chosen open submanifold; and
3. an inextendible curve in the maximal spacetime being studied.

Curvature regularity and a change of coordinates can diagnose a removable chart edge, but bounded scalar invariants alone do not prove extendibility. The extension must supply a manifold, an embedding, and a Lorentzian metric of specified regularity.

Quasiregular and topological obstructions

Incomplete curves can coexist with bounded curvature even when filling one point is not allowed within the chosen smooth topology. A flat cone illustrates the idea: away from its apex the metric is locally flat, radial geodesics reach the missing apex in finite length, yet the angular identification produces nontrivial holonomy. Depending on the cone angle and admissible category, adjoining a regular manifold point cannot reproduce the same circumference-to-radius relation with a smooth nondegenerate metric.

Misner-type quotients give a Lorentzian version in which local flatness coexists with global causal pathology and an incomplete region. The obstruction is produced by identification and causal structure rather than scalar curvature divergence. Such examples are often called quasiregular singularities, though terminology varies. They show why “zero curvature” is not equivalent to “harmless missing point.”

Topology can also block a chosen extension. If sequences approaching the apparent endpoint from distinct directions would need incompatible neighborhoods, adding one point may violate Hausdorffness or manifold structure. Adding several points can create nonunique extensions. An ideal-boundary construction records approach classes without pretending that they are ordinary physical events.

Genuine singular behavior needs extra evidence

For gravitational singularities, one often supplements incompleteness with one or more of the following:

• divergence of scalar polynomial curvature invariants;
• divergence of curvature components in a parallelly propagated frame;
• unbounded tidal distortion integrals along an incomplete observer;
• failure of every extension in a specified class, such as continuous or twice differentiable metrics; or
• a boundary classification tied to an admissible family of curves.

These criteria are inequivalent. A parallelly propagated component may diverge while all scalar invariants remain finite. A weak null singularity may permit a continuous metric extension while obstructing extensions with square-integrable connection or twice differentiable curvature. A finite tidal integral can let an idealized extended body retain finite volume even though curvature becomes unbounded.

The word singular should therefore be followed by the tested property. “Future null geodesically incomplete,” “curvature scalar divergent,” and “inextendible as a continuous Lorentzian metric” communicate distinct, checkable conclusions. An unspecified “singular point” communicates none of them.

Representing an absent endpoint

Several boundary constructions attach ideal elements to equivalence classes of curves, embeddings, or causal sets. Their purpose is to compare how curves leave the spacetime, not to smuggle new events into the original manifold. Scott and Szekeres first identify mutually covering boundary sets. Their equivalence classes are abstract boundary sets; only classes with singleton representatives are abstract boundary points, and the a-boundary is the set of those points Scott and Szekeres 1994, Definitions 20–22 and §§4–5.

The a-boundary construction itself depends on the smooth manifold and its envelopments, not on a metric, affine connection, or chosen curve family. The metric or connection, curve family, and differentiability threshold enter only in subsequent classifications of representatives, and only equivalence-invariant labels define classifications of a-boundary points.

A conformal completion is particularly effective for regular infinity; a causal completion uses limiting chronological pasts and futures; an abstract boundary compares embeddings. The same incomplete curve may approach a regular representative in one envelopment and a nonregular representative in another. The covering relation identifies their common a-boundary point, while the essential/removable classification tests whether a nonsingular covering set removes the apparent singularity. No attachment is neutral.

Exercise

Consider $M=\mathbb R^{1,1}\setminus\{(0,0)\}$ and the timelike geodesic $\gamma(\tau)=(\tau,0)$ for $\tau\lt0$. Decide whether (a) the curve is complete in $M$, (b) curvature diverges, and (c) $M$ is smoothly inextendible.

Solution

The proper-time parameter reaches $0$ after finite range, while the missing origin prevents continuation in $M$, so the geodesic is future incomplete. The metric is flat, hence curvature does not diverge. Finally, the inclusion into full Minkowski spacetime is a proper smooth isometric embedding, so $M$ is smoothly extendible. This single example disproves both “incomplete implies curvature blowup” and “incomplete implies inextendible.”

Spacetime Extensions and Maximality now makes the extension relation precise and distinguishes its several maximality notions.

References

• Scott, Susan M., and Peter Szekeres, “The Abstract Boundary—A New Approach to Singularities of Manifolds”, Journal of Geometry and Physics 13, §§2–5, pp. 223–253, 1994.
• Steinbauer, Roland, “The Singularity Theorems of General Relativity and Their Low Regularity Extensions”, Jahresbericht der Deutschen Mathematiker-Vereinigung 125, §2.3, pp. 8–9, 2023.