A Spacetime Singularity Need Not Be a Place

Calling a spacetime “singular” does not by itself identify a place inside it. In the classical singularity-theorem sense, the precise conclusion is that at least one timelike or null geodesic is inextendible in the spacetime despite having a finite endpoint of its affine-parameter range. Curvature blowup, failure of a coordinate chart, existence of an extension, and representation by an ideal boundary are separate questions.

The distinction matters because the same incomplete-curve diagnosis includes both removable holes in flat spacetime and invariantly singular black-hole interiors. The useful question is therefore not “Where is the singularity?” but “Which geometric property has failed, and what has actually been proved?”

Five independent geometric questions

Let $(M,g)$ be a time-oriented Lorentzian spacetime.

Causal geodesic incompleteness. A maximally extended, affinely parametrized causal geodesic $\gamma:(a,b)\to M$ is future incomplete when $b$ is finite, and past incomplete when $a$ is finite. For a unit timelike geodesic the affine parameter may be proper time; for a null geodesic it is an affine parameter, not a coordinate time. A spacetime is incomplete of a stated causal type and direction if at least one such geodesic exists.

Extension and inextendibility. A $C^k$ extension of $(M,g)$ is a same-dimensional Lorentzian spacetime $(\widetilde M,\widetilde g)$ with a $C^k$ metric and a proper isometric embedding

$$\iota:M\longrightarrow\widetilde M, \qquad \iota^*\widetilde g=g,$$

whose image is an open proper subset. “Inextendible” is meaningful only after the regularity class and any required matter fields or equations have been stated. Inextendibility of a geodesic within $M$ is not inextendibility of the spacetime.

Curvature behavior. Scalar polynomial invariants, curvature components in a parallelly propagated frame, and integrated tidal distortion are different diagnostics. Divergence of a scalar invariant is coordinate independent and can obstruct a regular $C^2$ endpoint. Bounded scalar invariants, however, neither control every freely falling frame component nor prove that an extension exists.

Chart failure. A chart has failed when its coordinate range ends, its Jacobian degenerates, or its metric components cease to be regular while another chart describes the same events smoothly. This is a statement about an atlas, not geodesic incompleteness or curvature. The definitions of incompleteness and extendibility, including the removable-point warning, are reviewed in Steinbauer 2023, §§2.1 and 2.3, arXiv v2 PDF pp. 4–5 and 8–9.

Ideal-boundary representation. A conformal, causal, or abstract boundary can represent ways in which curves leave $M$. An abstract-boundary element, for example, is an equivalence class of boundary representations across envelopments; its classification depends on the admissible curve family and differentiability class. It is not thereby an event of $M$ with a tangent space and metric. See Scott and Szekeres 1994, §§2–5, pp. 223–253.

Flat, incomplete, and smoothly extendible

The decisive counterexample is punctured Minkowski spacetime,

$$M=\mathbb R^{1,3}\setminus\{(0,0,0,0)\}, \qquad g=\eta.$$

The timelike line

$$\gamma(\tau)=(\tau,0,0,0), \qquad -\infty\lt\tau\lt0,$$

is a maximally extended geodesic in $M$. It reaches the missing origin after finite future proper time and cannot continue there within $M$, so $M$ is future timelike geodesically incomplete. Yet

$$R^a{}_{bcd}=0$$

everywhere, and the inclusion $M\hookrightarrow\mathbb R^{1,3}$ is a smooth proper isometric embedding. Thus this one example is incomplete, exactly flat, and smoothly extendible. It disproves all three implications “incomplete $\Rightarrow$ curvature blowup,” “incomplete $\Rightarrow$ inextendible,” and “incomplete $\Rightarrow$ a singular point belongs to spacetime.”

Schwarzschild: a regular horizon and a singular limit

For Schwarzschild spacetime set $m=GM$. Static coordinates make metric components singular at $r=2m$, but ingoing Eddington–Finkelstein coordinates give

$$ds^2 =-\left(1-\frac{2m}{r}\right)dv^2 +2\,dv\,dr+r^2d\Omega_2^2,$$

which is nondegenerate at the future horizon. The invariant comparison is

$$K=R_{abcd}R^{abcd}=\frac{48m^2}{r^6}, \qquad K\big|_{r=2m}=\frac{3}{4m^4}, \qquad K\longrightarrow\infty\quad(r\to0).$$

Therefore $r=2m$ is a regular null hypersurface in the extended geometry, although the static chart stops there. By contrast, approach to $r=0$ has scalar-curvature blowup and some causal geodesics reach that boundary in finite parameter; the divergence obstructs a $C^2$ extension through such an endpoint. Even here, $r=0$ is not an event in the Schwarzschild manifold at which the metric takes an infinite value. The horizon extension and the causal role of the $r=0$ boundary are exhibited in Bičák 2000, §§2.3–2.4, printed pp. 23–27.

What singularity theorems actually conclude

Classical singularity theorems combine their own regularity, convergence, initial or trapped-set, and global causal hypotheses to contradict causal geodesic completeness. Their conclusions retain a causal type and, when the theorem supplies one, a time direction: the representative Penrose theorem gives future null incompleteness, a representative Hawking theorem gives past timelike incompleteness, and the basic Hawking–Penrose theorem gives some timelike or null incompleteness without fixing the direction.

Precisely, the conclusion is that at least one inextendible causal geodesic has a finite affine-parameter endpoint (proper time for a unit timelike geodesic). It does not establish a singular point in $M$, curvature blowup, or inextendibility in any $C^k$ class. Those stronger properties require separate arguments. The typed classical statements and their proof pattern appear in Steinbauer 2023, §§3.1–3.5, especially Theorems 3.8, 3.9, and 3.11, arXiv v2 PDF pp. 9–20.

A quick diagnostic

Observation	What follows	What to check next
An inextendible causal geodesic has a finite affine endpoint	Incompleteness of that causal type and direction	Curvature, topology, and extensions in declared regularity classes
Metric components diverge at a coordinate value	The displayed chart is suspect	A regular overlapping chart, its Jacobian, and metric nondegeneracy
A scalar curvature invariant diverges along an approach	Invariant scalar-curvature blowup	Which geodesics reach the limit and which extension classes are obstructed
Selected scalar invariants remain bounded	Only those scalar tests pass	Parallelly propagated curvature, tidal integrals, holonomy, topology, and extension attempts
A $C^k$ isometric extension with proper image is constructed	The original spacetime is $C^k$-extendible	Whether incomplete curves continue and whether fields and equations also extend
An ideal endpoint is attached	A chosen completion represents an approach to $M$	The construction, equivalence relation, curve family, and regularity class

Common pitfalls

Calling a horizon a curvature singularity. Divergent static Schwarzschild components at $r=2m$ disappear in a horizon-penetrating chart, while $K$ stays finite. Coordinate components are not curvature invariants.

Using “maximal” without a category. Maximal analytic extension, $C^k$-inextendibility, and maximal globally hyperbolic development are different notions. Name the category before drawing a conclusion.

Turning a boundary representation into a physical event. An ideal endpoint can organize limiting curves without belonging to $M$. Tensor values there require an actual extension or additional boundary structure.

Where to continue

The complete Library treatment is Completeness, Extendibility, and Singularities. Its deeper pages separate geodesic completeness, incomplete curves without singular points, spacetime extensions and maximality, curvature blowup and singularity strength, and abstract boundaries. For the hypothesis-to-conclusion map, use Hawking–Penrose Singularity Theorems and What Singularity Theorems Conclude.

References

• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics: Selected Themes.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, 1–126. Lecture Notes in Physics 540. Springer, 2000. Open preprint: arXiv:gr-qc/0004016v1.
• Scott, Susan M., and Peter Szekeres. “The Abstract Boundary—A New Approach to Singularities of Manifolds.” Journal of Geometry and Physics 13 (1994): 223–253. Open manuscript: arXiv:gr-qc/9405063.
• Steinbauer, Roland. “The Singularity Theorems of General Relativity and Their Low Regularity Extensions.” Jahresbericht der Deutschen Mathematiker-Vereinigung 125, no. 2 (2023): 73–119. DOI. Open preprint: arXiv:2206.05939v2.