Inextendibility Criteria

An inextendibility proof must contradict a consequence of the proposed extension class. Curvature blowup is effective against $C^2$ extensions; connection nonintegrability targets a weak-solution class; timelike completeness and causal-boundary arguments can obstruct even $C^0$ extensions. No single test is universal. The correct method starts by fixing the regularity of the hypothetical larger metric and then asks which geometric structure survives at that threshold.

Required background. Analysis for Mathematical Relativity supplies compactness and weak regularity, while the Einstein equation supplies curvature and continuation criteria.

Helpful background. Causality and Global Structure supplies extension boundaries and maximizing curves; PDEs and Green Functions for Gravitation supplies weak-solution norms; and Black-Hole Geometry and Mechanics supplies model horizons.

The extension problem

Let $(M,g)$ be a connected time-oriented Lorentzian manifold. A proper $C^k$ extension consists of a connected manifold $\widetilde M$, a $C^k$ Lorentzian metric $\widetilde g$, and a $C^{k+1}$ isometric embedding $\iota:M\to\widetilde M$ whose image is proper. At low regularity, one specifies the differentiability of the embedding and metric separately. The future extension boundary is

$$\partial^+\iota(M) =\left\{p\in\partial\iota(M): \text{a future timelike curve in }\widetilde M \text{ reaches }p\text{ from }\iota(M) \right\}.$$

To prove inextendibility, assume such a triple exists and derive a contradiction near a boundary point accessible from $M$. The accessibility step is not automatic: a topological boundary point could be approached only in a wildly oscillatory or spacelike way. Global hyperbolicity supplies compact causal diamonds and maximizing curves that convert the topological extension into a controlled causal approach.

Timelike completeness obstructs continuous extension

The cleanest low-regularity theorem is independent of Einstein’s equation.

Theorem (Galloway–Ling–Sbierski). A smooth, time-oriented, globally hyperbolic, timelike-geodesically complete spacetime is $C^0$-inextendible.

The proof assumes a continuous extension and selects a boundary point reached by a timelike curve, reversing time orientation if the accessible boundary is in the past. In a sufficiently small coordinate neighborhood the continuous cone field is squeezed between two constant Minkowski cone fields. Global hyperbolicity gives a length-maximizing causal curve from an interior point to points approaching the boundary. The cone comparison and push-up argument show that an interior segment is a timelike geodesic with uniformly bounded proper time that exits every compact subset. Passing to the endpoint produces a future-inextendible timelike geodesic of finite length, contradicting timelike completeness Galloway–Ling–Sbierski 2018, Theorem 1 and §§2–3.

Both hypotheses matter. Without global hyperbolicity, maximizing curves and compact diamonds can fail. Without timelike completeness, the contradiction is absent. The theorem does not say that every null-complete spacetime is $C^0$-inextendible.

Curvature and connection obstructions

Suppose a scalar polynomial curvature invariant $I(g)$ tends to infinity along a sequence whose image approaches $p\in\widetilde M$. A $C^2$ extension has continuous Riemann tensor and hence bounded $I(\widetilde g)$ on a compact coordinate neighborhood of $p$. This contradiction rules out $C^2$ extension. The same architecture works with parallelly propagated curvature components if the curve and orthonormal frame have controlled limits.

At weaker thresholds, use the quantity the field equation actually requires. For a Christodoulou-type class, assume every candidate continuous extension has

$$\int_U \lvert\widetilde\Gamma\rvert^2\,d\mu_{\widetilde g}=\infty$$

in every neighborhood $U$ of the boundary point, in all admissible gauges or via an equivalent invariant formulation. Then no extension belongs to the stipulated Christoffel-$L^2$ weak-solution class. This does not contradict a bare $C^0$ extension, for which $\widetilde\Gamma$ need not exist.

Here the absolute value is a positive norm of the coordinate components, not a contraction with the indefinite Lorentzian metric, and the measure is the positive density $\sqrt{|\det\widetilde g|}\,d^4x$. On a relatively compact regular chart, bounded positive changes of this component norm do not change local square integrability. The regularity allowed for changes of chart is still essential because a connection has an inhomogeneous transformation law.

The limitation is crucial: divergence of coordinate Christoffel symbols in one chart is not invariant. A valid proof either restricts the admissible atlas and embedding regularity or identifies a geometric integral—such as mass inflation or tidal deformation—from which nonintegrability follows in every permitted extension.

Spacelike diameter and Schwarzschild

Curvature cannot be evaluated on a hypothetical continuous extension, so a $C^0$ proof needs metric and causal quantities. For a globally hyperbolic region $N$, define its spacelike diameter by

$$\operatorname{diam}_{\mathrm s}(N) =\sup_{\Sigma}\operatorname{diam}(\Sigma,d_{\Sigma}),$$

where the supremum ranges over Cauchy hypersurfaces of $N$ and $d_\Sigma$ is their induced Riemannian distance. A region contained in a sufficiently small continuous Lorentzian chart, with controlled cone slope and a regular foliation, has finite spacelike diameter.

Near the Schwarzschild $r=0$ boundary, however, one can choose points on Cauchy slices whose intrinsic separation grows without bound in a carefully selected globally hyperbolic subregion. If a $C^0$ extension existed, the same subregion would fit inside a finite-diameter extension chart, a contradiction. Sbierski combines this interior argument with causal accessibility of any candidate exterior boundary to prove:

The maximal analytic Schwarzschild spacetime is $C^0$-inextendible.

The construction and finite-diameter theorem appear in Sbierski 2018, Theorems 1.1 and 5.3, §§3–6; a streamlined boundary argument is given in Sbierski 2017, Theorem 3.1 and §§2–5. The proof is not merely “the Kretschmann scalar diverges.” That observation settles $C^2$, while the diameter contradiction reaches $C^0$.

Other invariant strategies

Several methods follow the same extension-first logic.

Causal mismatch. A continuous extension forces local chronological relations comparable to Minkowski space. If the original spacetime develops a causal configuration that no such chart can realize—often through diverging time separation or failure of a compactness property—the extension is impossible.

Holonomy or transport. Controlled loops or frames approaching the boundary may have incompatible limiting transport. This usually targets a class with a defined connection, not bare $C^0$.

Topology. An extension can be blocked by the topology of the end or by incompatible identifications, even when curvature is bounded. The proof must exclude a different embedding that resolves the apparent obstruction.

PDE continuation. If a local theorem continues the solution whenever a geometric norm $\mathcal N(t)$ remains finite, a finite maximal time implies $\limsup_{t\uparrow T}\mathcal N(t)=\infty$. This proves breakdown of that continuation package. Translating it into $C^k$ inextendibility requires an additional theorem that every $C^k$ extension would bound $\mathcal N$ in a compatible gauge.

Proof checklist

Before accepting an inextendibility argument, verify:

1. the extension metric, embedding, matter, and equation regularities;
2. a boundary point accessible by a controlled causal curve;
3. a consequence every extension in that class must satisfy;
4. an invariant or gauge-robust estimate contradicting that consequence;
5. uniformity on the generic data set if the result is used for censorship.

A divergent scalar proves neither step 2 nor a $C^0$ contradiction. A local coordinate singularity proves none of the five. Conversely, a causal-diameter argument can prove $C^0$ inextendibility without mentioning curvature at all.

Check the method

Why does timelike geodesic completeness plus global hyperbolicity rule out a $C^0$ extension, while null geodesic completeness alone does not follow from the same proof?

Solution

A continuous extension provides a boundary point accessible by a timelike curve. Local cone comparison and global hyperbolicity produce maximizing timelike segments whose limit is an inextendible timelike geodesic reaching the boundary in finite proper time. Timelike completeness contradicts that curve. The construction is timelike: the accessible boundary curve can be pushed strictly inside the cones, and its positive Lorentzian length is controlled. It need not yield a finite-affine-length null geodesic, so null completeness alone supplies no contradiction.

References

• Galloway, Gregory J., Eric Ling, and Jan Sbierski. “Timelike Completeness as an Obstruction to $C^0$-Extensions.” Communications in Mathematical Physics 359 (2018): 937–949. arXiv.
• Sbierski, Jan. “On the Proof of the $C^0$-Inextendibility of the Schwarzschild Spacetime.” Journal of Physics: Conference Series 968 (2018): 012012. arXiv.
• Sbierski, Jan. “The $C^0$-Inextendibility of the Schwarzschild Spacetime and the Spacelike Diameter in Lorentzian Geometry.” Journal of Differential Geometry 108 (2018): 319–378. arXiv.