Cosmic Censorship and Inextendibility

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Cosmic censorship becomes mathematically testable only after two independent choices are fixed. Weak censorship selects an asymptotic receiver and asks whether singular behavior is visible there for generic data. Strong censorship selects an extension regularity and asks whether the maximal globally hyperbolic development can be continued. This chapter makes the data topology, genericity, matter model, asymptotics, and regularity part of every statement.

Helpful background. The Einstein equation supplies constraints and evolution; Causality and Global Structure supplies null infinity, MGHDs, and Cauchy horizons; PDEs and Green Functions for Gravitation and Analysis for Mathematical Relativity supply data and weak-solution topologies; Black-Hole Geometry and Mechanics supplies charged and rotating models.

Six formulation steps

The pages proceed from definitions to proof tools and only then to dated status.

Function-Space Formulations of Weak Cosmic Censorship packages field system, asymptotically flat data, topology, generic subset, conformal boundary, and completeness/visibility conclusion into one claim.
Regularity-Dependent Formulations of Strong Cosmic Censorship distinguishes $C^0$ , Lipschitz, Christoffel- $L^2$ , and $C^2$ inextendibility and explains why their logical strength is reversed from their differentiability order.
Inextendibility Criteria develops curvature, connection, completeness, causal, and spacelike-diameter contradictions appropriate to those different extension classes.
Cauchy Horizons in Charged and Rotating Models follows exterior decay into interior blue shift and separates continuous horizon stability from higher-regularity blowup.
Cosmological Constant and Censorship compares null, spacelike, and timelike conformal boundaries and explains why AdS boundary conditions are part of the conjecture.
Theorem Status and Open Censorship Problems records exact proved, conditional, preprint, countervailing, and unresolved cells through 2026-08-12.

The ordering prevents a common mistake: choosing evidence first and adjusting the conjecture afterward. Here the formulation is fixed before a theorem is placed in its cell.

Two conjectures, no automatic implication

For asymptotically flat data, a schematic weak statement is

d\in\mathcal G_{\mathrm W} \quad\Longrightarrow\quad \mathscr I^+(\operatorname{MGHD}(d)) \text{ exists and is complete}.

A schematic strong statement at regularity $\mathcal R$ is

d\in\mathcal G_{\mathrm S} \quad\Longrightarrow\quad \operatorname{MGHD}(d) \text{ is }\mathcal R\text{-inextendible}.

Complete future null infinity can coexist with a continuously extendible inner Cauchy horizon, so WCC does not imply $C^0$ SCC. An MGHD can be inextendible while its visible exterior fails to possess complete null infinity, so SCC does not imply WCC. Even the data spaces $\mathcal G_{\mathrm W}$ and $\mathcal G_{\mathrm S}$ need not be identical.

Within SCC, the extension classes satisfy

C^0\text{-inextendible} \Longrightarrow C^{0,1}\text{-inextendible} \Longrightarrow C^2\text{-inextendible}.

The converses fail. The two-ended spherical Einstein–Maxwell–scalar theorems make this concrete: the Cauchy horizon is continuously extendible while generic solutions are $C^2$ -inextendible Luk–Oh 2019 I, Theorems 1.8–1.10 and Luk–Oh 2019 II, Theorems 1.1 and 1.5.

A theorem-status discipline

Every entry in the chapter’s status matrix records:

vacuum or exact matter model;
symmetry and topology of the initial hypersurface;
one-ended, two-ended, characteristic, or interior data;
weighted Sobolev, Hölder, BV, or other topology;
open dense, residual, finite-codimension, or another genericity notion;
asymptotic class and boundary condition;
$C^0$ , weak, Lipschitz, $C^2$ , completeness, or visibility conclusion;
peer-reviewed, conditional, linear, or preprint status.

For example, Christodoulou’s instability of spherical scalar naked endpoints is a BV theorem, while a 2026 preprint claims their open-set stability in a localized rough Hölder space. The claims are not contradictory because the topologies differ. Likewise, the 2025 peer-reviewed $C^0$ stability of a Kerr Cauchy-horizon segment and a 2026 conditional Lipschitz-instability preprint can both hold: they ask for different extension classes.

Review the chapter

Suppose a theorem proves that, for an open dense set of two-ended spherical Einstein–Maxwell–scalar data, every MGHD has complete future null infinity, a continuous Cauchy-horizon extension, and no $C^2$ extension. Classify the corresponding WCC, $C^0$ SCC, and $C^2$ SCC statements. Can the conclusion be transferred to one-ended vacuum collapse?

Solution

In the theorem’s exact data class, completeness of the designated $\mathscr I^+$ supports the selected WCC formulation. The continuous proper extension refutes $C^0$ inextendibility and hence $C^0$ SCC there. The generic absence of a $C^2$ extension proves the $C^2$ SCC formulation there. No result transfers automatically to one-ended vacuum data: the number of ends, matter model, symmetry, constraint space, tail mechanism, and topology all change.

Where the chapter hands off

The global meaning of naked endpoints, future predictability, and Cauchy horizons is developed in Cosmic Censorship and Determinism. The formation mechanisms that supply trapped surfaces are in Trapped Surfaces and Singularity Formation. The meaning of completeness and regularity at $\mathscr I^+$ is developed next in Conformal Null Infinity, which supplies the asymptotic receiver used in weak-censorship formulations. The current analytical landscape is synthesized in Van de Moortel 2025, §§2–7, but primary theorem pages retain the final authority for their hypotheses.

For dated assessment views, see the supporting Research records on Weak Cosmic Censorship and Strong Cosmic Censorship and Regularity. The definitions, theorem hypotheses, and proof boundaries remain in this chapter.

References

Luk, Jonathan, and Sung-Jin Oh. “Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data I: The Interior of the Black Hole Region.” Annals of Mathematics 190 (2019): 1–111. arXiv.
Luk, Jonathan, and Sung-Jin Oh. “Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data II: The Exterior of the Black Hole Region.” Annals of PDE 5 (2019): 6. arXiv.
Van de Moortel, Maxime. “The Strong Cosmic Censorship Conjecture.” arXiv:2501.13180 (2025). arXiv.