Math Conjectures

C2C^2 strong cosmic censorship

OPENLandmarkConjectureProposed c. 1979 · Canonical special case

Canonical statement

Fix a closed smooth 33-manifold Σ\Sigma and a smooth background Riemannian metric bb. Let Dvac(Σ)\mathcal D_{\mathrm{vac}}(\Sigma) be the space of smooth pairs (h,K)(h,K), with hh Riemannian and KK symmetric, satisfying Rh−∣K∣h2+(tr⁡hK)2=0R_h-|K|_h^2+(\operatorname{tr}_hK)^2=0 and div⁡hK−d(tr⁡hK)=0\operatorname{div}_hK-d(\operatorname{tr}_hK)=0. Give it the relative Fréchet topology induced by qm(γ,L)=∥γ∥Cm(b)+∥L∥Cm(b)q_m(\gamma,L)=\|\gamma\|_{C^m(b)}+\|L\|_{C^m(b)}, m=0,1,…m=0,1,\ldots, on differences (γ,L)=(h−h′,K−K′)(\gamma,L)=(h-h',K-K'). In every nonempty connected component of Dvac(Σ)\mathcal D_{\mathrm{vac}}(\Sigma), the data whose maximal globally hyperbolic development admits no proper isometric embedding into a connected Lorentzian manifold with C2C^2 metric form a residual set, meaning a countable intersection of open dense sets.
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Fix a closed smooth \(3\)-manifold \(\Sigma\) and a smooth background Riemannian metric \(b\). Let \(\mathcal D_{\mathrm{vac}}(\Sigma)\) be the space of smooth pairs \((h,K)\), with \(h\) Riemannian and \(K\) symmetric, satisfying \(R_h-|K|_h^2+(\operatorname{tr}_hK)^2=0\) and \(\operatorname{div}_hK-d(\operatorname{tr}_hK)=0\). Give it the relative Fréchet topology induced by \(q_m(\gamma,L)=\|\gamma\|_{C^m(b)}+\|L\|_{C^m(b)}\), \(m=0,1,\ldots\), on differences \((\gamma,L)=(h-h',K-K')\). In every nonempty connected component of \(\mathcal D_{\mathrm{vac}}(\Sigma)\), the data whose maximal globally hyperbolic development admits no proper isometric embedding into a connected Lorentzian manifold with \(C^2\) metric form a residual set, meaning a countable intersection of open dense sets.
PDE-008:[Penrose1979SingularitiesTimeAsymmetry][DafermosLuk2025KerrInterior][VanDeMoortel2026SCCSurvey]
Partial differential equationsMathematical physicsGeometry and topology

Notes

Strong cosmic censorship expresses the determinism of general relativity: for generic initial data, the maximal globally hyperbolic development should be inextendible, so that no observer's future is left undetermined by the data. The version recorded here fixes vacuum data on a closed 33-manifold and asks that, generically in the sense of a residual set, the development admit no proper isometric embedding into a spacetime with C2C^2 metric. The conjecture took its modern PDE form around Penrose's 1979 formulation [Penrose1979SingularitiesTimeAsymmetry].

The regularity threshold in the statement is now known to be essential. Dafermos and Luk proved the C0C^0-stability of the Kerr Cauchy horizon: for the relevant dynamical vacuum black-hole interiors the metric extends continuously across a Cauchy horizon even as curvature is expected to blow up [DafermosLuk2025KerrInterior]. This falsifies the C0C^0-inextendibility version and leaves the C2C^2 formulation as the central viable one. The conjecture has been established in important symmetry classes and matter models at suitable regularities; the current landscape is surveyed in [VanDeMoortel2026SCCSurvey].

Outside such special settings the conjecture remains open, and a resolution requires genericity arguments for unrestricted vacuum data.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.