Math Conjectures

Weak cosmic censorship (future-null-infinity form)

OPENIconicConjectureProposed 1969 · Standard version

Canonical statement

Let DAF\mathcal D_{\mathrm{AF}} be the space of smooth, complete vacuum initial data (h,K)(h,K) on R3\mathbb R^3 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. In fixed asymptotic coordinates, put r=∣x∣r=|x| and require ∂α(hij−δij)=O(r−1−∣α∣)\partial^\alpha(h_{ij}-\delta_{ij})=O(r^{-1-|\alpha|}) and ∂αKij=O(r−2−∣α∣)\partial^\alpha K_{ij}=O(r^{-2-|\alpha|}) for every multi-index α\alpha. Give DAF\mathcal D_{\mathrm{AF}} the relative weighted C∞C^\infty topology induced by the seminorms
qm(γ,L)=∑∣α∣≤msup⁡x(1+r)1+∣α∣∣∂αγ∣+∑∣α∣≤msup⁡x(1+r)2+∣α∣∣∂αL∣ q_m(\gamma,L)=\sum_{|\alpha|\le m}\sup_x(1+r)^{1+|\alpha|}|\partial^\alpha\gamma|+\sum_{|\alpha|\le m}\sup_x(1+r)^{2+|\alpha|}|\partial^\alpha L|
on differences (γ,L)=(h−h′,K−K′)(\gamma,L)=(h-h',K-K'). There is an open dense subset of DAF\mathcal D_{\mathrm{AF}} whose maximal globally hyperbolic developments have a conformal completion with complete future null infinity I+\mathscr I^+, meaning that every physical null geodesic ending at I+\mathscr I^+ has infinite affine length.
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Let \(\mathcal D_{\mathrm{AF}}\) be the space of smooth, complete vacuum initial data \((h,K)\) on \(\mathbb R^3\) satisfying \(R_h-|K|_h^2+(\operatorname{tr}_hK)^2=0\) and \(\operatorname{div}_hK-d(\operatorname{tr}_hK)=0\). In fixed asymptotic coordinates, put \(r=|x|\) and require \(\partial^\alpha(h_{ij}-\delta_{ij})=O(r^{-1-|\alpha|})\) and \(\partial^\alpha K_{ij}=O(r^{-2-|\alpha|})\) for every multi-index \(\alpha\). Give \(\mathcal D_{\mathrm{AF}}\) the relative weighted \(C^\infty\) topology induced by the seminorms \[ q_m(\gamma,L)=\sum_{|\alpha|\le m}\sup_x(1+r)^{1+|\alpha|}|\partial^\alpha\gamma|+\sum_{|\alpha|\le m}\sup_x(1+r)^{2+|\alpha|}|\partial^\alpha L| \] on differences \((\gamma,L)=(h-h',K-K')\). There is an open dense subset of \(\mathcal D_{\mathrm{AF}}\) whose maximal globally hyperbolic developments have a conformal completion with complete future null infinity \(\mathscr I^+\), meaning that every physical null geodesic ending at \(\mathscr I^+\) has infinite affine length.
PDE-007:[Penrose1969GravitationalCollapse][Christodoulou1999InstabilityNaked][Dafermos2005CosmicCensorship][Landsman2024WeakCosmicHistory]
Partial differential equationsMathematical physicsGeometry and topology

Notes

Weak cosmic censorship asserts, informally, that the singularities produced by generic gravitational collapse are hidden inside black holes rather than visible from far away. In the future-null-infinity formulation recorded here, one asks for an open dense set of smooth, asymptotically flat vacuum initial data whose maximal globally hyperbolic developments possess a complete future null infinity I+\mathscr I^+. The conjecture originates with Penrose's 1969 discussion of gravitational collapse [Penrose1969GravitationalCollapse]; its subsequent evolution as a precise statement is traced in [Landsman2024WeakCosmicHistory].

The genericity clause is essential: naked singularities do occur for exceptional data, and Christodoulou showed for the spherically symmetric scalar field that such examples are unstable, establishing the conjecture in that model [Christodoulou1999InstabilityNaked]. Beyond this, the statement is known in selected symmetry and matter classes and for small perturbations of Minkowski data, and it interacts with the uniqueness and interior problems for black holes [Dafermos2005CosmicCensorship]. Formulations in the literature differ in the admissible matter, the number of ends, decay rates, and the topology used to express genericity, so care is needed in comparing results.

No theorem yet covers generic, unrestricted vacuum collapse, and the conjecture remains open in that generality.

References (4)

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.