Cosmic censorship hypothesis

Summary

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Why it is open

Under a wide variety of circumstances, solutions of Einstein's equations with physically reasonable matter must develop singularities [4]. In 1969 Roger Penrose proposed that a "cosmic censor" prevents "naked" singularities, ones visible to a distant observer [1, 4, 12]. In Robert Wald's summary, the weak cosmic censorship conjecture says that, generically, the singularities of gravitational collapse are contained in black holes, and the strong conjecture that, generically, "timelike singularities never occur, so that even an observer who falls into a black hole will never 'see' the singularity" [4]. In current mathematical work the weak form says that generic asymptotically flat initial data have a complete future null infinity, so that distant observers can live out their lives unaffected [4, 12]. Mihalis Dafermos and Jonathan Luk note that the modern weak and strong formulations "are not logically related in the way these adjectives would suggest" [13].

A resolution would be a proof or a disproof of a precise version, and making the conjecture precise is part of the problem. The word "generic" has to be defined; it was added to the weak conjecture after Demetrios Christodoulou found naked singularities in a model, and the answer may depend on how regular the initial data and the allowed extensions of spacetime are required to be [12]. For the strong conjecture, the regularity required of an extension is central: Dafermos and Luk's work suggests that a spacetime may be extendable with a continuous metric but not with a smoother one [13]. In 1997 Wald noted that mathematical techniques were not yet able to attack a general proof directly [4]. If the weak conjecture fails, Wald wrote, "we would come face-to-face with the laws of quantum gravity" wherever collapse produced a naked singularity [4].

What is known

Early evidence. Global existence is proven for nearly flat initial data, so the weak conjecture holds where gravity is too weak to form singularities; in 1997 Wald described the remaining evidence as largely "anecdotal" or "circumstantial", or confined to special cases [4].

Naked singularities in a model, and their instability (1994–1999). For a spherically symmetric scalar field coupled to gravity, Christodoulou constructed in 1994 self-similar solutions that are asymptotically flat and have an incomplete future null infinity, that is, naked singularities [2, 12]. Their initial data are smooth except where they meet the past light cone of the singularity, where a derivative is only Hölder continuous; whether smooth, or even twice-differentiable, data can lead to a naked singularity in this model remained open in a 2025 review [12]. In 1999 he proved that a general class of such naked singularities is unstable: adding any non-zero combination of two specific perturbations to the initial data restores a complete future null infinity [3, 12]. Yakov Shlapentokh-Rothman reads later work by Jaydeep Singh as consistent with the self-similar examples being stable against more regular perturbations, and concludes that weak cosmic censorship "may depend sensitively on the precise functional framework" [12].

Without matter (2023). Igor Rodnianski and Shlapentokh-Rothman constructed the exterior region of a naked singularity for the Einstein vacuum equations (gravity without matter) in 3+1 dimensions; with a companion interior construction, the review states that vacuum naked singularities exist [10, 12]. Their data are N times differentiable for any chosen N, except on the past light cone of the singularity, where one quantity is only Hölder continuous [12].

Other settings. In five dimensions, simulations by Luis Lehner and Frans Pretorius indicate that unstable black strings end in a naked singularity without fine-tuning, which they call "a generic violation of cosmic censorship" [5]. With anti-de Sitter boundary conditions, Toby Crisford and Jorge Santos presented what they call "the first counterexample to the weak cosmic censorship conjecture in four spacetime dimensions" [6]. From a study of linear perturbations of charged black holes in a universe with a positive cosmological constant, Vitor Cardoso and colleagues conjectured that strong cosmic censorship is violated for nearly extremal charge [7].

Strong cosmic censorship and rotating black holes (2017–2025). Inside a rotating Kerr black hole lies a Cauchy horizon, beyond which initial data no longer determine the solution [13]. Dafermos and Luk proved that, for data of the kind expected just inside a dynamical rotating black hole, spacetime extends across part of a Cauchy horizon with a continuous metric [13]. They write that, with planned further work and a proof that the Kerr exterior is stable, "as is widely expected", this would show the continuous-metric (C⁰) formulation to be false, while their proof suggests that such horizons are generically "weak null singularities", so that a formulation due to Christodoulou could still hold [13]. In a charged, spherically symmetric model where continuous extensions do exist, Luk and Sung-Jin Oh proved in a two-part series the version that forbids twice-differentiable (C²) extensions for generic two-ended data [8]. Sergiu Klainerman and Jérémie Szeftel proved the nonlinear stability of the Kerr family for small angular momentum, in a series whose last part is joint with Elena Giorgi [9, 11].

Recent claims. On 6 October 2026 OpenAI released a repository of manuscripts "produced by an internal OpenAI model", including a family of three manuscripts dated 23 September 2026 that the repository summarises as proving "local strong cosmic censorship near each fixed rotating subextremal Kerr bridge" [14]. The principal manuscript claims that, near each fixed rotating Kerr solution below extremality, the two-ended vacuum initial data whose developments can be extended with a continuous metric and locally square-integrable connection form a meagre set (small in the sense of Baire category); it places this at the threshold that, it says, Christodoulou identified for formulating strong cosmic censorship, and asserts nothing for the non-rotating or extremal cases or outside these neighbourhoods [14]. The repository says its results are "at different stages of verification" and that "some of the unformalized results could have issues"; its formalization catalogue lists no Lean proof for this family, and its history recorded no revision or withdrawal of these manuscripts by 14 October 2026 [14].

Main approaches

Symmetric model problems. Study collapse where symmetry reduces the equations, chiefly a spherically symmetric scalar field. Strongest evidence: rigorous naked-singularity examples and their instability [2, 3], and the twice-differentiable strong conjecture in a charged model [8]. Main difficulty: in vacuum, spherical symmetry allows only the static Schwarzschild solution, so these results cannot be transferred directly [12].

Trapped surfaces. Show that concentrated gravitational radiation forms a trapped surface, one whose area shrinks along both families of light rays, a sign of strong gravity; in the scalar-field model, Christodoulou's instability proof works by forcing such surfaces to form [12]. Strongest evidence: Christodoulou's proof that trapped surfaces form in vacuum from initial data free of them, and its extensions [12]. Main difficulty: there is still no vacuum theorem as strong as the spherically symmetric criterion, and Shlapentokh-Rothman writes that "it is far from clear that such an analogue can exist without symmetry assumptions" [12].

Stability of black holes, outside and inside. Prove that perturbed black holes settle down outside, and determine how singular their interiors become [9, 13]. Strongest evidence: Kerr exterior stability for small angular momentum [9, 11] and continuous-metric stability of the Kerr Cauchy horizon [13]. Main difficulty: the exterior proof is restricted to small angular momentum, a restriction its authors trace to one family of estimates [9], and the interior answer depends on the regularity demanded [13].

Constructing counterexamples. Build naked singularities directly, by self-similarity or by simulation [5, 10]. Strongest evidence: vacuum naked singularities [10] and violations in other settings [5, 6]. Main difficulty: the vacuum examples do not come from smooth data and their stability is open [12]; the simulated violations lie in five dimensions or under anti-de Sitter boundary conditions [5, 6].

Open sub-questions

1. Stability of vacuum naked singularities. Are the vacuum naked singularities of Rodnianski and Shlapentokh-Rothman stable or unstable, and does the answer depend on regularity as in Singh's scalar-field work [10, 12]?
2. Smooth data. Can smooth, or even twice-differentiable, initial data lead to a naked singularity, for the spherically symmetric scalar field or in vacuum [12]?
3. Higher dimensions. Do asymptotically flat naked singularities exist in higher dimensions, and how generic are they [5, 12]?
4. Kerr for all rotation rates. Can the nonlinear stability of the Kerr exterior be proven beyond small angular momentum [9, 11]?
5. Christodoulou's formulation near Kerr. For generic data near Kerr, do Cauchy horizons become weak null singularities across which no extension has locally square-integrable connection, as Dafermos and Luk's work suggests and a 2026 manuscript claims locally [13, 14]?
6. Positive cosmological constant. Is strong cosmic censorship violated for nearly extremal charged black holes with a positive cosmological constant, and does the answer change when rough initial data are allowed [7, 12]?

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Key references

1. Penrose, R. (1969). Gravitational collapse: the role of general relativity. Rivista del Nuovo Cimento, Numero Speciale 1, 252–276; reprinted as a "Golden Oldie" in General Relativity and Gravitation 34(7), 1141–1165 (2002). 10.1023/A:1016578408204 · Checked 14 Oct 2026
2. Christodoulou, D. (1994). Examples of naked singularity formation in the gravitational collapse of a scalar field. Annals of Mathematics 140(3), 607–653. 10.2307/2118619 · Checked 14 Oct 2026
3. Christodoulou, D. (1999). The instability of naked singularities in the gravitational collapse of a scalar field. Annals of Mathematics 149(1), 183–217. 10.2307/121023 · Checked 14 Oct 2026
4. Wald, R. M. (1999). Gravitational collapse and cosmic censorship. In Black Holes, Gravitational Radiation and the Universe, Springer, 69–86. 10.1007/978-94-017-0934-7_5 · Checked 14 Oct 2026
5. Lehner, L. & Pretorius, F. (2010). Black strings, low viscosity fluids, and violation of cosmic censorship. Physical Review Letters 105(10), 101102. 10.1103/PhysRevLett.105.101102 · Checked 14 Oct 2026
6. Crisford, T. & Santos, J. E. (2017). Violating the weak cosmic censorship conjecture in four-dimensional anti–de Sitter space. Physical Review Letters 118(18), 181101. 10.1103/PhysRevLett.118.181101 · Checked 14 Oct 2026
7. Cardoso, V., Costa, J. L., Destounis, K., Hintz, P. & Jansen, A. (2018). Quasinormal modes and strong cosmic censorship. Physical Review Letters 120(3), 031103. 10.1103/PhysRevLett.120.031103 · Checked 14 Oct 2026
8. Luk, J. & Oh, S.-J. (2019). 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(1), 1–111. 10.4007/annals.2019.190.1.1 · Checked 14 Oct 2026
9. Klainerman, S. & Szeftel, J. (2023). Kerr stability for small angular momentum. Pure and Applied Mathematics Quarterly 19(3), 791–1678. 10.4310/pamq.2023.v19.n3.a1 · Checked 14 Oct 2026
10. Rodnianski, I. & Shlapentokh-Rothman, Y. (2023). Naked singularities for the Einstein vacuum equations: the exterior solution. Annals of Mathematics 198(1), 231–391. 10.4007/annals.2023.198.1.3 · Checked 14 Oct 2026
11. Giorgi, E., Klainerman, S. & Szeftel, J. (2024). Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes. Pure and Applied Mathematics Quarterly 20(7), 2865–3849. 10.4310/pamq.241128023033 · Checked 14 Oct 2026
12. Shlapentokh-Rothman, Y. (2025). Weak cosmic censorship, trapped surfaces, and naked singularities for the Einstein vacuum equations. Comptes Rendus. Mécanique 353, 379–410 (review article). 10.5802/crmeca.284 · Checked 14 Oct 2026
13. Dafermos, M. & Luk, J. (2025). The interior of dynamical vacuum black holes I: the C⁰-stability of the Kerr Cauchy horizon. Annals of Mathematics 202(2), 309–630. 10.4007/annals.2025.202.2.1 · Checked 14 Oct 2026
14. OpenAI (2026). Generic future inextendibility with square-integrable connection near a fixed Kerr spacetime. Manuscript dated 23 September 2026, in the repository openai/math (family 264, "Strong cosmic censorship near two-ended Kerr data", with two companion manuscripts on C¹ and C² inextendibility), released 6 October 2026, not peer reviewed. Source · Checked 14 Oct 2026

Page reviewed 14 Oct 2026

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Open, as listed by Wikipedia · Unsolved problems in physics. Checked 08 Oct 2026.

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