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Strong cosmic censorship near two-ended Kerr data

Claims typical vacuum perturbations of a rotating Kerr black hole cannot be smoothly continued past the inner horizon.

3 papers September 23, 2026 Explained by AI

Strong cosmic censorship asks whether Einstein's equations predict the future uniquely from initial data. An exact rotating black hole fails this, because spacetime can be continued past its inner horizon in many ways. Three papers claim that for typical small vacuum perturbations of such a black hole, no continuation exists past that point, even at low smoothness. The claims are local to each fixed black hole, and no Lean check exists.

First step
1979
Roger Penrose, strong formulation of cosmic censorship tied to the inner-horizon blueshift
Significance
Major
Strong cosmic censorship near Kerr is a central question in mathematical general relativity, and these papers claim a generic form without symmetry.
Trust
Claimed, not checked
Field
Mathematical physics
3 papers

The explanation

A rotating black hole in vacuum is described by the Kerr solution. It has a mass M and a rotation a, with 0 < a < M for the subextremal case. Such a black hole has two horizons, at r = M ± √(M² − a²). Take M = 1 and a = 0.6. The outer horizon sits at r = 1.8 and the inner one at r = 0.2. The inner one is a Cauchy horizon. Past it, the initial data no longer decide what happens.

Here is the background. Einstein's equations turn an initial snapshot of space into a spacetime. Choquet-Bruhat and Geroch showed there is a largest such spacetime, unique up to relabeling. It is called the maximal globally hyperbolic development. For exact Kerr, the interior contains a Cauchy horizon, so the solution can be continued past it. That continuation is not fixed by the data, which breaks determinism. Strong cosmic censorship hopes this is not typical. Penrose linked it to the infinite blueshift at an inner horizon. Light falling in piles up there, which may destroy the smoothness needed to continue.

The answer depends on how smooth a continuation must be. Dafermos and Luk showed that a continuous extension across part of the Cauchy horizon persists under perturbation. So the metric itself can survive. The question becomes whether its derivatives can. Luk and Oh proved a generic C² version, but only in a spherically symmetric model with matter. Here C² means two continuous derivatives. Removing the symmetry is the main obstacle.

The three papers claim results in this setting, near each fixed rotating Kerr black hole. The data are smooth, two-ended vacuum data with no symmetry condition. Genericity is measured in the sense of Baire category. A meagre set is a negligible, thin set; its complement is generic. One paper claims that data admitting a C² future extension are meagre. Another claims a dense, generic set of data with no future C¹ extension. The third goes lower. It claims extensions whose metric is continuous but whose derivatives are only square-integrable also form a meagre set. So a typical perturbation leaves no continuation at these levels of smoothness.

How sure is this? There is no Lean formalization for these papers. The claims rest on three long papers. No specialist review of them has been published. The results are local to each fixed Kerr black hole. They do not rule out continuous extensions, which Dafermos and Luk showed can exist.

What it doesn't show. This does not prove strong cosmic censorship in general, rule out continuous (C⁰) extensions, or describe what forms at the inner horizon.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. There is no Lean check. The claims rest on three long papers with no published specialist review, and they are local near each Kerr black hole.

Words used

Kerr solution
The exact solution of Einstein's vacuum equations for a rotating black hole with mass M and rotation a.
Subextremal
Rotation smaller than mass, 0 < a < M, so the black hole has two separate horizons.
Cauchy horizon
A boundary past which the initial data no longer determine the spacetime.
Maximal globally hyperbolic development
The largest spacetime that the initial data determine uniquely through Einstein's equations.
Strong cosmic censorship
The conjecture that, for typical data, that largest spacetime cannot be continued, so physics stays deterministic.
C¹, C²
Smoothness levels: one or two continuous derivatives of the metric.
Meagre set
A thin, negligible set in the sense of Baire category; data outside it are called generic.

The original papers (3)

All quoted lines (9)
  1. Four\‘es-Bruhat proved local existence and geometric uniqueness; Choquet-Bruhat and Geroch established a maximal globally hyperbolic development, unique up to an isometry preserving the initial data.
    Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes Back to text
  2. Their subextremal interiors contain Cauchy horizons, so the exact solutions admit continuations beyond their maximal Cauchy developments.
    Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes Back to text
  3. Penrose's discussion of the strong formulation relates this question to global hyperbolicity and to the infinite blueshift at an inner horizon.
    Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes Back to text
  4. Dafermos and Luk prove persistence of a continuously extendible Cauchy-horizon portion from suitable Cauchy data already inside a black hole.
    Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes Back to text
  5. In that model, Luk and Oh proved a generic C2C^2 inextendibility theorem for admissible complete two-ended asymptotically flat data, combining exterior lower bounds with an interior instability argument.
    Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes Back to text
  6. The data are smooth, complete, two-ended vacuum data with no symmetry condition.
    Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes Back to text
  7. Nevertheless, we prove that data admitting any C2C^2 future extension are meagre in a weighted smooth neighborhood of its bridge data.
    Quantitative Near-Kerr Evolution and Generic C2 Future Inextendibility Back to text
  8. For each fixed Kerr spacetime with mass M>0M>0 and rotation 0<a<M0<\mathfrak a<M, we prove that a dense GδG_\delta set of nearby smooth, complete two-ended vacuum data has a maximal globally hyperbolic development with no future C1C^1 extension.
    Generic C1 Future Inextendibility Near Rotating Subextremal Kerr Spacetimes Back to text
  9. For each fixed rotating subextremal Kerr background with mass M>0M>0 and rotation 0<∣a∣<M0<|\mathfrak a|<M, we prove that the smooth vacuum data near its complete two-ended bridge whose full maximal globally hyperbolic development admits a future C0∩Wloc1,2C^0\cap W^{1,2}_{\mathrm{loc}} extension form a meagre set in the weighted smooth topology.
    Generic Future Inextendibility with Square-Integrable Connection Near a Fixed Kerr Spacetime Back to text

Other sources

  1. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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