The $H^1$ (Christodoulou) inextendibility of the Cauchy horizon of generic rotating vacuum black holes

The source paper (Sbierski, Annals of PDE 2023) proved the blue-shift instability of the Kerr Cauchy horizon for linearised gravity — the spin $+2$ Teukolsky field becomes non-square-integrable at the Cauchy horizon for generic decay along the event horizon — and posed the problem of promoting it to the nonlinear vacuum Einstein equations, across the full sub-extremal range, with a coordinate-invariant inextendibility conclusion in the class $g\in C^0$, $\partial g\in L^2_{\mathrm{loc}}$. Later work resolved large parts of this program: Price's-law decay for the Teukolsky equation now covers the full sub-extremal range $|a|<M$ (Millet; Ma–Zhang for slow rotation), and in 2026 Luk–Sbierski, and independently Gurriaran, proved — conditional on Price's-law-type horizon decay assumptions — that generic rotating vacuum black holes develop a weak null singularity at the Cauchy horizon, across which the metric is continuously extendible but not Lipschitz extendible. The open core is the strongest surviving formulation: whether the metric connection is generically non-square-integrable at the Cauchy horizon, i.e. whether the maximal globally hyperbolic development of the generic class admits no $C^0$ Lorentzian extension with locally square-integrable Christoffel symbols — the Christodoulou ($H^1$) formulation of strong cosmic censorship for rotating vacuum black holes. The question is two-sided: in the positive-cosmological-constant analogue the $H^1$ formulation is expected to fail near extremality.

Background

The Kerr spacetimes form the two-parameter family $(M,a)$ of stationary rotating vacuum solutions of Einstein's equation $\mathrm{Ric}(g)=0$; in the sub-extremal range $0<|a|<M$ they model the expected final states of astrophysical black holes. Inside the event horizon $\mathcal{H}^+$ (the boundary of the region from which no signal escapes), the sub-extremal Kerr interior contains a further null hypersurface, the Cauchy horizon $\mathcal{CH}^+$, across which the Kerr metric extends smoothly — but non-uniquely — as a solution. This non-uniqueness threatens the deterministic character of general relativity, and Penrose's strong cosmic censorship conjecture asserts that such extendibility is non-generic. The mechanism expected to save determinism is the blue-shift instability: radiation decaying along $\mathcal{H}^+$ at an inverse-polynomial (Price's-law) rate is infinitely blue-shifted for observers approaching $\mathcal{CH}^+$, heuristically driving mass inflation, i.e. blow-up of curvature at the Cauchy horizon (Poisson–Israel). Because Dafermos–Luk proved the $C^0$-stability of the Kerr Cauchy horizon — for perturbations with suitable decay the interior metric remains continuously extendible across $\mathcal{CH}^+$ — the $C^0$-inextendibility formulation of strong cosmic censorship fails, and any generic singularity must be a weak null singularity: the metric stays $C^0$ while the connection and curvature blow up. The surviving formulations are distinguished by regularity: the Christodoulou, or $H^1$, formulation demands inextendibility of the maximal globally hyperbolic development (the unique maximal solution spawned by given initial data) as a $C^0$ Lorentzian metric with locally square-integrable Christoffel symbols, $\Gamma\in L^2_{\mathrm{loc}}$ (equivalently $g\in C^0$ with $\partial g\in L^2_{\mathrm{loc}}$), which precludes even weak-solution extensions; Lipschitz ($C^{0,1}$) inextendibility sits strictly between this and the classical $C^2$ formulation, since Lipschitz metrics form a proper subclass of $C^0$ metrics with locally square-integrable derivatives. On the linear level, gravitational perturbations of Kerr are governed by the spin $s=\pm2$ Teukolsky equation for the extreme curvature components of the linearized field in the Newman–Penrose formalism (which projects the curvature onto a principal null frame of the background). Sbierski (2023) proved the blue-shift instability for linearised gravity: if the spin $+2$ Teukolsky field satisfies suitable integrated upper and lower decay bounds along the event horizon, then it fails to be square-integrable at the Cauchy horizon with respect to a frame regular there. Those bounds were initially available only for slowly rotating Kerr (Ma–Zhang); Millet subsequently proved Price's-law decay for the full sub-extremal range $|a|<M$, and Gurriaran derived precise oscillatory blow-up asymptotics of the Teukolsky field in the interior of general sub-extremal Kerr. In 2026 the nonlinear promotion of this program was completed at the Lipschitz level: Luk–Sbierski, and independently Gurriaran, proved that for characteristic initial data on a dynamical event horizon settling down to a sub-extremal strictly rotating Kerr horizon, with Price's-law-type upper bounds and a lower bound on the $l=2$ Teukolsky mode, a weak null singularity forms at the Cauchy horizon, across which the metric is continuously extendible but not Lipschitz extendible — the latter via Sbierski's coordinate-invariant criterion that curvature blow-up at a weak null singularity implies $C^{0,1}_{\mathrm{loc}}$-inextendibility. Since ruling out Lipschitz extensions is strictly weaker than ruling out all $C^0\cap W^{1,2}_{\mathrm{loc}}$ extensions, the strongest open question left by this chain is whether the connection at the Cauchy horizon is itself generically non-square-integrable.

Problem Statement

Determine whether the maximal globally hyperbolic development of the generic class of perturbed rotating vacuum black holes — concretely, the solutions constructed by Luk–Sbierski and Gurriaran from, respectively, smooth characteristic data on a dynamical event horizon settling down to a sub-extremal strictly rotating Kerr spacetime ($0<|a|<M$) with Price's-law-type upper bounds and a lower bound on the $l=2$ Teukolsky mode, and smooth initial data on a hypersurface slightly inside a dynamical black hole settling down to a sub-extremal Kerr spacetime and satisfying a precise nonlinear Price's-law-type estimate, in both classes the Cauchy horizon being now known to be a weak null singularity with Lipschitz-inextendible metric — is inextendible across the Cauchy horizon as a $C^0$ Lorentzian metric with locally square-integrable Christoffel symbols ($g\in C^0$, $\partial g\in L^2_{\mathrm{loc}}$): that is, determine whether any extension of this regularity exists, in a coordinate-invariant sense, regardless of the coordinates or frame chosen for the extension. Inextendibility in this class is equivalent to the connection components at the weak null singularity failing to be square-integrable, so that no weak-solution extension across the Cauchy horizon exists; this is the Christodoulou ($H^1$) formulation of strong cosmic censorship for rotating vacuum black holes. Extendibility means the existence, for solutions within the generic class, of a $C^0\cap W^{1,2}_{\mathrm{loc}}$ Lorentzian extension across the Cauchy horizon whose first derivatives are locally square-integrable. The source cluster also leaves open two auxiliary questions — deriving the horizon-decay assumptions from generic asymptotically flat Cauchy data (tied to the open exterior stability of Kerr beyond slow rotation) and computing the pole/zero structure at $\omega=0$ of the Teukolsky reflection coefficient $\mathfrak{R}_{\mathcal{H}_r^+,0l}(\omega)$.

References

1. Jan Sbierski, Instability of the Kerr Cauchy Horizon Under Linearised Gravitational Perturbations, Annals of PDE 9, Art. 7 (2023), DOI 10.1007/s40818-023-00146-9, arXiv:2201.12295, https://arxiv.org/abs/2201.12295
2. Jan Sbierski, A note on the instability of the Kerr Cauchy horizon under linearised gravitational perturbations (2026), arXiv:2604.06283, https://arxiv.org/abs/2604.06283
3. Jonathan Luk and Jan Sbierski, The formation of a weak null singularity in the interior of generic rotating black holes (2026), arXiv:2604.04877, https://arxiv.org/abs/2604.04877
4. Sebastian Gurriaran, Non-linear instability of the Kerr Cauchy horizon near $i_+$ (2026), arXiv:2603.17911, https://arxiv.org/abs/2603.17911
5. Sebastian Gurriaran, Precise Asymptotics of the Spin $+2$ Teukolsky Field in the Kerr Black Hole Interior, Communications in Mathematical Physics 406, Art. 152 (2025), DOI 10.1007/s00220-025-05332-3, arXiv:2409.02670, https://arxiv.org/abs/2409.02670
6. Jan Sbierski, Lipschitz inextendibility of weak null singularities from curvature blow-up, Inventiones mathematicae 243, 961–991 (2026), DOI 10.1007/s00222-025-01387-0, arXiv:2409.18838, https://arxiv.org/abs/2409.18838
7. Pascal Millet, Optimal decay for solutions of the Teukolsky equation on the Kerr metric for the full subextremal range $|a|<M$ (2023), arXiv:2302.06946, https://arxiv.org/abs/2302.06946
8. Siyuan Ma and Lin Zhang, Sharp Decay for Teukolsky Equation in Kerr Spacetimes, Communications in Mathematical Physics 401, 333–434 (2023), DOI 10.1007/s00220-023-04640-w, https://doi.org/10.1007/s00220-023-04640-w
9. Siyuan Ma and Lin Zhang, Precise late-time asymptotics of scalar field in the interior of a subextreme Kerr black hole and its application in Strong Cosmic Censorship conjecture, Transactions of the American Mathematical Society (2023), DOI 10.1090/tran/8957, arXiv:2208.08702, https://arxiv.org/abs/2208.08702
10. Mihalis Dafermos and Jonathan Luk, The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon, Annals of Mathematics 202, 309–630 (2025), DOI 10.4007/annals.2025.202.2.1, https://doi.org/10.4007/annals.2025.202.2.1
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12. Peter Cameron and Jan Sbierski, On the uniqueness of continuous spacetime extensions in 1+1 dimensions with applications to weak null singularities (2025), arXiv:2511.13422, https://arxiv.org/abs/2511.13422