Penrose's strong cosmic censorship conjecture asserts that, for generic initial data, the maximal globally hyperbolic development of the Einstein equations is inextendible in a suitable regularity class. For charged black holes with positive cosmological constant Λ\Lambda, the Christodoulou–Chruściel (H1H^1) formulation — the weakest regularity in which extensions beyond the Cauchy horizon still count as weak solutions — is in doubt: exponential Price-law decay of perturbations in the black hole exterior can outweigh the blueshift instability at the Cauchy horizon near extremality. Rossetti (Ann. Henri Poincaré 2025) proved, from spherically symmetric characteristic data for the Einstein–Maxwell–charged-Klein–Gordon (EMcKG) system with Λ>0\Lambda>0 satisfying an assumed exponential Price-law bound of rate ss, that a Cauchy horizon exists with C0C^0 extensions, and, in the regime s>K−s>K_- and ρ=K−/K+<2\rho=K_-/K_+<2, that mass inflation is prevented and the metric extends with continuous metric, Christoffel symbols in Lloc2L^2_{\mathrm{loc}} and scalar field in Hloc1H^1_{\mathrm{loc}} — suggesting a violation of the H1H^1 conjecture. Whether the C0C^0 and H1H^1 conjectures actually hold for generic spacelike initial data remains open: the verdict hinges on (i) a rigorous sharp Price law (spectral gap) for the charged scalar field on Reissner–Nordström–de Sitter exteriors and its nonlinear, generic-data extension, (ii) the complete mass-inflation threshold classification in the (s,ρ)(s,\rho) plane, and (iii) the weak-solution and uniqueness status of the H1H^1 extensions. The problem is to resolve these conjectures for this system.

Background

Strong cosmic censorship (SCC), conjectured by Penrose, asserts that the future maximal globally hyperbolic development (MGHD) — the largest spacetime uniquely determined by initial data for the Einstein equations — is, for generic initial data, inextendible as a Lorentzian manifold whose metric lies in a specified regularity class. The charged black hole solutions threaten this: the Reissner–Nordström–de Sitter (RNdS) spacetime, i.e. the static spherically symmetric charged black hole with positive cosmological constant Λ>0\Lambda>0, admits a smooth future extension across an inner null boundary, the Cauchy horizon CH+\mathcal{CH}^+, so Laplacian determinism fails there unless dynamical perturbations destroy the horizon. The expected destruction mechanism is the blueshift instability: radiation arriving at CH+\mathcal{CH}^+ is infinitely blueshifted, inflating the Hawking mass (in spherical symmetry, the renormalized monopole part of the mass) and blowing up curvature — the mass inflation scenario, established for Λ=0\Lambda=0 by Dafermos and successors. For Λ>0\Lambda>0, however, scalar perturbations decay exponentially along the event horizon (Price's law), at a rate ss governed by the spectral gap of the wave operator, i.e. by the infimum of −Im⁡ω-\operatorname{Im}\omega over the quasinormal modes ω\omega — the resonant frequencies of the black hole exterior. Near extremality, when the surface gravities (the exponential rates of null coordinates near the horizons) K−K_- of the Cauchy horizon and K+K_+ of the event horizon satisfy ρ=K−/K+→1\rho=K_-/K_+\to 1, the blueshift weakens, and linear analyses (Costa–Franzen; Hintz–Vasy) show that waves on RNdS remain H1H^1-bounded at CH+\mathcal{CH}^+ precisely when s>K−s>K_- and ρ<2\rho<2.

The regularity class determines the formulation of SCC. The Christodoulou–Chruściel Hloc1H^1_{\mathrm{loc}} class — continuous metric with Christoffel symbols locally square-integrable, so that the metric itself is Hloc1H^1_{\mathrm{loc}} — is the weakest regularity in which the Einstein equations still make sense distributionally and extensions count as weak solutions; local well-posedness at this regularity is nevertheless not known.

The model problem couples the Einstein equations with the Maxwell field and a complex scalar field ϕ\phi of charge q≠0q\neq 0 and mass m≥0m\geq 0 obeying the charged Klein–Gordon equation (the Einstein–Maxwell–charged-Klein–Gordon, EMcKG, system), under spherical symmetry. Dafermos proved for Λ=0\Lambda=0 that the Reissner–Nordström Cauchy horizon is C0C^0-stable while mass inflation generically renders it weakly singular; Van de Moortel proved the analogous statement for the charged Klein–Gordon field with Λ=0\Lambda=0, providing the bootstrap framework of the present problem. For Λ>0\Lambda>0, Costa–Girão–Natário–Silva treated the uncharged massless scalar from characteristic data with an exponential Price-law bound: mass inflation occurs when the decay is too slow, while in the no-inflation regime the spacetime admits extensions of H1H^1 regularity. Rossetti (the source paper) generalized this to the charged, possibly massive, scalar field: local well-posedness of the characteristic problem, existence of CH+\mathcal{CH}^+ and C0C^0 extensions, and — when s>K−s>K_- and ρ<2\rho<2 — boundedness of the renormalized Hawking mass and extensions with continuous metric, Christoffel symbols in Lloc2L^2_{\mathrm{loc}} and ϕ∈Hloc1\phi\in H^1_{\mathrm{loc}}, matching the threshold suggested by the linear theory and covering near-extremal black holes.

These are conditional results: the exponential Price-law bound of rate ss is an assumption on data prescribed on the event horizon, whereas the conjectures concern generic data on a spacelike Cauchy hypersurface. The decisive exterior input — a rigorous sharp value of ss (the spectral gap of RNdS, expected to satisfy s>K−s>K_- near extremality on the basis of numerical quasinormal-mode analyses by Cardoso–Costa–Destounis–Hintz–Jansen) and its realization for generic spacelike data in the nonlinear charged-field exterior — is missing, as are the complete (s,ρ)(s,\rho) classification of mass inflation and the weak-solution/uniqueness theory of the extensions. Numerical work on Kerr–Newman–de Sitter (Davey–Dias–Sola Gil) likewise indicates SCC violation in a near-extremal region of parameter space, while rough-data results of Dafermos–Shlapentokh-Rothman show that smooth no-inflation data are non-generic within rougher data classes, motivating the finite-codimension genericity discussion.

Problem Statement

Let Λ>0\Lambda>0, scalar charge q≠0q\neq 0 and mass m≥0m\geq 0, and consider the spherically symmetric Einstein–Maxwell–charged-Klein–Gordon system. Let Σ\Sigma be a regular, compact (without boundary) spacelike hypersurface, and prescribe initial data on Σ\Sigma that are generic within the spherically symmetric class (in the source's convention: an open and dense subset of the moduli space of initial data in a suitable topology, interpretable as a finite-codimension condition). The problem is to decide the strong cosmic censorship conjectures for this system — Conjectures 1.1 and 1.2 of Rossetti — namely, to prove either:

(A) [SCC holds] for generic such data, the future maximal globally hyperbolic development (MGHD) cannot be locally extended as a Lorentzian manifold with continuous metric (C0C^0 formulation, Conjecture 1.1), nor with continuous metric and Christoffel symbols in Lloc2L^2_{\mathrm{loc}} (the Christodoulou–Chruściel Hloc1H^1_{\mathrm{loc}} class, Conjecture 1.2); or

(B) [SCC fails] for generic such data, in a nonempty open region of black-hole parameter space (expected near extremality, ρ=K−/K+<2\rho=K_-/K_+<2), the MGHD is locally extendible across its Cauchy horizon in the above classes.

The conjectures are currently resolved only conditionally, from characteristic data on the event horizon with an assumed exponential Price-law bound of rate ss; the source identifies a rigorous Price law as the open input decisive for turning these conditional results into a definitive answer, in the form appropriate to the branch proved: (i) a rigorous exponential Price law with sharp rate ss for the charged scalar field in the black hole exterior — the spectral gap of the Klein–Gordon operator on sub-extremal (in particular near-extremal) Reissner–Nordström–de Sitter spacetimes and its nonlinear extension with charged matter — together with the genericity, relative to spacelike Cauchy data, of the decay regime with s>K−s>K_- (on branch (B)), or generic lower-bound/blow-up mechanisms replacing it (on branch (A)). Two linked open questions are further discussed by the source, as context rather than as part of the conjectures themselves: (ii) the nonlinear classification of the mass-inflation/no-mass-inflation threshold in the (s,ρ)(s,\rho) plane — sufficiency of s>K−s>K_- and ρ<2\rho<2 is the known conditional result, and the source describes this range as "expected to be sharp", noting the complementary instability (lower-bound) analysis as a future direction that is in any case not needed for branch (B); and (iii) on the extendibility branch, the weak-solution status and uniqueness of the constructed Hloc1H^1_{\mathrm{loc}} extensions, which bears on whether extendibility constitutes an actual failure of determinism at weak-solution regularity — the source's Remark 5.7 observes that the Gajic–Lin procedure for weak extensions beyond CH+\mathcal{CH}^+ (which reveals infinitely many weak solutions in the asymptotically flat extremal case) is not obstructed by the cosmological constant or by sub-extremality, while local well-posedness at Hloc1H^1_{\mathrm{loc}} regularity remains unknown. The removal of spherical symmetry is explicitly outside these conjectures and is not required.

References

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