Strong cosmic censorship at the $W^{1,2}_{loc}$ level for nonlinearly perturbed Reissner–Nordström black holes without symmetry (Christodoulou's formulation)

The source paper (Luk–Oh, arXiv:1501.04598) proved that the Cauchy horizon of a subextremal Reissner–Nordström black hole is linearly unstable under generic scalar perturbations and explicitly posed the nonlinear follow-up problems: pointwise Price's-law lower bounds on the event horizon, the spherically symmetric $W^{1,2}_{loc}$ blow-up conjecture (its Conjecture 1.23), and the generic metric-level $W^{1,2}_{loc}$-inextendibility statement without symmetry. Later literature resolved the first two points — precise pointwise late-time asymptotics on the horizon by Angelopoulos–Aretakis–Gajic, and the spherical scalar-field-level conjecture in the classical $C^2$ formulation by Luk–Oh. The surviving open core is Christodoulou's $W^{1,2}$ formulation of strong cosmic censorship for charged black holes at zero cosmological constant without symmetry assumptions: for generic small nonlinear perturbations of subextremal Reissner–Nordström initial data in the Einstein–Maxwell system, is the maximal globally hyperbolic development inextendible as a Lorentzian manifold whose metric is continuous with locally square-integrable Christoffel symbols? The question is existential and admits two definitive answer directions — a proof that such a generic inextendible class exists, or a proof that no such generic class exists.

Background

The Reissner–Nordström spacetime is the unique static, spherically symmetric charged black hole solution of the Einstein–Maxwell equations, parametrized by mass $M$ and charge $Q$; in the subextremal range $|Q|<M$ its exterior ends at an event horizon $r=r_+$ (the one-way membrane of the black hole), while deeper inside an inner Cauchy horizon sits at $r=r_-$. The Cauchy horizon is the locus where the deterministic evolution of initial data breaks down: it is a null hypersurface across which the Reissner–Nordström metric extends smoothly, but non-uniquely, so that initial data on an ordinary spacelike slice do not determine the spacetime beyond it. This is the threat posed to determinism that Penrose's strong cosmic censorship conjecture (introduced in Penrose's 1969 collapse paper, reprinted in Gen. Rel. Grav. 34 (2002)) is designed to remove: for generic initial data, the maximal globally hyperbolic development — the largest region uniquely determined by the data, i.e. the largest one in which causality holds — should be inextendible, so that the extension ambiguity never occurs for generic data. The conjecture's truth depends on the regularity demanded of a hypothetical extension. In the classical $C^2$ formulation, the metric of the extension must be twice continuously differentiable, matching the regularity at which the Einstein equations are classically well posed. Christodoulou's low-regularity formulation, quoted as Conjecture 3 in Dafermos–Luk (arXiv:1710.01722), asks for inextendibility even as a Lorentzian manifold with continuous metric and Christoffel symbols (the connection coefficients, built from first derivatives of the metric) merely locally square-integrable — the regularity level conventionally denoted "$W^{1,2}$". Such an extension has a metric that is continuous and lies in the Sobolev class $W^{1,2}_{loc}$ (the space of functions whose weak first derivatives are square-integrable on compact sets); since in four dimensions a general $W^{1,2}_{loc}$ metric need not be continuous, this is a strictly smaller class of extensions than all $W^{1,2}_{loc}$ metrics, and the corresponding inextendibility statements are inequivalent. This is roughly the weakest regularity at which the Einstein equations retain a determinate weak-solution structure, making it the critical threshold for the conjecture. The physical mechanism expected to enforce inextendibility is mass inflation, identified heuristically by Poisson and Israel: the Cauchy horizon is a blueshift surface, meaning infalling radiation is exponentially blueshifted (its local energy grows like $e^{\kappa_- v}$ in advanced time $v$) as it approaches the inner horizon, so that even decaying perturbations of the exterior are amplified into a divergent effective mass function and a curvature blow-up localized at the Cauchy horizon. The needed exterior input is Price's law: perturbations of a black hole settle to a stationary state at a polynomial rate along the event horizon (heuristically $|\partial_v\phi|\sim v^{-p}$, $p>1/2$, for a scalar field, where $\partial_v$ is the derivative in advanced time along the horizon). Rigorous progress came in stages. Dafermos (CPAM 58 (2005) 445) studied the spherically symmetric Einstein–Maxwell–real-scalar-field model, proving that the metric extends continuously to the Cauchy horizon (so $C^0$-stability holds there) yet the spacetime is $C^2$-inextendible, with mass inflation realized rigorously — establishing the $C^2$ formulation in this symmetric model and showing simultaneously that low-regularity formulations are genuinely different questions. Luk–Oh (arXiv:1501.04598) then proved the linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations on the fixed background: if the scalar field obeys suitable pointwise lower bounds along the event horizon (their bound (1.19), a two-sided pointwise Price-law estimate), then generic solutions develop infinite nondegenerate energy at the Cauchy horizon; since those pointwise bounds were unproven, the paper itself established only an $L^2$-averaged lower divergence (its Corollary 1.9) and stated the open problems studied here. In the vacuum (uncharged, rotating Kerr) case, the subsequent program has run at other regularity levels: Dafermos–Luk (arXiv:1710.01722) proved $C^0$-stability of the Kerr Cauchy horizon (so the $C^0$ formulation of strong cosmic censorship is false), Sbierski and Gurriaran proved linear/linearised instability at the Kerr Cauchy horizon, and recent work (Sbierski arXiv:2409.18838; Luk–Sbierski arXiv:2604.04877; Hintz arXiv:2606.28253 on the nonlinear stability of the Kerr exterior) targets Lipschitz inextendibility, i.e. failure of extensions with metric of bounded slope. Since Lipschitz metrics form a strictly smaller class of extensions than continuous metrics with locally square-integrable Christoffel symbols, Lipschitz-inextendibility is a strictly weaker conclusion.

Problem Statement

Fix the Einstein–Maxwell system, with vanishing cosmological constant $\Lambda=0$, in four spacetime dimensions without any symmetry assumption. (The Einstein–Maxwell system coupled to a real massless scalar field is a related but distinct problem: the two systems have different data spaces, and neither a proof nor a refutation for one transfers automatically to the other.) Let $\mathcal{D}$ be a smooth two-ended asymptotically flat initial-data set on a Cauchy hypersurface that is globally close, in a fixed standard smooth topology for asymptotically flat initial data (a weighted smooth/Sobolev topology of the kind used in Luk–Oh I/II and Dafermos–Luk), to subextremal Reissner–Nordström data with $|Q|<M$, and let $(\mathcal{M},g)$ denote its maximal globally hyperbolic development. The question, which is Christodoulou's $W^{1,2}$ formulation of strong cosmic censorship specialized to perturbations of Reissner–Nordström and is the surviving open core of Luk–Oh (arXiv:1501.04598, Conjecture 1.15 and the discussion following it; that discussion proposes inextendibility against the wider class of all $W^{1,2}_{loc}$ metrics, whereas the continuous-metric, $L^2_{loc}$-Christoffel class is Christodoulou's variant quoted as Conjecture 3 in Dafermos–Luk, arXiv:1710.01722): does there exist a generic (open and dense in that topology, as in the spherically symmetric results of Luk–Oh I/II) class of such initial data for which $(\mathcal{M},g)$ is inextendible as a Lorentzian manifold with continuous metric and Christoffel symbols in $L^2_{loc}$ — that is, $(\mathcal{M},g)$ admits no nontrivial extension whatsoever whose metric is continuous with locally square-integrable Christoffel symbols? This is a global inextendibility statement and does not presuppose that the development contains a Cauchy horizon; in particular, data whose developments terminate in a singularity without any Cauchy horizon are inextendible in this sense. Being existential, the question is answered in full either by a proof that such a generic class exists for the Einstein–Maxwell system, or by a refutation establishing that no such generic class exists.

References

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