The only known naked singularities of the 3+1-dimensional Einstein vacuum equations without symmetry assumptions are the twisted self-similar spacetimes constructed by Rodnianski and Shlapentokh-Rothman (exterior, Annals of Mathematics 2023) and Shlapentokh-Rothman (interior, 2022), glued into a maximal globally hyperbolic development with an incomplete future null infinity. For these spacetimes the metric is known to extend Hoelder-continuously, after a suitable rescaling, to the inner Cauchy horizon {u=0,v>0}\{u=0, v>0\}, but the sharp asymptotics of the metric, shear, and curvature there -- and hence any quantitative discontinuity or inextendibility statement across that horizon -- are unknown. For the companion Einstein-scalar field system, An and Wu (2026) proved that non-spherically symmetric perturbations of Christodoulou's self-similar naked singularity retain the naked-singularity structure and are C1,κ/(1−κ)+C^{1,\kappa/(1-\kappa)+}-inextendible across the inner Cauchy horizon, flagging the vacuum analogue as a delicate open question. The problem posed here is to determine the precise singular structure of the inner Cauchy horizon of the vacuum naked singularities and to establish whether these spacetimes are extendible or inextendible across it in an explicitly identified regularity class, thereby fixing the status of strong cosmic censorship at the inner Cauchy horizon of the known vacuum naked singularities.

Background

In general relativity, spacetime is a 3+13+1-dimensional Lorentzian manifold (M,g)(\mathcal{M},g); in the absence of matter the Einstein vacuum equations demand that gg be Ricci-flat, Ric(g)=0\mathrm{Ric}(g)=0. The natural framework for gravitational collapse without symmetry is the double-null gauge, in which g=−2Ω2(du⊗dv+dv⊗du)+γAB(dθA−bAdu)⊗(dθB−bBdu)g=-2\Omega^2(du\otimes dv+dv\otimes du)+\gamma_{AB}(d\theta^A-b^Adu)\otimes(d\theta^B-b^Bdu) with u,vu,v optical (null) functions; Ω\Omega is the lapse and bb the shift, and the geometry of the two-spheres Su,vS_{u,v} of the foliation is captured by Ricci coefficients such as the shear χ^\hat\chi (the trace-free part of the outgoing null expansion, measuring how the spheres distort along the outgoing null direction e4=Ω−1∂ve_4=\Omega^{-1}\partial_v, with e3=Ω−1∂ue_3=\Omega^{-1}\partial_u denoting the dual ingoing null direction) and by curvature components such as α\alpha (the leading part of the Riemann tensor with both legs along e4e_4).

Weak cosmic censorship, proposed by Penrose in 1969, asserts that for generic regular asymptotically flat initial data the singularities of collapse are hidden from distant observers; a precise formulation is that future null infinity I+\mathscr{I}^+, the terminal manifold of the outgoing null geodesics, is complete. A maximal globally hyperbolic development (the largest spacetime uniquely determined by the data, within which the equations are well-posed) arising from regular data and having an incomplete I+\mathscr{I}^+ is called a naked singularity. Strong cosmic censorship, in the formulation of Christodoulou (1999), asserts that for generic data the maximal development is inextendible, i.e.\ admits no extension as a solution of the equations; failure would occur at a Cauchy horizon, a boundary hypersurface of the development across which the solution could otherwise be continued. Singular Cauchy horizons across which the metric extends continuously while curvature blows up are also central to the modern theory of black hole interiors (Dafermos and Luk, 2025).

The first rigorous naked singularities were constructed by Christodoulou (1994) for the spherically symmetric Einstein-scalar field system: continuously self-similar solutions (admitting a homothetic vector field KK with LKg=2g\mathcal{L}_K g=2g, i.e.\ invariance under overall rescaling) with self-similarity parameter κ∈(0,1/3)\kappa\in(0,1/3) whose I+\mathscr{I}^+ is incomplete. For those exact solutions the Cauchy horizon emanating from the singularity is regular -- the solution extends as a self-similar solution across it -- while the naked singularities themselves are unstable (Christodoulou 1999; Li and Liu 2022; An 2025). For the vacuum equations, Birkhoff's theorem (every spherically symmetric vacuum solution is isometric to Schwarzschild) blocks any direct analogue, and no reduction to an autonomous phase-plane system is available. Breaking this barrier, Rodnianski and Shlapentokh-Rothman (2023) introduced twisted self-similarity -- self-similar solutions for which the homothetic vector field is tangent but not normal to the past light cone of the singularity, handled in a renormalized coordinate v^\hat v near that cone -- and constructed the exterior region of a naked singularity for the 3+13+1-dimensional Einstein vacuum equations from such data, without symmetry assumptions. Shlapentokh-Rothman (2022) then constructed the interior region, solving a novel class of degenerate equations of mixed elliptic-hyperbolic type, and glued the two regions; the resulting spacetime is the maximal development of asymptotically flat characteristic data and has an incomplete I+\mathscr{I}^+, giving the first and so far only naked singularities of the vacuum equations without symmetry. In the double-null coordinates of these constructions the exterior region is {u<0, v>0}\{u<0,\,v>0\} (up to rescaling of uu), and the future boundary of the development consists of the incomplete I+\mathscr{I}^+ together with the inner Cauchy horizon {u=0, v>0}\{u=0,\,v>0\}. It is known only that, after a suitable rescaling, the metric extends Hoelder-continuously to {u=0}∖{v^=0}\{u=0\}\setminus\{\hat v=0\}; the sharp behavior there is expected, but not proven, to be at best Hoelder continuity of Ωχ^\Omega\hat\chi and non-square-integrability of α\alpha.

For the Einstein-scalar field system, An and Wu (2026) recently completed the analogous program beyond spherical symmetry: prescribing non-spherically symmetric, approximately self-similar data on the incoming cone and small perturbations on an outgoing cone, they proved global existence of the resulting solutions, showed that the metric and scalar field extend continuously to the inner Cauchy horizon {u=0, v>0}\{u=0,\,v>0\} while the second null derivative (Ω−1e3)2ϕ(\Omega^{-1}e_3)^2\phi blows up at a quantified rate as u→0u\to 0, and deduced that the spacetime is inextendible across the inner Cauchy horizon in the Hoelder class C1,κ/(1−κ)+C^{1,\kappa/(1-\kappa)+}: the failure of weak cosmic censorship in its strict (non-generic) formulation is accompanied by a quantitative strong-censorship mechanism at the horizon. Their proof leans essentially on the explicit asymptotics of Christodoulou's background solution. In the vacuum setting the corresponding information about the non-explicit twisted self-similar background is unavailable, and they state that establishing the corresponding discontinuity and inextendibility statements across the inner Cauchy horizon remains a delicate open question -- a question already posed by Rodnianski and Shlapentokh-Rothman, who asked in Remark 1.2 for a systematic study of the precise behavior of the solution along {u=0}\{u=0\} for the naked-singularity spacetimes. Their paper also asks, separately and for the different family of global Fefferman-Graham self-similar solutions with data near the round metric, whether the cone {u=0}\{u=0\} is shear-free and thus locally modeled on a Fefferman-Graham solution (Fefferman-Graham spacetimes being the classical self-similar Ricci-flat metrics generated by formal power series around a null cone).

Problem Statement

Let (M,g)(\mathcal{M},g) be a naked-singularity spacetime for the 3+13+1-dimensional Einstein vacuum equations obtained by gluing the exterior solution of Rodnianski and Shlapentokh-Rothman to the interior solution of Shlapentokh-Rothman, in the double-null coordinates (u,v,θA)(u,v,\theta^A) of those constructions, so that (M,g)(\mathcal{M},g) is the maximal globally hyperbolic development of asymptotically flat characteristic data, its future boundary consists of an incomplete future null infinity together with the inner Cauchy horizon {u=0, v>0}\{u=0,\,v>0\}, and, after a suitable rescaling, the metric extends Hoelder-continuously to {u=0}∖{v^=0}\{u=0\}\setminus\{\hat v=0\}.

Determine the fine structure of this inner Cauchy horizon, specifically:

(1) Asymptotics. Establish sharp asymptotics, as u→0u\to 0 with v>0v>0 fixed, for the metric components, the null connection coefficients (lapse Ω\Omega, shear χ^\hat\chi, shift bb), and the curvature components (in particular α\alpha); in particular decide whether χ^\hat\chi and α\alpha remain bounded, blow up, or fail to be integrable along the horizon.

(2) Extendibility dichotomy. Prove either (a) a quantitative inextendibility statement -- (M,g)(\mathcal{M},g) admits no extension as a solution of the Einstein vacuum equations across {u=0}\{u=0\} within an explicitly identified regularity class, the vacuum analogue of the C1,κ/(1−κ)+C^{1,\kappa/(1-\kappa)+}-inextendibility of An and Wu -- or (b) an extendibility statement, exhibiting an extension across the horizon and identifying the regularity class in which it is achieved.

These deliverables jointly settle a single scientific question: whether strong cosmic censorship, in a quantitative formulation, holds at the inner Cauchy horizon of the known vacuum naked singularities, or whether that horizon is regular and extendible as it is for Christodoulou's exact self-similar background in the spherically symmetric Einstein-scalar field system.

References

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