The source paper (Sbierski, Duke Math. J. 2022) proved that the future development of generic, sufficiently small, spherically symmetric perturbations of two-ended subextremal Reissner–Nordström initial data for the Einstein–Maxwell–scalar field system — spacetimes the author calls Dafermos–Luk–Oh (DLO) spacetimes, building on Luk–Oh's proof of the C2C^2-formulation of strong cosmic censorship — is future Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendible, using unboundedness of a local causal holonomy. The paper explicitly leaves open the time-dual statement: it is expected, but not proved, that Luk–Oh's analysis extends to show that the past Cauchy horizon of the past development is generically singular, which would yield generic past Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility and hence the generic full Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility of the whole maximal development — the Cloc0,1C^{0,1}_{\mathrm{loc}}-formulation of strong cosmic censorship for this model class. This past-direction statement is the surviving precise open core: the companion question about removing the sign assumptions from the spherically symmetric inextendibility theorem was resolved in a stronger, symmetry-free form by Sbierski's 2025 Inventiones paper, and the Schwarzschild Cloc0,1C^{0,1}_{\mathrm{loc}} statement attributed to future work is, as the source itself notes, already implied by the earlier C0C^0-inextendibility of Schwarzschild.

Background

The strong cosmic censorship conjecture in general relativity asserts, roughly, that the Einstein equations are generically deterministic: the maximal globally hyperbolic development (MGHD) — the unique largest solution determined by given initial data, in which causality is well-posed — of generic initial data should be inextendible, so that no classical evolution can be continued beyond it as a solution. For the exact charged Reissner–Nordström (RN) black hole the conjecture fails dramatically: the MGHD ends not at a spacelike singularity but at a Cauchy horizon, a null boundary across which the metric extends smoothly, so that beyond it the future is no longer determined by the data. Dafermos showed for the spherically symmetric Einstein–Maxwell–scalar field system that the Cauchy horizon of small perturbations of RN is only weakly singular (a 'weak null singularity': the metric extends continuously but curvature blows up along it), and Luk–Oh proved, for generic and sufficiently small spherically symmetric perturbations of two-ended asymptotically flat subextremal RN initial data, that the MGHD is C2C^2-inextendible — the C2C^2-formulation of strong cosmic censorship for this class. Here 'two-ended' means the initial hypersurface has two asymptotically flat ends joined through an Einstein–Rosen bridge, as for the t=0t=0 slice of the exterior of the RN spacetime with a bifurcate event horizon. How low can the assumed regularity of a hypothetical extension be pushed before inextendibility fails? The answer matters: Dafermos–Luk proved that the Cauchy horizon in the interior of dynamical vacuum black holes is C0C^0-stable, so the C0C^0-formulation of strong cosmic censorship fails in the vacuum (Kerr) regime; Cloc0,1C^{0,1}_{\mathrm{loc}} (locally Lipschitz) regularity — where the metric's coordinate expressions are Lipschitz, so the Levi–Civita connection still exists as an L∞L^\infty object — is the critical threshold. Sbierski's source paper introduced a 'local causal holonomy' (the parallel transport of a vector around small causal loops, computed at the level of the connection rather than curvature) and showed it becomes unbounded near cosmological big-bang singularities and near spherically symmetric weak null singularities, giving a general theorem (Theorem 4.1, with a technical hypothesis requiring, along sequences, a positive sign of ∂ur⋅∂vr\partial_u r \cdot \partial_v r in double-null coordinates, where rr is the area radius) implying Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility. Beyond unboundedness of the local causal holonomy, established geometric criteria that force Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility of a weak null singularity include blow-up of metric-coefficient first derivatives in double-null gauge (the source paper's weak-null-singularity criterion, Theorem 4.3) and the symmetry-free curvature-blow-up criterion of Sbierski's 2025 Inventiones paper (Invent. math. 243 (2025) 961–991, arXiv:2409.18838, Theorem 3.14 there). Applied on top of Luk–Oh's estimates, this upgraded their C2C^2-result to a Cloc0,1C^{0,1}_{\mathrm{loc}}-formulation of strong cosmic censorship — but only for the future development (Theorem 4.31). The past development of the same data, ending at a past Cauchy horizon, was left untouched: the paper states (footnote attributing the expectation to private communication with J. Luk) that with additional work Luk–Oh's analysis should extend to show the past Cauchy horizon is generically singular, after which the time-dual inextendibility theorem would give generic past Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility and, combining both time directions, generic full Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility of the whole spacetime.

Problem Statement

For generic (in the sense of Luk–Oh: generic and sufficiently small) spherically symmetric perturbations of asymptotically flat two-ended subextremal Reissner–Nordström initial data for the Einstein–Maxwell–scalar field system, prove that the past Cauchy horizon of the past development is singular — in the sense of satisfying quantified blow-up conditions compatible with an established Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility criterion — and deduce the time-dual inextendibility statement: the past development is past Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendible. Combined with the known generic future Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility (Theorem 4.31 of the source paper), this should establish the generic full Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility of the whole maximal globally hyperbolic development — the Cloc0,1C^{0,1}_{\mathrm{loc}}-formulation of strong cosmic censorship for this model class. The statement concerns the past development as a whole, including all of its boundary components (the past Cauchy horizon, past null infinity, and the exterior), for the class of admissible data and the genericity notion specified above, and for both asymptotically flat ends.

References

  1. Jan Sbierski, On holonomy singularities in general relativity and the Cloc0,1C^{0,1}_{\mathrm{loc}}-inextendibility of space-times, Duke Mathematical Journal 171 (2022), no. 14. DOI: 10.1215/00127094-2022-0040, https://doi.org/10.1215/00127094-2022-0040; arXiv:2007.12049, https://arxiv.org/abs/2007.12049 (source paper)
  2. Jan Sbierski, Lipschitz inextendibility of weak null singularities from curvature blow-up, Inventiones mathematicae 243 (2025), 961–991. DOI: 10.1007/s00222-025-01387-0, https://doi.org/10.1007/s00222-025-01387-0; arXiv:2409.18838, https://arxiv.org/abs/2409.18838
  3. Jonathan Luk and Sung-Jin Oh, Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. The interior of the black hole region, Annals of Mathematics (2) 190 (2019), no. 1, 1–111. DOI: 10.4007/annals.2019.190.1.1, https://doi.org/10.4007/annals.2019.190.1.1; arXiv:1702.05715, https://arxiv.org/abs/1702.05715
  4. Jonathan Luk and Sung-Jin Oh, Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data II. The exterior of the black hole region, Annals of PDE 5 (2019), no. 1, Article 6. DOI: 10.1007/s40818-019-0062-7, https://doi.org/10.1007/s40818-019-0062-7; arXiv:1702.05716, https://arxiv.org/abs/1702.05716
  5. Jan Sbierski, The C0C^0-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry, Journal of Differential Geometry 108 (2018), 319–378. DOI: 10.4310/jdg/1518490820, https://doi.org/10.4310/jdg/1518490820
  6. Mihalis Dafermos and Jonathan Luk, The interior of dynamical vacuum black holes I: The C0C^0-stability of the Kerr Cauchy horizon, Annals of Mathematics (2) 202 (2025), no. 2, 309–630. DOI: 10.4007/annals.2025.202.2.1, https://doi.org/10.4007/annals.2025.202.2.1; arXiv:1710.01722, https://arxiv.org/abs/1710.01722
  7. Maxime Van de Moortel, The Strong Cosmic Censorship conjecture, Comptes Rendus. Mécanique 353 (2025), no. G1, 415–454. DOI: 10.5802/crmeca.271, https://doi.org/10.5802/crmeca.271; arXiv:2501.13180, https://arxiv.org/abs/2501.13180
  8. Peter Cameron and Jan Sbierski, On the uniqueness of continuous spacetime extensions in 1+1 dimensions with applications to weak null singularities, preprint (2025, revised 2026). arXiv:2511.13422, https://arxiv.org/abs/2511.13422
  9. Maxime Van de Moortel, The coexistence of null and spacelike singularities inside spherically symmetric black holes, preprint (2025). arXiv:2504.12370, https://arxiv.org/abs/2504.12370