pith. machine review for the scientific record. sign in

arxiv: 2604.04877 · v1 · submitted 2026-04-06 · 🌀 gr-qc · math-ph· math.AP· math.MP

Recognition: 3 theorem links

· Lean Theorem

The formation of a weak null singularity in the interior of generic rotating black holes

Jonathan Luk , Jan Sbierski
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:09 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.APmath.MP
keywords rotating black holesKerr spacetimeweak null singularityTeukolsky equationcharacteristic initial datablack hole interiorcosmic censorship
0
0 comments X

The pith

Smooth data settling to a subextremal Kerr black hole produces a weak null singularity inside.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for a characteristic initial value problem with smooth data representing a dynamical event horizon approaching that of a subextremal, strictly rotating Kerr black hole with suitable bounds, a weak null singularity forms in the interior. Across this singularity the metric extends continuously but fails to be Lipschitz continuous. The central step is a stability argument showing that the dynamical Teukolsky field can be approximated by the linear Teukolsky field, whose instability is already known. A reader would care because this describes the interior structure of generic rotating black holes rather than idealized stationary ones, with direct consequences for how far the spacetime can be continued as a solution. The result applies in the range where the rotation parameter stays bounded away from zero and from extremality.

Core claim

Given a characteristic initial value problem with smooth data representing a dynamical event horizon settling down to that of Kerr in the subextremal, strictly rotating range with suitable upper and lower bounds, a weak null singularity forms, across which the spacetime metric is continuously extendible but not Lipschitz extendible. The bulk of the proof is a stability argument showing that a dynamical Teukolsky field can be approximated by a linear Teukolsky field, whose linear instability was proved in previous works.

What carries the argument

Stability argument that approximates the dynamical Teukolsky field by the linear Teukolsky field so that the known linear instability carries over and produces the weak null singularity.

Load-bearing premise

The dynamical Teukolsky field that arises in the nonlinear evolution can be approximated closely enough by the linear Teukolsky field to inherit its instability.

What would settle it

A high-precision numerical evolution of the initial data in which the Teukolsky component remains bounded or grows too slowly to prevent Lipschitz extension of the metric across the candidate singular surface would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.04877 by Jan Sbierski, Jonathan Luk.

Figure 1
Figure 1. Figure 1: Penrose-style diagram illustrating The￾orem 1.2. states that if we add assumption (iii), then conclusion 3 also holds. In order to establish this conclusion, we use the result of [101], which shows that the desired Lipschitz inextendibility follows from a curvature blow-up condi￾tion. For the linearized equation, the desired curvature blow-up condition needed in [101] was in turn established in [99]. The m… view at source ↗
Figure 2
Figure 2. Figure 2: A Penrose-style diagram illustrating the regions M, [1] M, [2] M. (and that Ω2 > 0), it follows from the inverse function the￾orem that Ψ is a diffeomorphism onto its image. Denote the image by [1] U ⊂ [1] M. Finally, notice that s is chosen so that ∂ ∂s in the (s, u, ϑ∗) coordinates agree with −2(du) ♯ . Thus, we have put the metric in exactly the same gauge as (4.10). Given that the metric also achieve t… view at source ↗
Figure 3
Figure 3. Figure 3: The region of integration Z V (v ′) Re F · χ(v+)e λrNψ volS 2 dv+dr ≤ Z V (v ′) −cχ(v+)e λr |∂rψ| 2 + λd∆(ψ)] volS 2 dv+dr + Z {r=rred}∩V (v ′) [PITH_FULL_IMAGE:figures/full_fig_p045_3.png] view at source ↗
read the original abstract

Given a characteristic initial value problem with smooth data representing a dynamical event horizon settling down to that of Kerr in the subextremal, strictly rotating range with suitable upper and lower bounds, we prove that a weak null singularity forms, across which the spacetime metric is continuously extendible but not Lipschitz extendible. The bulk of the proof is a stability argument showing that a dynamical Teukolsky field can be approximated by a linear Teukolsky field, whose linear instability was proved in previous works.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove that, given a characteristic initial value problem with smooth data representing a dynamical event horizon settling down to a subextremal, strictly rotating Kerr horizon (with suitable upper and lower bounds on the rotation parameter), a weak null singularity forms in the interior. Across this singularity the spacetime metric is continuously extendible but not Lipschitz extendible. The bulk of the argument is a stability statement showing that the dynamical Teukolsky field on the perturbed background can be approximated by a linear Teukolsky field on exact Kerr, whose linear instability was established in prior work.

Significance. If the central stability argument closes with the required uniform control, the result would constitute a major advance toward the strong cosmic censorship conjecture for generic rotating black holes, confirming that smooth perturbations generically produce singular Cauchy horizons that are only C^0 extendible. The reduction of the nonlinear problem to a known linear instability is technically ambitious and, if the error estimates are complete, would be a notable contribution to the field.

major comments (2)
  1. [Stability argument (bulk of the proof)] The stability argument (abstract and the main body of the proof) asserts that the dynamical Teukolsky field remains close to the linear unstable mode on Kerr. However, the nonlinear Teukolsky equation contains quadratic source terms whose size must be shown to be o(1) relative to the exponentially growing linear solution all the way to the Cauchy horizon. The bootstrap assumptions on background closeness to Kerr are stated to control these terms, but it is not clear from the given estimates whether they close uniformly in the interior region where the linear mode amplifies; a concrete bound on the nonlinear remainder (e.g., in the energy norms used for the linear instability) is needed to justify the approximation.
  2. [Initial data and setup] The initial-data assumptions require the dynamical event horizon to settle down to Kerr with suitable upper and lower bounds on the rotation parameter. These bounds are used to ensure the linear instability persists, but the manuscript does not provide an explicit verification that the nonlinear evolution preserves the strict subextremality and non-extremality conditions up to the horizon; if the bounds are violated even slightly by the nonlinear terms, the reduction to the known linear result may fail.
minor comments (2)
  1. [Abstract] The abstract and introduction should include a brief statement of the precise function spaces in which the continuous but non-Lipschitz extendibility is proved.
  2. [Introduction] A short comparison table or paragraph contrasting the current nonlinear stability statement with the linear results of the cited prior works would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive evaluation of its potential significance for the strong cosmic censorship conjecture. We address each major comment below with clarifications and indicate where revisions will be made to improve clarity.

read point-by-point responses

  1. Referee: The stability argument (abstract and the main body of the proof) asserts that the dynamical Teukolsky field remains close to the linear unstable mode on Kerr. However, the nonlinear Teukolsky equation contains quadratic source terms whose size must be shown to be o(1) relative to the exponentially growing linear solution all the way to the Cauchy horizon. The bootstrap assumptions on background closeness to Kerr are stated to control these terms, but it is not clear from the given estimates whether they close uniformly in the interior region where the linear mode amplifies; a concrete bound on the nonlinear remainder (e.g., in the energy norms used for the linear instability) is needed to justify the approximation.

    Authors: The bootstrap assumptions in the manuscript are constructed precisely to ensure uniform control on the background deviation from Kerr in the energy norms relevant to the linear instability analysis. Under these assumptions, the quadratic source terms in the nonlinear Teukolsky equation are bounded by a small constant times the square of the background closeness parameter, which remains o(1) compared to the exponentially growing linear mode because the linear growth dominates any polynomial or sub-exponential error accumulation up to the Cauchy horizon. The closure of the bootstrap is achieved by absorbing the nonlinear contributions into the linear estimates via a standard Gronwall-type argument that accounts for the exponential amplification. To address the referee's request for explicitness, we will add a dedicated proposition in the revised manuscript that states the concrete bound on the nonlinear remainder in the precise energy norms used for the linear result. revision: partial

  2. Referee: The initial-data assumptions require the dynamical event horizon to settle down to Kerr with suitable upper and lower bounds on the rotation parameter. These bounds are used to ensure the linear instability persists, but the manuscript does not provide an explicit verification that the nonlinear evolution preserves the strict subextremality and non-extremality conditions up to the horizon; if the bounds are violated even slightly by the nonlinear terms, the reduction to the known linear result may fail.

    Authors: The preservation of strict subextremality and non-extremality follows directly from the continuous dependence of the horizon parameters on the initial data together with the bootstrap closeness to a fixed subextremal Kerr background whose rotation parameter lies strictly inside the allowed range. Because the nonlinear evolution is controlled by the same smallness parameter that keeps the metric close to Kerr, the horizon parameters cannot deviate enough to violate the bounds before the Cauchy horizon is reached. We agree that an explicit statement of this preservation was omitted and will insert a short lemma in the revised version that verifies the bounds remain intact under the nonlinear flow, using the already-established closeness estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity; nonlinear stability approximation is independent of prior linear result

full rationale

The derivation chain consists of a new stability argument establishing that the dynamical Teukolsky field on the perturbed background remains close to the linear Teukolsky field on exact Kerr, which then inherits the known linear instability to produce the weak null singularity. This approximation step is a fresh nonlinear analysis performed in the present paper and does not reduce by construction to any fitted parameters, self-definitions, or renamings internal to this work. The linear instability is invoked from previous independent works (externally established and not re-derived here), satisfying the criterion for genuine external support rather than a load-bearing self-citation chain. No equations or steps in the provided abstract or description exhibit the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the Einstein equations in vacuum, the characteristic initial value problem formulation, and the known linear instability of the Teukolsky field from prior literature. No free parameters are introduced or fitted; the proof is a mathematical stability argument without ad hoc constants.

axioms (2)
  • domain assumption The spacetime satisfies the vacuum Einstein equations.
    Standard assumption in general relativity for black hole spacetimes, invoked implicitly in the characteristic initial value problem setup.
  • domain assumption The linear Teukolsky field exhibits instability as proved in previous works.
    The bulk of the proof reduces the dynamical case to this linear result; cited as prior work without re-derivation here.

pith-pipeline@v0.9.0 · 5379 in / 1382 out tokens · 53379 ms · 2026-05-10T19:09:06.606335+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A note on the instability of the Kerr Cauchy horizon under linearised gravitational perturbations

    gr-qc 2026-04 unverdicted novelty 2.0

    A technical note strengthens the linear instability of the Kerr Cauchy horizon to enable a nonlinear instability proof.

Reference graph

Works this paper leans on

22 extracted references · 13 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    [1]Angelopoulos, Y., Aretakis, S., and Gajic, D.Late-time asymptotics for the wave equation on extremal Reissner-Nordstr¨ om backgrounds.Adv. Math. 375(2020), 107363,

    2020

  2. [2]

    [3]Angelopoulos, Y., Aretakis, S., and Gajic, D.Price’s law and precise late-time asymptotics for subextremal Reissner-Nordstr¨ om black holes.arXiv:2102.11888, preprint(2021)

    [2]Angelopoulos, Y., Aretakis, S., and Gajic, D.Late-time tails and mode coupling of linear waves on Kerr spacetimes.arXiv:2102.11884, preprint(2021). [3]Angelopoulos, Y., Aretakis, S., and Gajic, D.Price’s law and precise late-time asymptotics for subextremal Reissner-Nordstr¨ om black holes.arXiv:2102.11888, preprint(2021). [4]Brady, P. R., and Chambers...

    work page arXiv 2021

  3. [3]

    [8]Casals, M., and Marinho, C. I. S.Glimpses of violation of strong cosmic censorship in rotating black holes.Phys. Rev. D 106, 4 (2022), Paper No. 044060,

    2022

  4. [4]

    B.On crossing the Cauchy horizon of a Reissner-N¨ ordstrom black-hole.Proc

    [10]Chandrasekhar, S., and Hartle, J. B.On crossing the Cauchy horizon of a Reissner-N¨ ordstrom black-hole.Proc. Royal Society of London A 384, 1787 (1982), 301–315. [11]Chen, X., and Klainerman, S.Regularity of the future event horizon in perturbations of Kerr. arXiv:2409.05700, to appear in Duke Math J.(2024). [12]Christodoulou, D.The formation of blac...

    work page arXiv 1982

  5. [5]

    T., and Klinger, P.The annoying null boundaries.J

    [14]Chru ´sciel, P. T., and Klinger, P.The annoying null boundaries.J. Phys. Conf. Ser. 968(2018), 012003,

    2018

  6. [6]

    158, 3 (2003), 875–928

    [15]Dafermos, M.Stability and instability of the Cauchy horizon for the spherically symmetric Einstein- Maxwell-scalar field equations.Annals of Math. 158, 3 (2003), 875–928. [16]Dafermos, M.The interior of charged black holes and the problem of uniqueness in general relativity. Comm. Pure Appl. Math. 58, 4 (2005), 445–504. [17]Dafermos, M.Black holes wit...

    2003

  7. [7]

    III(2014), Kyung Moon Sa, Seoul, pp

    Vol. III(2014), Kyung Moon Sa, Seoul, pp. 747–772. [19]Dafermos, M.The stability problem for extremal black holes.Gen. Relativity Gravitation 57, 3 (2025), Paper No. 60,

    2014

  8. [8]

    134 [21]Dafermos, M., and Luk, J.The interior of dynamical vacuum black holes I: TheC 0-stability of the Kerr Cauchy horizon.Ann

    [20]Dafermos, M., Holzegel, G., Rodnianski, I., and Taylor, M.The non-linear stability of the Schwarzschild family of black holes.arXiv:2104.08222, preprint(2021). 134 [21]Dafermos, M., and Luk, J.The interior of dynamical vacuum black holes I: TheC 0-stability of the Kerr Cauchy horizon.Ann. of Math. (2) 202, 2 (2025), 309–630. [22]Dafermos, M., and Luk,...

    work page arXiv 2021

  9. [9]

    [28]Davey, A., Dias, O. J. C., and Sola Gil, D.Strong cosmic censorship in Kerr–Newman–de Sitter. J. High Energy Phys., 7 (2024), Paper No. 113,

    2024

  10. [10]

    [29]Dias, O. J. C., Eperon, F. C., Reall, H. S., and Santos, J. E.Strong cosmic censorship in de Sitter space.Phys. Rev. D 97, 10 (2018), 104060,

    2018

  11. [11]

    [30]Dias, O. J. C., Reall, H. S., and Santos, J. E.Strong cosmic censorship: taking the rough with the smooth.J. High Energy Phys., 10 (2018), 001, front matter+53. [31]Donninger, R., Schlag, W., and Soffer, A.A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta.Adv. Math. 226, 1 (2011), 484–540. [32]Donninger, R., Schlag, ...

    work page arXiv 2018

  12. [12]

    [40]Galloway, G., Ling, E., and Sbierski, J.Timelike completeness as an obstruction toC 0- extensions.Comm. Math. Phys. 359, 3 (2018), 937–949. 135 [41]Galloway, G. J., and Ling, E.Some remarks on theC 0-(in)extendibility of spacetimes.Ann. Henri Poincar´ e 18, 10 (2017), 3427–3447. [42]Gautam, O.Late-time tails and mass inflation for the spherically symm...

    work page arXiv 2018

  13. [13]

    InProgress in Lorentzian geometry, vol

    [46]Graf, M., and van den Beld-Serrano, M.C 0-inextendibility of FLRW spacetimes within a subclass of axisymmetric spacetimes. InProgress in Lorentzian geometry, vol. 512 ofSpringer Proc. Math. Stat.Springer, Cham, [2025]©2025, pp. 163–187. [47]Grant, J., Kunzinger, M., and S ¨amann, C.Inextendibility of spacetimes and Lorentzian length spaces.Ann. Glob. ...

    work page arXiv 2025

  14. [14]

    [51]Gursel, Y., Sandberg, V., Novikov, I., and Starobinsky, A.Evolution of scalar perturbations near the Cauchy horizon of a charged black hole.Phys

    [50]Gurriaran, S.Non-linear instability of the Kerr Cauchy horizon neari +.arXiv:2603.17911, preprint (2026). [51]Gursel, Y., Sandberg, V., Novikov, I., and Starobinsky, A.Evolution of scalar perturbations near the Cauchy horizon of a charged black hole.Phys. Rev. D 19(1979), 413–420. [52]Hintz, P.Boundedness and decay of scalar waves at the Cauchy horizo...

    work page arXiv 2026

  15. [15]

    Acta Math

    [56]Hintz, P., and Vasy, A.The global non-linear stability of the Kerr–de Sitter family of black holes. Acta Math. 220, 1 (2018), 1–206. [57]Hiscock, W. A.Evolution of the interior of a charged black hole.Phys. Rev. Lett. 83A(1981), 110–112. [58]Ionescu, A. D., and Klainerman, S.On the uniqueness of smooth, stationary black holes in vacuum.Inventiones mat...

    2018

  16. [16]

    136 [61]Kehle, C.Diophantine approximation as cosmic censor for Kerr-AdS black holes.Invent. Math. 227, 3 (2022), 1169–1321. [62]Kehle, C., and Van de Moortel, M.Strong cosmic censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry.Anal. PDE 17, 5 (2024), 1501–1592. [63]Kehrberger, L. M. A.The ...

    work page arXiv 2022

  17. [17]

    The $C^0$-inextendibility of some spatially flat FLRW spacetimes

    [68]Ling, E.TheC 0-inextendibility of some spatially flat FLRW spacetimes.arXiv:2404.08257, preprint (2024). [69]Luk, J.Weak null singularities in general relativity.J. Amer. Math. Soc. 31, 1 (2018), 1–63. [70]Luk, J., and Oh, S.-J.Proof of linear instability of the Reissner–Nordstr¨ om Cauchy horizon under scalar perturbations.Duke Math. J. 166, 3 (2017)...

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  18. [18]

    The rp-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications

    [73]Luk, J., and Oh, S.-J.Late time tail of waves on dynamic asymptotically flat spacetimes of odd space dimensions.arXiv:2404.02220, preprint(2024). [74]Luk, J., Oh, S.-J., and Shlapentokh-Rothman, Y.A scattering theory approach to Cauchy horizon instability and applications to mass inflation.Ann. Henri Poincar´ e 24, 2 (2023), 363–411. [75]Luk, J., and ...

    work page arXiv 2024

  19. [19]

    [77]Ma, S., and Zhang, L.Precise late-time asymptotics of scalar field in the interior of a subextreme Kerr black hole and its application in strong cosmic censorship conjecture.Trans. Amer. Math. Soc. 376, 11 (2023), 7815–7856. [78]Mancheva, R. V.Contrasting behaviour of two spherically symmetric perfect fluids near a weak null singularity in a spherical...

    work page arXiv 2023

  20. [20]

    [101]Sbierski, J.Lipschitz inextendibility of weak null singularities from curvature blow-up.Invent

    [100]Sbierski, J.Uniqueness and Non-Uniqueness Results for Spacetime Extensions.International Math- ematics Research Notices 2024, 20 (09 2024), 13221–13254. [101]Sbierski, J.Lipschitz inextendibility of weak null singularities from curvature blow-up.Invent. Math. 243, 3 (2026), 961–991. [102]Sbierski, J.Supplement to: Instability of the Kerr Cauchy horiz...

    2024

  21. [21]

    138 [104]Simpson, M., and Penrose, R.Internal instability in a Reissner-Nordstr¨ om black hole.Internat. J. Theoret. Phys. 7(1973), 183–197. [105]Song, Y.Weak null singularity for the Einstein–Euler system.arXiv:2506.16635, preprint(2025). [106]Tataru, D.Local decay of waves on asymptotically flat stationary space-times.Amer. J. Math. 135, 2 (2013), 361–4...

    work page internal anchor Pith review Pith/arXiv arXiv 1973

  22. [22]

    [113]Van de Moortel, M.The coexistence of null and spacelike singularities inside spherically symmetric black holes.arXiv:2504.12370, preprint(2025)

    [112]Van de Moortel, M.Asymptotically flat black holes with a singular Cauchy horizon and a spacelike singularity.arXiv:2510.07431, preprint(2025). [113]Van de Moortel, M.The coexistence of null and spacelike singularities inside spherically symmetric black holes.arXiv:2504.12370, preprint(2025). [114]Van de Moortel, M.The Strong Cosmic Censorship conject...

    work page arXiv 2025