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arxiv: 2604.27976 · v1 · submitted 2026-04-30 · 🌀 gr-qc · hep-th

Near--extremal gravitational collapse in 4+1 dimensions: Schwarzschild--de--Sitter space

Maciej Dunajski , Sebastian J. Szybka This is my paper

Pith reviewed 2026-05-07 07:07 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords gravitational collapsenear-extremal black holesSchwarzschild-de Sitterhigher-dimensional gravitynumerical relativitycosmological horizonthird law of black hole thermodynamics
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The pith

Numerical evolution in 4+1 dimensions with positive cosmological constant produces black holes exceeding 99% of the extremal mass.

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

The paper examines gravitational collapse of radially symmetric waves in five-dimensional Einstein gravity with a positive cosmological constant. Starting from regular initial data that includes a cosmological horizon, the spacetime evolves to form a black hole whose mass surpasses 99 percent of the value at which the black hole horizon and cosmological horizon would coincide. This outcome is consistent with characteristic gluing constructions and supplies numerical evidence that the third law of black hole thermodynamics need not prevent extremality when a cosmological horizon is present.

Core claim

Evolution of a regular initial data with cosmological horizon leads to a formation of a black hole with mass exceeding 99% of the extremal value corresponding to the black hole and cosmological horizons coinciding. The results fit within the framework of characteristic gluing, and present some evidence that the third law of black hole thermodynamics may not hold in the cosmological context, where the extremality corresponds to the maximal mass of the Schwarzschild black hole in de-Sitter space.

What carries the argument

Numerical evolution of radially symmetric gravitational waves on regular initial data containing a cosmological horizon, within 4+1 dimensional Einstein gravity with positive Lambda, producing a Schwarzschild-de Sitter black hole near the extremal limit.

If this is right

  • A black hole mass can reach more than 99 percent of the maximum value set by the coincidence of event and cosmological horizons.
  • Characteristic gluing provides a consistent analytic framework for the near-extremal horizon formation observed numerically.
  • The third law of black hole thermodynamics does not necessarily prohibit reaching extremality in the presence of a cosmological constant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that the thermodynamic interpretation of extremality changes when a cosmological horizon is present, because the bound is a maximum mass rather than a zero-temperature limit.
  • Similar collapse simulations in 3+1 dimensions with positive Lambda could test whether the high degree of extremality is specific to five dimensions.

Load-bearing premise

The numerical evolution accurately captures the continuum physics without dominant discretization or gauge artifacts, and the selected initial data is representative rather than specially tuned to drive the system toward extremality.

What would settle it

A simulation with substantially higher resolution or with qualitatively different but still regular initial data that produces a final black hole mass well below 99 percent of the extremal value would falsify the central numerical claim.

Figures

Figures reproduced from arXiv: 2604.27976 by Maciej Dunajski, Sebastian J. Szybka.

Figure 1
Figure 1. Figure 1: The squashing function B for weak initial data view at source ↗
Figure 2
Figure 2. Figure 2: The Hawking mass M for weak initial data. the center) propagate outwards. The coordinate system is singular at the cosmological horizon rc = 2.449 where the coordinate speed of light of a radial signal vanishes, so we cannot follow radiation through the cosmological horizon. The time evolution of the Hawking mass M shown in view at source ↗
Figure 3
Figure 3. Figure 3: The squashing function B for intermediate initial data view at source ↗
Figure 4
Figure 4. Figure 4: The function A for intermediate initial data. The two zeros of A correspond to the apparent rs = 0.074 and the cosmological rc = 2.433 horizons view at source ↗
Figure 5
Figure 5. Figure 5: The Hawking mass M for intermediate initial data. 7 view at source ↗
Figure 6
Figure 6. Figure 6: The Hawking mass M for intermediate initial data. Dashed lines indicate the apparent horizon mass, M = 0.0055, the late-time plateau at M = 0.0058, and the position of the apparent horizon view at source ↗
Figure 7
Figure 7. Figure 7: The squashing function B for strong initial data view at source ↗
Figure 8
Figure 8. Figure 8: The function A for strong initial data. The apparent horizon forms at rs = 1.645 where A vanishes. The cosmological horizon is located at rc = 1.815 and corresponds to the second zero of A. 8 view at source ↗
Figure 9
Figure 9. Figure 9: The Hawking mass M for strong initial data. 5 Characteristic gluing The Kehle–Unger existence proof [16] of the gravitational collapse to extreme RN metric is based on the characteristic gluing method developed in [1]. A null cone in Minkowski or Schwarzschild space–time, is glued to an extremal horizon in RN. The resulting null cone is then extended to a four–dimensional region of a space–time using the r… view at source ↗
Figure 10
Figure 10. Figure 10: Characteristic gluing and the space–time extension. A de–Sitter null cone in view at source ↗
Figure 11
Figure 11. Figure 11: Outgoing and ingoing expansions θ+ = 3r −1 rv, θ− = 3r −1 ru on the gluing surface u = 0 with the extremal SdS horizon at u = 0, v = 1, and Λ = 0.1, m = 15. back from the AdS boundary, so there is no dispersion and a black hole forms for any size of initial data. This is consistent with the non–linear instability of AdS. In our paper we have focused on Λ > 0, where both the dispersion and the collapse can… view at source ↗
Figure 12
Figure 12. Figure 12: V (r) in red and A0(r) in blue with m = 1,Λ = 0.1. The zeroes rs and rc of A0 correspond to the black hole and the cosmological horizons. Linearising (2.2c) with B(r, t) = β(r, t) around the static solution (2.3) gives β¨ − 1 r 3 A0(r 3A0β ′ ) ′ + 8 r 2 A0β = 0. (A.25) Setting x = Z A0(r) −1 dr, β = e −iktr −3/2u(x) (A.26) turns (A.25) into the Schr¨odinger equation − d 2u dx2 + V (r(x))u = k 2u, (A.27) w… view at source ↗
read the original abstract

We numerically study a formation of near extremal horizons from a gravitational collapse of radially symmetric gravitational waves in $4+1$ dimensions within the framework of pure Einstein gravity with positive cosmological constant. Evolution of a regular initial data with cosmological horizon leads to a formation of a black hole with mass exceeding $99\%$ of the extremal value corresponding to the black hole and cosmological horizons coinciding. We demonstrate how our results fit within the framework of characteristic gluing, and present some evidence that the third law of black hole thermodynamics may not hold in the cosmological context, where the extremality corresponds to the maximal mass of the Schwarzschild black hole in de--Sitter space.

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

3 major / 3 minor

Summary. The manuscript reports a numerical investigation of radially symmetric gravitational wave collapse in 4+1-dimensional Einstein gravity with a positive cosmological constant. The evolution of regular initial data containing a cosmological horizon results in the formation of a black hole whose final mass exceeds 99% of the extremal mass at which the black-hole and cosmological horizons coincide. The authors interpret this outcome using characteristic gluing and suggest that this provides evidence against the validity of the third law of black hole thermodynamics in the cosmological setting.

Significance. Should the numerical findings prove robust, this would constitute a notable result in numerical general relativity, demonstrating that near-extremal Schwarzschild-de Sitter black holes can form dynamically from non-tuned initial data. The direct integration of the Einstein equations without auxiliary parameters or self-referential definitions is a methodological strength. The potential implication for the third law in de Sitter space could stimulate further theoretical work on black hole thermodynamics with cosmological horizons.

major comments (3)
  1. [Results (§4)] The claim that the black hole mass exceeds 99% of the extremal value is presented without any reported convergence tests, grid resolutions, or quantitative error bars on the mass ratio M_f/M_ext. Given that the horizons approach each other in the near-extremal regime, making the apparent horizon location and mass extraction sensitive to discretization errors, this omission undermines confidence in the precise 99% threshold. Richardson extrapolation or at least a comparison of results at different resolutions should be included.
  2. [Numerical Methods (§3)] The description of the characteristic gluing and the horizon extraction procedure does not include validation tests for accuracy in the regime where the black hole and cosmological horizons nearly coincide. Specific tests, such as the convergence of the extracted mass as the grid is refined, are necessary to support the central quantitative claim.
  3. [Discussion] The evidence offered for the possible violation of the third law is tied to the high extremality but lacks a direct comparison to the expected thermodynamic behavior or a calculation of the surface gravity approaching zero; this part of the argument would benefit from more explicit quantification.
minor comments (3)
  1. [Abstract] The abstract is clear but could briefly mention the dimensionality (4+1) for completeness, although it is in the title.
  2. [Introduction] A few additional references to previous numerical studies of gravitational collapse in de Sitter space would help contextualize the work.
  3. [Figures] The evolution plots would be improved by overlaying results from multiple grid resolutions to visually demonstrate convergence.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the constructive major comments. We agree that additional numerical validation and explicit quantification will strengthen the manuscript. We address each point below and will revise the paper accordingly to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Results (§4)] The claim that the black hole mass exceeds 99% of the extremal value is presented without any reported convergence tests, grid resolutions, or quantitative error bars on the mass ratio M_f/M_ext. Given that the horizons approach each other in the near-extremal regime, making the apparent horizon location and mass extraction sensitive to discretization errors, this omission undermines confidence in the precise 99% threshold. Richardson extrapolation or at least a comparison of results at different resolutions should be included.

    Authors: We agree that convergence tests are necessary to support the quantitative claim of M_f/M_ext > 99%. In the revised manuscript we will add a new paragraph in §4 (or a short appendix) that specifies the grid resolutions employed in the primary runs, presents the extracted mass ratio at each resolution, and demonstrates that the value remains above 99% with only small variations between successive refinements. This direct comparison will provide a practical error estimate and address the sensitivity of the horizon finder in the near-extremal regime. While we did not perform Richardson extrapolation in the original study, the resolution comparison will be sufficient to establish robustness of the reported threshold. revision: yes

  2. Referee: [Numerical Methods (§3)] The description of the characteristic gluing and the horizon extraction procedure does not include validation tests for accuracy in the regime where the black hole and cosmological horizons nearly coincide. Specific tests, such as the convergence of the extracted mass as the grid is refined, are necessary to support the central quantitative claim.

    Authors: We acknowledge that the numerical methods section would benefit from explicit validation tests focused on the near-coincident horizon regime. In the revised version we will expand §3 to include a dedicated validation subsection. This will report the convergence of the extracted black-hole mass and the locations of both horizons under successive grid refinements, confirming that the characteristic gluing procedure and apparent-horizon finder remain accurate when the horizons approach each other. These tests will directly support the reliability of the central mass-ratio measurement. revision: yes

  3. Referee: [Discussion] The evidence offered for the possible violation of the third law is tied to the high extremality but lacks a direct comparison to the expected thermodynamic behavior or a calculation of the surface gravity approaching zero; this part of the argument would benefit from more explicit quantification.

    Authors: We agree that more explicit quantification will clarify the thermodynamic implications. In the revised discussion we will add an explicit computation of the surface gravity κ of the final black hole, using the analytic SdS relation evaluated at the numerically extracted mass and the fixed cosmological constant. We will show that κ is correspondingly small for M_f > 0.99 M_ext and will briefly contrast this with the standard third-law expectation in asymptotically flat space, noting the distinct role of the cosmological horizon in the dynamical formation process. This addition will make the argument more quantitative without altering the original interpretation. revision: yes

Circularity Check

0 steps flagged

Numerical evolution of Einstein equations yields independent result; no circular reduction

full rationale

The paper's central claim follows from direct numerical integration of the 5D Einstein equations with positive cosmological constant, starting from regular initial data possessing a cosmological horizon. The extremal mass M_ext is defined analytically as the value at which the black-hole and cosmological horizons of the static Schwarzschild-de Sitter solution coincide; this definition is external to the simulation and does not depend on the evolved data. The reported mass ratio (>99% of M_ext) is an output of the time evolution, not a fitted parameter or a quantity defined in terms of itself. Characteristic gluing is invoked only for post-hoc interpretation of the final state and does not enter the definition or extraction of the mass ratio. No self-citation chain, ansatz smuggling, or renaming of known results is required to obtain the stated numerical outcome. The derivation chain is therefore self-contained against external benchmarks (the Einstein equations and the analytic SdS metric) and receives score 0.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Einstein equations with positive cosmological constant in five dimensions under spherical symmetry, plus the assumption that the numerical scheme converges to the continuum solution. No new entities are postulated.

free parameters (1)
  • initial gravitational wave amplitude and profile
    Parameters of the radially symmetric waves are chosen to produce collapse but are not specified numerically in the abstract.
axioms (2)
  • domain assumption Einstein field equations with positive cosmological constant in 4+1 dimensions
    Standard background for the model; invoked throughout the numerical evolution.
  • domain assumption Radial symmetry of the gravitational waves and spacetime
    Reduces the problem to effectively 1+1 dimensions for numerical evolution.

pith-pipeline@v0.9.0 · 5417 in / 1612 out tokens · 68667 ms · 2026-05-07T07:07:54.919604+00:00 · methodology

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Czimek, S

    Aretakis, S. Czimek, S. and Rodnianski, I. (2024) The Characteristic Gluing Problem for the Einstein Vacuum Equations: Linear and Nonlinear Analysis, Annales Henri Poincare 25, 3081–3205

    work page 2024

  2. [2]

    and Hawking, S

    Bardeen, B., Carter, B. and Hawking, S. W. (1973) The four laws of black hole mechan- ics, Commun. Math. Phys.31, 161

    work page 1973

  3. [3]

    Chmaj, T, and Schmidt, B

    Bizo´ n, P. Chmaj, T, and Schmidt, B. G. (2005) Critical behavior in vacuum gravitational collapse in 4+1 dimensions. Phys. Rev. Lett.95071102

    work page 2005

  4. [4]

    and Rostworowski, A

    Bizo´ n, P. and Rostworowski, A. (2017) Gravitational turbulent instability ofAdS5. Acta. Phys. Polon.B48, 1375

    work page 2017

  5. [5]

    and Lemos, J

    Cardoso, V. and Lemos, J. P. S. (2003) Quasinormal modes of the near extremal Schwarzschild-de Sitter black hole. Phys. Rev.D 67084020

    work page 2003

  6. [6]

    (2025) Symmetries of extremal horizonsarXiv:2512.10200

    Colling, A. (2025) Symmetries of extremal horizonsarXiv:2512.10200

    work page arXiv 2025

  7. [7]

    and Holzegel, G

    Dafermos, M. and Holzegel, G. (2006) On the nonlinear stability of higher-dimensional triaxial Bianchi IX black holes. Adv.Theor.Math.Phys.10, 503-523

    work page 2006

  8. [8]

    (2025) The stability problem for extremal black holes

    Dafermos, M. (2025) The stability problem for extremal black holes. Gen.Rel.Grav.57, 60

    work page 2025

  9. [9]

    (2023) Equivalence principle, de-Sitter space, and cosmological twistors

    Dunajski M. (2023) Equivalence principle, de-Sitter space, and cosmological twistors. arXiv:2304.08574. Internat. J. Modern Phys.D 32, 20214. 15

    work page arXiv 2023

  10. [10]

    Dunajski, and J

    Dunajski M. and Lucietti, J. (2026) Intrinsic rigidity of extremal horizons, arXiv:2306.17512. Journal of Differential Geometry.132, 179-201

    work page arXiv 2026

  11. [11]

    and Pretorius, F

    Gelles, Z. and Pretorius, F. (2026) A Nonlinear Endpoint of Charged Horizon Instabil- itiesarXiv:2602.11256

    work page arXiv 2026

  12. [12]

    S., and Santos J

    Gadioux, M., Reall H. S., and Santos J. E. (2025) Formation of extremal Reissner- Nordstr¨ om black holes: insights from numerics.arXiv:2512.10008

    work page arXiv 2025

  13. [13]

    and Perry, M

    Ginsparg, P. and Perry, M. J. (1983) Semiclassical perdurance of de Sitter space. Nuclear PhysicsB222

    work page 1983

  14. [14]

    Gundlach, D

    Gundlach, C., Hilditch, D. Mart´ ın-Garc´ ıa, J. M. (2025) Critical Phenomena in Gravi- tational CollapsearXiv:2507.07636

    work page arXiv 2025

  15. [15]

    (1986) Third Law of Black-Hole Dynamics

    Israel, W. (1986) Third Law of Black-Hole Dynamics. A Formulation and Proof. Phys. Rev. Lett.57, 397

    work page 1986

  16. [16]

    Gravitational collapse to extremal black holes and the third law of black hole ther- modynamics

    Kehle, C. and Unger, R. (2022) Gravitational collapse to extremal black holes and the third law of black hole thermodynamics.arXiv: 2211.15742

    work page arXiv 2022

  17. [17]

    (1918) ¨Uber die physikalischen Grundlagen der Einsteinschen Rela- tivit¨ atstheorie Ann

    Kottler, F. (1918) ¨Uber die physikalischen Grundlagen der Einsteinschen Rela- tivit¨ atstheorie Ann. Phys. (Leipzig)56, 410

    work page 1918

  18. [18]

    S., Jhingan, S., Chamorro, A

    Deshingkar, S. S., Jhingan, S., Chamorro, A. and Joshi, P. S. (2001) Gravitational Collapse and Cosmological Constant. Phys.Rev.D63124005

    work page 2001

  19. [19]

    (2012) On the Local Existence for the Characteristic Initial Value Problem in General Relativity

    Luk, J. (2012) On the Local Existence for the Characteristic Initial Value Problem in General Relativity. International Mathematics Research Notices20

    work page 2012

  20. [20]

    (2000) Shapiro, S

    Markovic, D. (2000) Shapiro, S. L. Gravitational collapse with a cosmological constant Phys. Rev.D 61, 084029

    work page 2000

  21. [21]

    Murata, K., Reall, H. S. and Tanahashi, N. (2013) What happens at the horizon(s) of an extreme black hole? CQG30, 235007

    work page 2013

  22. [22]

    (1999) The structure of the extreme Schwarzschild-de Sitter space-time

    Podolsky, J. (1999) The structure of the extreme Schwarzschild-de Sitter space-time. Gen. Rel. Grav.311703-1725

    work page 1999

  23. [23]

    Schutz, B. F. and Will, C. M. (1985) Black hole normal modes - A semianalytic approach. Astrophysical Journal291, L33-L36

    work page 1985

  24. [24]

    Szybka, S. J. and Chmaj, T. (2008) Fractal Threshold Behavior in Vacuum Gravitational Collapse. Phys. Rev. Lett.100, 101102

    work page 2008

  25. [25]

    Tapia del Moral, M (2026) Gravitational collapse to extremal black holes with Λ>0: counterexamples to the third law without charge.In prepraration

    work page 2026

  26. [26]

    (1974) Disk-Accretion onto a Black Hole

    Thorne, K. (1974) Disk-Accretion onto a Black Hole. II. Evolution of the Hole. Astro- physical Journal,191, pp. 507-520

    work page 1974

  27. [27]

    xAct: Efficient tensor computer algebra for the Wolfram Language.http://www.xact. es/ 16

    work page