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This paper demonstrates numerically that five-dimensional vacuum gravity admits solutions in which a Schwarzschild black hole absorbs gravitational waves and becomes an extremal rotating Myers-Perry black hole in finite time — the first vac

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2026-08-03 07:09 UTC pith:CUU7TMQI

load-bearing objection Strong numerical evidence for the first vacuum third-law counterexample; analytically incomplete but worth a serious referee. the 2 major comments →

arxiv 2601.20955 v2 pith:CUU7TMQI submitted 2026-01-28 gr-qc hep-th

Violation of the third law of black hole mechanics in vacuum gravity

John R. V. Crump , Maxime Gadioux , Harvey S. Reall , Jorge E. Santos This is my paper
classification gr-qc hep-th MSC 83C5783C0565M70 PACS 04.25.D-04.70.-s
keywords third law of black hole mechanicsvacuum gravityextremal Myers-Perry black holecharacteristic gluingfive-dimensional gravitynumerical relativityfinite-time extremalitygravitational collapse
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims to have found, by numerical construction, solutions of five-dimensional vacuum general relativity that describe a Schwarzschild black hole becoming exactly extremal in finite time as it absorbs gravitational waves — a direct violation of the third law of black hole mechanics in pure vacuum, with no matter fields involved. A second class of solutions forms an extremal rotating black hole from initial data that contains no black hole at all. Both constructions use a characteristic-gluing method on a highly symmetric five-dimensional geometry, in which the metric reduces to two coordinates and the gluing problem becomes a finite set of matching conditions. The authors solve these conditions numerically to residuals below 1e-20 and verify convergence under mesh refinement and polynomial order. If the solutions are genuine, the third law is false in vacuum independently of any matter model, settling a question left open by matter-based counterexamples.

Core claim

On the paper's own terms, the central discovery is the existence of C^2 (or smoother) vacuum spacetimes with SU(2)×Z4 symmetry in five dimensions whose future development contains an extremal Myers-Perry black hole with two equal rotation parameters, reached in finite time. In type (i) solutions the initial data outside the apparent horizon coincide exactly with a Schwarzschild spacetime, and the transition to the extremal horizon is effected by a band of ingoing gravitational radiation on a null hypersurface; in type (ii) solutions the initial data are Minkowski-like and contain no horizon. The gluing matches the transverse derivatives of the metric across the null hypersurface to prescribe

What carries the argument

The load-bearing device is characteristic gluing: data on a null hypersurface are chosen to coincide with Schwarzschild data on one segment and with extremal Myers-Perry horizon data on another, and the Einstein equations are used to propagate both sides; the problem reduces to matching finitely many transverse derivatives (B_j, ∂_U^j Φ, and J) at the junction. The metric ansatz — the SU(2)×Z4-invariant double-null form — reduces a five-dimensional vacuum problem to a two-dimensional effective system with a complex scalar Φ and a U(1) gauge field A, which supplies the mechanism for transporting angular momentum. The numerical engine is a neural-network parameterisation of the gluing function

Load-bearing premise

The construction's soundness rests on the numerical residual of 1e-20 for the gluing conditions being a true zero of an exact solution, together with the transfer of the C^2-regularity-at-the-origin argument from the 4D setting to this 5D one; if either fails, the spacetimes may not exist.

What would settle it

Run the same characteristic gluing at substantially higher resolution (more spectral elements, higher polynomial degree, and independent extended-precision Newton solves) and check whether residuals and the h/p-convergence rates continue to decrease to zero; alternatively, evolve the glued initial data with a fully independent 5D Cauchy code and test whether the claimed extremal horizon actually forms and the spacetime is smooth across the gluing surface. A solution that is merely numerical would show stalled convergence or finite constraint violation away from C.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • The third law of black hole mechanics is violated in vacuum gravity in five dimensions, so the law cannot be rescued by restricting to smooth matter or by charge-to-mass bounds.
  • Extremal rotating black holes can form from gravitational waves alone in finite time, meaning vacuum collapse does not require matter to reach extremality.
  • The gluing method extends to subextremal Myers-Perry black holes with arbitrary parameters, so near-extremal and extremal horizons are accessible endpoints of vacuum evolution, not isolated points.
  • A two-stage gluing combining type (i) and type (ii) should yield a vacuum solution that starts from non-black-hole initial data, passes through a Schwarzschild phase, and ends at an extremal black hole.
  • The 5D construction is intended as a stepping stone toward the conjectured 4D extremal Kerr construction, where the symmetry reduction is absent and the problem is cohomogeneity-3.

Where Pith is reading between the lines

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

  • If the gluing solutions are exact, then the usual no-finite-time-extremality results must rely on additional structure (matter equations or symmetry) that pure vacuum in higher dimensions evades; one concrete test is whether a fully nonlinear Cauchy evolution of the type (ii) initial data produces trapped surfaces, which the paper itself flags as an open question.
  • The effective scalar/gauge description of the vacuum metric may transfer intuition from charged-scalar collapse to pure gravity: the 'charge' here is angular momentum, and the extremality bound is encoded in the equality J = r_+^3, so the third-law question is really about whether angular momentum can be delivered in finite time without violating the area/mass balance.
  • A sharper falsification target would be to compute the actual constraint residual on a full 5D grid, not just on the gluing hypersurface, at ever higher resolution; if it saturates above zero, the claimed spacetime does not exist. This is a check the authors do not perform in the paper.
  • The type (ii) solutions, if genuine, would also bear on the critical-collapse threshold in vacuum: determining whether they contain trapped surfaces would show whether they sit at the black-hole/dispersion boundary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper presents a numerical construction of solutions to five-dimensional vacuum Einstein gravity, with SU(2) x Z4 symmetry, that interpolate in finite time from a Schwarzschild (or Minkowski) region to an extremal Myers-Perry (EMP) black hole. The construction uses characteristic gluing along a null hypersurface C: the authors parameterize the gluing fields B and Phi by neural-network functions, reduce the C^k matching problem to a finite-dimensional system of 3k+3 equations, and solve it by gradient descent followed by a quasi-Newton method in extended precision. They report residual norms below 1e-20 and verify h- and p-refinement convergence of the discontinuous Galerkin discretization. Type (i) solutions (Schwarzschild to EMP) are claimed to be C^2 everywhere; type (ii) solutions (Minkowski to EMP) are claimed to be C^4 away from the origin and C^2 at r=0, and hence to describe formation of an EMP black hole from vacuum initial data with no pre-existing black hole.

Significance. If the construction is correct, the paper would provide the first vacuum counterexample to the third law of black hole mechanics, resolving a conjecture of Kehle and Unger and showing that the third law fails independently of any matter model. The paper also demonstrates a 5D analogue of the characteristic gluing method in a setting that is genuinely cohomogeneity-2. The manuscript is notable for shipping reproducible code, providing explicit EMP matching values, proving the monotonicity of the Hawking mass in this symmetry class, and including convergence tests with scaling exponents that match DG theory. The main caveat is that the central existence statement is certified numerically rather than by an analytic theorem; the paper is transparent about the numerical nature of the demonstration, but the title and abstract make a definitive physical claim.

major comments (2)
  1. [Numerical approach; SM 'Convergence plots'] The central existence claim rests on a numerical root of a finite-dimensional system. The final residual <1e-20 is the discrete residual of the gluing conditions after DG integration, not a bound on the continuum residual. The h/p convergence tests compare solutions within the same discretization family; at the highest resolutions the differences f and g are only about 1e-14 to 1e-16, so the continuum gluing residual is not certified at the 1e-20 level. More importantly, no interval-Newton, Newton-Kantorovich, or other computer-assisted argument proves that the discrete root corresponds to an exact solution of the infinite-dimensional matching problem. Since the headline is a violation of the third law, this is load-bearing. I recommend either adding an independent solver or interval-arithmetic certificate, or explicitly and consistently limiting the claim to numerical evidence in the ti
  2. [SM 'Regularity at the origin'] The type (ii) result requires a C^2 metric at r=0. The chain of reasoning is: the data are H^{k+1}, the Cauchy-stability argument of Kehle-Unger [4] yields a regular origin, propagation of regularity gives H^{k+1} including r=0, and Sobolev embedding gives C^{k-2}. This chain is transferred from [4], which concerns a different theory (4D Einstein-Maxwell with a massless charged scalar). The manuscript does not verify the hypotheses needed to apply [4] to the 5D SU(2) x Z4 vacuum system, nor does it give the precise theorem from [4] being invoked. If any of these steps fails, the type (ii) spacetime would not be a regular C^2 solution. This does not affect the type (i) claim, but it is part of the abstract's second result, so it should be settled or explicitly labelled as conjectural.
minor comments (6)
  1. [Eq. (5)] The expression '- 1/3 log 2Q(V)' is easily misread as -1/3 log(2Q(V)), which would diverge at V=0 since Q(0)=0. Please write -(log 2) Q(V)/3 with parentheses, e.g., -(1/3)(log 2)Q(V).
  2. [Numerical approach] The count of 3k+3 matching conditions is not fully consistent between the summary ('J, B_j and derivative_U^j Phi with 0 <= j <= k') and the final Newton solve ('B_j, derivative_U^j Phi with 1 <= j <= k, and J' plus r(0) and derivative_U r(1)). Clarify which conditions are automatic from the ansatz (e.g., Phi(1)=0) and how r(0) and derivative_U r(1) enter the count.
  3. [Eq. (24)] The definition of g uses a componentwise ratio z/z_ref. If any component of the reference vector vanishes, the measure is undefined. State that this does not occur for the presented solutions or use an alternative relative-error definition.
  4. [SM 'Convergence plots'] The fitted exponents in Figs. 6 and 7 are based on only six and five data points, respectively. Report uncertainties in the fitted exponents and justify that the asymptotic convergence regime has been reached. Also, the convergence plots are shown only for type (ii); state where the analogous type (i) plots can be found.
  5. [Eq. (6)] The symbol p is used both for the number of hidden units in the neural network and, in the SM, for the polynomial order (elsewhere n). Rename one to avoid confusion.
  6. [Numerical approach] The statement that 'a set of parameters theta provides a solution when L(theta)=0' is followed by a numerical tolerance of 1e-20. Define the norm of the residual and specify how the 1e-20 criterion is measured, including the precision used in the final quasi-Newton solve.

Circularity Check

0 steps flagged

No significant circularity: the gluing construction solves for free data against independently derived Myers–Perry target values, and the cited self-work is peripheral.

full rationale

The paper's central claim is that a finite-dimensional gluing system has a solution, found by tuning neural-network parameters so that the transverse-derivative matching conditions on the characteristic surface C hold. The target quantities (B_j, J, Phi=0) are not taken from the numerical solution: they are computed in the SM directly from the explicit equal-angular-momenta Myers–Perry metric in Boyer–Lindquist coordinates, giving closed-form expressions (SM Eq. 16) for the EMP values. The matching conditions are the equations to be solved, not a fit to the conclusion. The only self-citations ([6], [12]) are used for peripheral context: a known charge-to-mass bound and comments about non-uniqueness or a Raychaudhuri analogy. Neither is load-bearing for the existence claim. The Kehle–Unger work [4] is external and is cited for the characteristic-gluing framework and the Cauchy-stability argument used to extend type-(ii) solutions to r=0; it is not a self-citation. The numerical certification is a question of rigor rather than circularity: a small discrete residual and empirical h/p convergence are evidence for an exact continuum root, but even if one doubts this, the derivation does not reduce to its own inputs by construction. No step was found in which a predicted quantity is defined in terms of the fitted parameters, or in which a self-citation is invoked to forbid alternatives.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The construction introduces no new fields or entities. The free parameters are boundary data and computational coefficients. The main epistemic load is carried by the assumption that the numerical matching is exact, which is not proven analytically.

free parameters (4)
  • initial Schwarzschild horizon radius r_i = type (i): r_i/r_+ ≈ 8.909e-4
    Free boundary datum in the characteristic gluing; fixes initial black-hole mass. It is a legitimate family parameter, not a fit to external data.
  • final EMP horizon radius r_+ = set to unity by scaling
    Sets the overall scale; dimensionless ratios are reported relative to it.
  • profile widths δ1, δ2 = (0.1, 0.1) for type (i); (0.05, 0.1) for type (ii)
    Shape parameters in the mollifier P(V); chosen by hand and affect numerical conditioning and location of derivative discontinuities.
  • neural-network weights and layer sizes (χ, μ, κ; p=(2,2,3) or (4,16,8)) = p=(2,2,3) for k=2; p=(4,16,8) for k=4; weights solved to residual <1e-20
    Computational degrees of freedom tuned to make gluing constraints vanish; no physical meaning.
axioms (4)
  • standard math The Einstein vacuum equations restricted to the SU(2)×Z4 ansatz are equivalent to the reduced system (9a)-(9i) for the functions (r, B, Φ, A_U, Ω).
    The SM derives these by direct computation; this is a symmetry reduction, not an independent physical postulate.
  • domain assumption Local existence for characteristic initial value problems for the Einstein equations holds for the null data used.
    Invoked in the gluing construction: 'The characteristic initial value problem ... admits a local solution [13].'
  • domain assumption The characteristic gluing framework of Aretakis-Czimek-Rodnianski and the Cauchy-stability/regularity-loss argument of Kehle-Unger transfer to this 5D vacuum setting, giving the claimed differentiability (C^k across C; C^{k-2} at r=0 for type (ii)).
    The paper adapts [4,11] without proving the transfer in detail; this is the main analytic scaffolding.
  • ad hoc to paper A numerical solution of the 3k+3 gluing equations with residual below 1e-20 and matching h/p convergence rates is an exact solution of the matching problem.
    No rigorous error bound or interval-arithmetic certificate is given; the existence claim rests on numerical convergence.

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Cite this review

Pith. "Pith review of Violation of the third law of black hole mechanics in vacuum gravity." pith.science (2026). https://pith.science/paper/CUU7TMQI

@misc{pith2026260120955,
  author       = {Pith},
  title        = {Pith review of: Violation of the third law of black hole mechanics in vacuum gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUU7TMQI}},
  note         = {Machine review of arXiv:2601.20955}
}
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read the original abstract

We demonstrate numerically the existence of solutions of five-dimensional vacuum gravity describing the formation, in finite time, of an extremal rotating black hole from a pre-existing Schwarzschild black hole. This is the first example of a violation of the third law of black hole mechanics in vacuum gravity and demonstrates that the third law is false independently of any matter model. We also demonstrate the existence of solutions describing the formation, in finite time, of an extremal rotating black hole from vacuum initial data that does not contain a black hole.

Figures

Figures reproduced from arXiv: 2601.20955 by Harvey S. Reall, John R. V. Crump, Jorge E. Santos, Maxime Gadioux.

Figure 1
Figure 1. Figure 1: FIG. 1. Penrose diagrams for gluing constructions of type (i) (left) and (ii) (right). The dark regions are exactly isometric [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Type (i) solution: plots of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Type (ii) solution: plots of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Log–log plots of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Log plots of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.