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REVIEW 3 major objections 6 minor 3 cited by

For each scalar-field profile tested, gluing to an extremal Reissner-Nordstrom black hole works only above a minimum black-hole mass and, for massive scalars, below a maximum mass-to-charge ratio.

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2026-08-03 17:16 UTC pith:6M5AAKTG

load-bearing objection Useful numerical companion to Kehle-Unger: the C0 thresholds look solid and the qualitative picture holds, but the headline C2 numbers in Table 1 should be read as upper bounds, not proven minima, and the paper deserves a serious referee. the 3 major comments →

arxiv 2512.10008 v3 pith:6M5AAKTG submitted 2025-12-10 gr-qc

Formation of extremal Reissner-Nordstr\"om black holes: insights from numerics

Maxime Gadioux , Harvey S. Reall , Jorge E. Santos This is my paper
classification gr-qc MSC 83C5783C2283C75
keywords extremal black holescharacteristic gluingthird law of black hole mechanicscharged scalar fieldReissner-Nordstromnumerical relativitymass-to-charge ratiocosmological constant
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.

This paper numerically implements the characteristic gluing construction that was used to prove that an extremal Reissner-Nordstrom black hole can form in finite time from gravitational collapse of a charged scalar field. It asks how large the final black hole must be and how light the scalar must be for the construction to work. It finds that, for each scalar-field profile, extremal gluing requires a minimum dimensionless mass eM, ranging from about 10 to 2900 depending on the profile and the required smoothness. It also finds that adding a scalar mass allows gluing only up to a finite mass-to-charge ratio, always below 1, with the maximum varying strongly with the Ansatz. If these thresholds are correct, they turn the existence proof into concrete quantitative conditions and show that violating the third law of black hole mechanics is possible but requires substantial, though finite, fine-tuning.

Core claim

The paper performs characteristic gluing numerically for several scalar-field profiles on a null cone, connecting a flat or (A)dS sphere to a Reissner-Nordstrom horizon. It finds that extremal gluing (q=1) works only above a minimum dimensionless mass eM that depends on the profile and the desired smoothness: e.g., 15.96 for the C0 even profile at Lambda=0, 720 for the C1 even profile, and 4417 for the C2 even profile; other profiles give different values. A scalar mass m permits gluing only up to a maximum m/e, always well below 1, with values such as 0.2059 (C0 even) and 0.0272 (C1 even) at Lambda=0. A positive cosmological constant lowers the required mass, a negative one raises it. Gluin

What carries the argument

The central object is the characteristic gluing along an outgoing null cone C, with free scalar profile rho(V) e^{-iV}. Charge balance reduces to Q/(e r_+^2) = I(alpha), where I(alpha)=int xi^2 Im(Phi partial_V Phi) dV and xi=r/r_+ solves the null Raychaudhuri equation. Because I depends only on the profile parameters, its maximum over the region where r>0 sets the minimum mass via (eM)_min = 1/I_max at q=1. For C^k gluing, 2k transverse derivatives of Phi must vanish at the horizon, producing a multi-parameter shooting problem solved numerically.

Load-bearing premise

The thresholds rely on numerical searches that assumed the root-finder captured all gluing solutions, that the maximum of I occurs where it was computed, and that gluing data can be extended to a full spacetime by existing theorems.

What would settle it

Run the same Ansatz and ODE integration below the quoted (eM)_min, say below 15.96 for the C0 even profile, and find any parameter set with Q(1)=qM, the required transverse derivatives of Phi vanishing, inf r>0, and sup partial_U r<0: that would refute the threshold. Alternatively, find a valid extremal gluing with m/e above the quoted maximum for that Ansatz.

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

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If this is right

  • For each profile tested, extremal gluing has a finite minimal eM; below it no solution satisfies the required r>0 and partial_U r<0 conditions.
  • A positive cosmological constant lowers the minimal mass while a negative one raises it.
  • Including a scalar mass gives a maximum m/e for which q=1 can be reached; beyond it no extremal gluing exists even at very large eM.
  • Higher smoothness classes demand substantially larger masses and can exhibit multiple solution branches, as seen in C^2.
  • Some gluing solutions pass through a temporarily superextremal horizon before reaching the final stationary black hole.

Where Pith is reading between the lines

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

  • If these thresholds are true minima, optimizing the scalar profile could push eM_min considerably lower, and it is an open numerical question whether it can reach zero to form arbitrarily small extremal black holes.
  • The strong Ansatz dependence of the threshold, paired with a universal final extremal state, evokes critical collapse and suggests possible scaling relations between profile families and minimal mass.
  • The temporary superextremal phase indicates gluing data routinely overshoot the final charge-to-mass ratio, which invites numerical evolution to test whether such horizons are stable or seed instabilities.
  • The m/e maxima remain well below 1, so the known no-go bound at m/e=1 might be sharpenable; varying the phase or adding many parameters could move toward it.

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

3 major / 6 minor

Summary. The paper presents a numerical study of characteristic gluing for constructing spherically symmetric spacetimes that form exactly extremal Reissner–Nordström black holes in finite time, following the existence proof of Kehle and Unger. For several scalar-field Ansätze (even and odd bump functions, polynomials, and a modified even family), the authors solve the constraint/transport equations along the gluing null cone to accuracy C^0, C^1, and C^2, and determine, for each Ansatz and regularity class, the smallest value of eM for which gluing to an extremal RN horizon appears possible, together with the largest scalar mass-to-charge ratio m/e for which such gluing is found. They also study the effect of a cosmological constant. The results are summarized in Table 1, and the paper additionally reports that the dynamical horizon can be temporarily superextremal during the gluing.

Significance. If the numerics are reliable, this is a valuable quantitative companion to Kehle–Unger: it shows that the proof's large-eM regime is not necessary in practice, that the threshold depends strongly on the Ansatz and on regularity, and that mass-to-charge thresholds are well below the rigorous bound m/e < 1. The paper is transparent about the limitations of the C^2 search, and it gives concrete profiles that could seed future explicit spacetime constructions. The main scientific value is in the C^0 and C^1 results; the C^2 entries currently have the status of upper/lower bounds rather than established extrema. No code or data are shipped, and no residual tolerances or error bars are reported, which limits reproducibility for a numerical paper.

major comments (3)
  1. [§3.4 and Table 1] The C^2 entries in Table 1 — (eM)_min = 4417, 2076 and max m/e = 0.00992 — are presented as established extrema, but the text explicitly states that 'there may well exist solutions we have not found' and that Fig. 12 likely omits branches. These values are therefore only bounds obtained from the branches the solver happened to find: for (eM)_min they are upper bounds on the true infimum, and for max m/e they are lower bounds on the true supremum. The abstract and Table 1 should be reworded accordingly, or an independent global search (e.g. continuation/random multistart with documented success rates) should be added.
  2. [§3.3.2 and §3.4] The claim that 'we have not found any examples' of multiple C^1 solutions is used to infer uniqueness of the branch and hence to interpret the C^1 minima as true minima. This is a numerical observation, not a proof, and the C^2 experience shows that extra branches can appear at higher k. The C^1 (eM)_min values (720, 2909, 78.5, 239) should similarly be stated as 'the smallest value found' unless the search over the parameter space is made exhaustive or its completeness is quantified. The distinction matters because the paper's central claim is that gluing is possible only above these thresholds.
  3. [Appendix A and §3.1] The numerical accuracy of the thresholds is not quantified. The text reports step size 0.001 and 'regularly checked . . . convergence tests', but no residual norms, no tolerance for the Broyden solves, no estimated error bars for (eM)_min, Imax, or the critical m/e values, and no data/code are provided. Since the paper's contribution is quantitative, at least the final thresholds should be accompanied by an estimate of discretization and root-finding error. This is not fatal to the qualitative conclusions, but it prevents the reader from assessing how many significant digits in Table 1 are meaningful.
minor comments (6)
  1. [Abstract and §1] The abstract says 'gluing is possible only if the final black hole mass is large enough.' Given the numerical, Ansatz-dependent nature of the evidence, 'only if' is too strong; 'for each of the Ansätze studied, gluing was found to be possible only when eM exceeded a certain value' would be more accurate. The same overstatement appears in the Discussion.
  2. [Eq. (18)–(20)] The notation K = 2k+1 for the number of basis functions is easily confused with the regularity order k. Since the paper uses both k and K in close proximity, a different symbol for the number of parameters would improve readability.
  3. [Fig. 12] The figure relies on colors (black/blue/green) to label branches. The colors will not survive grayscale printing. Please add line styles or explicit labels to all branch plots.
  4. [§2.2, Eq. (5)] The equations are written in a slightly unusual mixed notation, with ∂_V A_U rather than ∂_V A_U after Eq. (5e). Throughout, the gauge choice A_V = 0 should be stated before Eq. (5) is used, not later in §2.3.
  5. [§3.2.5] The discussion of superextremal horizons is interesting but would benefit from a short explanation of why the quasilocal definition is gauge-invariant, since the renormalised mass ϖ is defined via the induced metric on the sphere. Currently the paper only states the formula (25).
  6. [Spelling] Typo in the Introduction: 'produce spactimes' should be 'produce spacetimes'.

Circularity Check

0 steps flagged

No circularity: numerical thresholds are outputs of a shooting problem, not fitted inputs renamed as predictions; self-citations are contextual only.

full rationale

The derivation chain is self-contained. The gluing data are found by solving the constraint ODEs (5) for the scalar profile parameters, and the thresholds in Table 1 are read off from the maximum of the functional I(alpha) (Eq. 21) and from the sign of sup dU r, both of which are outputs of the numerical integration rather than inputs fitted to the reported eM or m/e bounds. For C^0 gluing, (eM)_min = 1/I_max is an analytic consequence of (23), with I_max computed from the chosen Ansatz; the Ansatz-dependence is explicit, not a disguised universal. The phase e^{-iV} is adopted following [3] and is not claimed to be derived, so no ansatz is smuggled in as a result. Refs [12] and [13] are self-citations to Reall, but they are used only as external consistency checks (m/e >= 1 impossibility) and do not support the numerical upper bounds. The Section 3.4 admission that Broyden root-finding may miss C^2 branches is a completeness caveat that makes the C^2 values plausible upper bounds, but it is an evidentiary/robustness limitation, not a circular reduction. No equation is defined in terms of the quantity it is said to predict.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central numerical claims depend on the chosen Ansatz families, the shooting parameters, and the extension theorems of [3] and [14]. The only explicitly tuned parameter is γ in the modified even Ansatz; all other inputs are standard or imported from prior work.

free parameters (1)
  • γ (modified even Ansatz shape parameter) = 0.36 (C^0), 0.48 (C^1)
    Chosen to maximize I_max (Figure 6). The reported (eM)_min for the modified Ansatz depends on this value; no principle fixes γ a priori.
axioms (5)
  • domain assumption The characteristic initial value problem for the Einstein-Maxwell-charged scalar system is well-posed (Luk [14]) and the Cauchy stability argument of Kehle-Unger [3] extends local solutions to the whole spacetime.
    Invoked in Section 2.1 to justify that gluing data on C yields a global spacetime; the paper does not reproduce these proofs and its numerical solutions lie outside the large-eM regime where [3] is proven.
  • standard math The horizon sphere data set of Reissner-Nordström is gauge-equivalent to the lapse-normalised data with ∂_V r=0, Q=qM (Proposition 2, Section 2.3), i.e. Birkhoff's theorem.
    Used to set the target values at V=1.
  • standard math Minkowski/(A)dS sphere data with m=0, Q=0 are gauge-equivalent to the lapse-normalised data (Proposition 1).
    Used to set initial values at V=0.
  • domain assumption The mass-charge inequality m/e≥1 prevents extremal RN formation in finite time (Reall [12]).
    Used as a consistency check in Sections 3.2-3.4 and Discussion; the paper's numerical upper bounds lie well below 1.
  • ad hoc to paper The scalar field profile on C can be taken as Φ=ρ e^{-iV} with ρ real and compactly supported; this Ansatz family is rich enough to satisfy the gluing constraints.
    Guided by [3]; the paper acknowledges the results depend on this choice and does not prove the Ansatz is exhaustive.

pith-pipeline@v1.3.0-alltime-deepseek · 25119 in / 12681 out tokens · 116113 ms · 2026-08-03T17:16:35.024718+00:00 · methodology

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Cite this review

Pith. "Pith review of Formation of extremal Reissner-Nordstr\"om black holes: insights from numerics." pith.science (2026). https://pith.science/paper/6M5AAKTG

@misc{pith2026251210008,
  author       = {Pith},
  title        = {Pith review of: Formation of extremal Reissner-Nordstr\"om black holes: insights from numerics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6M5AAKTG}},
  note         = {Machine review of arXiv:2512.10008}
}
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read the original abstract

An extremal Reissner-Nordstr\"om black hole can form in finite time in the gravitational collapse of a massless charged scalar field. The proof of this is based on the method of characteristic gluing, which involves making an Ansatz for the scalar field at the horizon. We perform a numerical investigation of the characteristic gluing procedure for several different Ans\"atze. In each case, gluing is possible only if the final black hole mass is large enough. We find that the minimum required mass varies significantly for different Ans\"atze. We also consider the effect of including a mass term for the scalar field. In this case, for each Ansatz we determine the maximum mass-to-charge ratio for the scalar field such that gluing is possible. Analogous results are obtained for a non-zero cosmological constant.

Figures

Figures reproduced from arXiv: 2512.10008 by Harvey S. Reall, Jorge E. Santos, Maxime Gadioux.

Figure 1
Figure 1. Figure 1: Setup for characteristic gluing with zero cosmological constant along the outgoing null cone C ≡ {0} × [0, 1]. The metric and matter content in regions R1 and R2 are known and are given by Q1 and Q2, respectively. Data must be imposed on C such that the Einstein equations are satisfied. In the spherically-symmetric situation considered in this paper, R1 can be extended to r = 0, the centre of symmetry, to … view at source ↗
Figure 2
Figure 2. Figure 2: Left: Characteristic gluing with a positive cosmological constant. The Cauchy slice (red) is complete and extends to past null infinity. Right: Characteristic gluing with a negative cosmological constant. The Cauchy slice extends to the timelike boundary. Once R1 and R2 have been appropriately glued, the null cone C can be extended into a full 4-dimensional region of spacetime, thanks to a result of Luk [1… view at source ↗
Figure 3
Figure 3. Figure 3: Left: The function I for the even (solid black), odd (dashed blue) and polynomial (dash-dotted red) Ans¨atze. Right: qmax against eMImax for different values of ΛM2 : from thickest to thinnest, ΛM2 = 0.17, 0.12, 0.1, 0, −0.5, −2. Notice that for ΛM2 > 1/9, there is a lower bound on the charge of the final black hole. compactly supported χi and er 2 +/Q is large then an argument of Kehle and Unger establish… view at source ↗
Figure 4
Figure 4. Figure 4: Left: sup ∂U r (in units M = 1) against ΛM2 for the gluing solution achieved with the minimal value of eM, with m = 0. The different curves represent different dimensionless charges: q = 1 (solid), q = 0.9 (dashed) and q = 0.4 (dotted). The dot denotes the maximum value of ΛM2 (which depends on q). In all cases we have sup ∂U r < 0 so the candidate solution is a valid solution. Right: Profiles ρ for α1 = α… view at source ↗
Figure 5
Figure 5. Figure 5: Left: Maximum charge qmax as a function of eM for m/e = 0 (solid blue), 0.2 (dashed red) and 0.5 (dash-dotted green), with the even Ansatz. The blue curve attains extremality at eM ≈ 15.96, denoted by a star. The latter two curves exhibit a kink when the limiting condition sup ∂U r < 0 becomes important. The kink is more prominent for larger m/e; the insets show that of the red and green curves. After the … view at source ↗
Figure 6
Figure 6. Figure 6: Imax versus γ for the modified even Ansatz with parameter γ. The maximum is reached at γ ≈ 0.36. When the cosmological constant is positive, the upper bound on m/e increases slightly. For example, setting the cosmological constant to the maximum value of ΛM2 = 2/9, with the polynomial Ansatz extremality is possible up to m/e ≈ 0.3797. This is an approximately 37% rise in the upper bound compared with the a… view at source ↗
Figure 7
Figure 7. Figure 7: Renormalised Hawking mass ϖ/M (black) and charge Q/M (dashed blue) over time. Left: Plots for a C 0 solution with the even Ansatz leading to the formation of an extremal black hole. Right: Plots for a C 0 solution with the odd Ansatz leading to the formation of a black hole with q = 0.9. In both cases, the horizon is temporarily superextremal. is integrated backwards in time starting at V = 1, ∂V r remains… view at source ↗
Figure 8
Figure 8. Figure 8: Left: Boundary of the restricted parameter space for the even Ansatz. Only one half is shown; the other can be obtained by a reflection through the origin. Colours indicate the value of I. Large values are in bluer tones, and tend to accumulate in the region where α ≈ (±β, 0, 0), corresponding to a scalar field profile for which the first compactly supported pulse has much larger amplitude than the two sub… view at source ↗
Figure 9
Figure 9. Figure 9: Left: Profiles ρ providing solutions to the C 1 gluing problem for the even (solid black), odd (dashed blue), polynomial (dash-dotted red) and modified (γ = 0.48, dotted black) Ans¨atze for q = 1 and Λ = m = 0. The value of eM used was (approximately) the smallest that allowed q = 1 to be reached, namely 720, 2909, 78.5 and 239, respectively. Right: Profiles for the even Ansatz with eM = 720, for q = 0.1, … view at source ↗
Figure 10
Figure 10. Figure 10: C 1 gluing with a massive scalar field. Left: Maximum charge qmax as a function of eM for Λ = 0 and m/e = 0 (solid blue), 0.01 (dashed red, lying nearly on top of the blue curve), 0.07 (dash-dotted green) and 0.35 (dotted black). The former two curves attain extremality at eM ≈ 720 and 727, respectively, denoted by stars. The latter two curves exhibit a kink when the limiting condition sup ∂U r becomes im… view at source ↗
Figure 11
Figure 11. Figure 11: Left: Minimum value of eM required to reach extremality with the polynomial Ansatz as a function of m/e. The curve diverges for sufficicntly large m/e. Right: Same data but for the odd Ansatz. The main plot shows only the segment where sup ∂U r is the limiting condition. Here, eM appears to blow up as m/e is increased. The inset shows the data for lower mass-to-charge ratios. When inf r is the limiting co… view at source ↗
Figure 12
Figure 12. Figure 12: Left: inf r versus q of various solution branches at eM = 600, m/e = 0, ΛM2 = 0. It is likely that there exist more solutions than presented here. Right: Profiles for C 2 solutions which form an extremal black hole at eM = 4417 (black) and eM = 6198 (blue), with m = 0 = Λ. of the parameters is slowly varied with the others kept fixed, each solution changes continuously, and we can trace the solutions as f… view at source ↗
Figure 13
Figure 13. Figure 13: Minimum value (eM)min necessary to reach extremality as a function of m/e, for two branches of the even Ansatz in the C 2 problem. Here Λ = 0. Left: The branch ceases to exist at the point. Beyond, we find inf r < 0. Right: Here the branch initially has decreasing (eM)min but eventually diverges to infinity. The sudden increase arises due to sup ∂U r approaching zero. 4 Discussion The gluing solutions are… view at source ↗

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Forward citations

Cited by 3 Pith papers

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

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

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

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Reference graph

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