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This paper claims that the nonlinear endpoint of the charged Aretakis instability is a dynamical extremal black hole whose horizon develops divergent energy density and constant charge density — hair — and that this threshold solution is un

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2026-08-03 00:10 UTC pith:3Q65DLIO

load-bearing objection First fully nonlinear treatment of the charged Aretakis instability with back-reaction; the scaling laws and horizon hair look credible, the 'visible from future null infinity' implication is speculative and the authors know it. the 4 major comments →

arxiv 2602.11256 v2 pith:3Q65DLIO submitted 2026-02-11 gr-qc

A Nonlinear Endpoint of Charged Horizon Instabilities

Zachary Gelles , Frans Pretorius This is my paper
classification gr-qc MSC 83C5783C7583-08 PACS 04.70.-s04.25.D-04.40.Nr
keywords charged scalar fieldextremal black holesAretakis instabilitycritical phenomenaReissner-Nordströmhorizon hairgravitational collapsenumerical relativity
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 asks what happens when a charged scalar field is scattered off a nearly super-extremal charged black hole, with the metric, Maxwell field, and scalar evolved together nonlinearly. By fine-tuning the initial charge to a threshold value, the authors construct dynamical extremal black holes — spacetimes that become exactly extremal only in the limit of infinite advanced time. They find that the horizon of this threshold solution develops 'hair': the energy density diverges along the horizon while the charge density tends to a constant. The same critical scaling laws emerge for three different one-parameter families of initial data, supporting the claim that dynamical extremal black holes are universal threshold solutions in the sense of critical gravitational collapse, with the horizon hair as a family-dependent exception. A sympathetic reader should care because this is the first fully nonlinear endpoint of the charged Aretakis instability, and it suggests that near-threshold spacetimes without a black hole could exhibit arbitrarily large curvature visible from infinity.

Core claim

The central claim is that the nonlinear endpoint of the charged horizon instability is a dynamical extremal black hole whose event horizon carries hair: as the initial charge approaches the critical value Q*, the energy density along the horizon grows without bound while the charge density approaches a constant value. The authors establish this by evolving the Einstein–Maxwell–Klein–Gordon system in spherical symmetry, fine-tuning a one-parameter family of super-extremal initial data until the apparent horizon time V_trap diverges. Near threshold, they observe the scaling V_trap ∝ |Q0 − Q*|^{-1/2}, 1 − Q/M ∝ |Q0 − Q*|, and 1 − r/M ∝ |Q0 − Q*|^{1/2}, with the same exponents for three distinct

What carries the argument

The central object is the dynamical extremal Reissner–Nordström solution arising as the threshold of black hole formation in the Einstein–Maxwell–Klein–Gordon system. The mechanism that carries the argument is the fine-tuned scattering of charged wave packets off a super-extremal RN background, combined with the identification of the threshold through the divergence of the apparent-horizon formation time V_trap. The critical scaling laws follow from the near-horizon conformal structure of the extremal geometry, and the horizon hair is quantified by the horizon charge density ∂_r Q, which tends to a family-dependent constant as Q0 → Q*. The interior growth is explained through a Doppler facto

Load-bearing premise

The central claim that the horizon energy density diverges and the interior curvature blows up rests on the assumption that the power-law growth seen for V up to 400 continues to infinite V and is not contaminated by the pre-existing singular Reissner–Nordström geometry at U_max, since the simulations cannot evolve to exact extremality and stop before reaching the singularity.

What would settle it

Run the same fine-tuned scattering with a regularized initial slice — for instance, adding an outgoing scalar pulse or a small black hole on the ingoing null surface so that the interior is not singular RN — and check whether the Ricci scalar still grows without bound near the would-be horizon; if the growth disappears, the interior singularity is an artifact of the background, not a universal threshold feature. Alternatively, a convergence study at V_max > 400 should show whether V_trap keeps following the |Q0−Q*|^{−1/2} power law or turns over.

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

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If this is right

  • If the threshold solution is universal as claimed, then the 1/2 critical exponent for V_trap and V_diss should appear for any one-parameter family of initial data that crosses the extremal threshold, in both charged and neutral scalar collapse and in other matter models.
  • The divergence of energy density and higher curvature gradients along the dynamical extremal horizon implies that linearized Aretakis-type instability is not regulated away by backreaction; it persists nonlinearly.
  • On the dispersive side of threshold, the same growth occurs along a would-be horizon, so curvature can become arbitrarily large in spacetimes that never form a trapped surface — meaning observers at future null infinity could see large curvature.
  • The family-dependent horizon charge density constitutes a genuine hair parameter of dynamical extremal black holes, alongside mass and charge.
  • The scaling V_diss ∼ |Q0−Q*|^{-1/2} on both sides of threshold links the critical exponent to the surface gravity of the near-extremal black hole, κ ∼ sqrt(1−Q/M), connecting critical collapse to near-extremal transient instability.

Where Pith is reading between the lines

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

  • If the interior curvature blow-up is confirmed with regularized initial data (e.g., an outgoing pulse on the initial null surface), it would constitute a naked singularity arising from the dispersive side of the threshold — a concrete counterexample to cosmic censorship in this matter model.
  • Since the same 1/2 exponent appears in uncharged scalar collapse, charged scalar collapse, and Vlasov collapse, the near-horizon conformal symmetry may force this exponent universally, suggesting an analytic derivation for charged fields analogous to existing ones for neutral fields.
  • A testable extension: vary the scalar charge coupling eQ0 and measure the horizon charge density and energy-density growth; the paper's framework predicts the hair value and Aretakis growth rates should be continuous functions of eQ0 while the critical exponents remain fixed.
  • If the 'would-be interior' growth is indeed universal, then fine-tuned super-extremal spacetimes without black holes could serve as astrophysical sources of high-curvature signatures at null infinity, a signature that future numerical codes with larger V domains could look for.

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

4 major / 4 minor

Summary. This paper studies the nonlinear, spherically symmetric Einstein–Maxwell–Klein–Gordon system in double-null coordinates, building on the Murata–Reall–Tanahashi construction. Starting from super-extremal Reissner–Nordström initial data and fine-tuning a charged scalar pulse, the authors present evidence for a threshold Q* at which the trapped-region formation time Vtrap diverges, yielding a dynamical extremal black hole. They report universal near-threshold scalings Vtrap ∝ |Q0 − Q*|^{-1/2}, 1 − Q/M ∝ |Q0 − Q*|, and 1 − r/M ∝ |Q0 − Q*|^{1/2} across three one-parameter families. They also report Aretakis-like growth of energy density and curvature gradients on the (would-be) horizon on both sides of the threshold, and a family-dependent horizon charge density ('hair'). The most speculative claim is that the Ricci scalar grows without bound just inside the (would-be) horizon, implying that, on the dispersive side, large curvatures could be visible from future null infinity.

Significance. If the central results hold, the paper would be a substantial extension of MRT to charged scalar fields, with implications for the nonlinear fate of the Aretakis instability, for charged-horizon hair, and for critical phenomena in gravitational collapse. The manuscript has clear strengths: it compares with independently derived analytic exponents [10,11], checks three distinct initial-data families, and explicitly identifies the main caveats—notably the finite simulation domain and the singular ingoing initial slice. However, the numerical support for the headline interior-singularity claim is explicitly acknowledged by the authors to be incomplete, and the scaling exponents are presented without quantitative fitting uncertainties or convergence data. The result is plausible but, as written, the strongest physical implications outrun the evidence.

major comments (4)
  1. [§III A, Eq. (14), Figs. 3–4] The critical exponent 1/2 is described as fit 'by eye,' with no reported uncertainties on Q*, Vtrap, or the fitted slope, and no resolution-convergence comparison is shown in the paper despite the statement that the exponents converge. Because Eq. (14) is the main quantitative evidence for the claimed universal threshold, the fit needs to be made reproducible: specify the fitting window, propagate the bisection uncertainty in Q*, and provide a convergence study in ΔV and in C from Eq. (9).
  2. [§III A, event-horizon proxy at Vmax=400] For |Q0 − Q*| = ε, Eq. (14) predicts Vtrap ∝ ε^{-1/2}; with Vmax = 400 this implies Vtrap > Vmax for ε ≲ 6 × 10^{-6}. Yet Figs. 4 and 7 extend to ε ∼ 10^{-7}. It is unclear how an 'apparent horizon on the largest V' can be assigned for runs in which no trapped surface forms before Vmax, and how Q* determined by bisection is independent of Vmax. The finite-domain truncation therefore appears to be conflated with a genuine threshold; this needs to be addressed explicitly, e.g. by extrapolating in Vmax or by restricting the claimed scalings to resolvable runs.
  3. [§IV C, Figs. 13 and 18; abstract] The claim that interior Ricci curvature grows without bound and would be visible from future null infinity is the most distinctive new physical implication, but the paper itself states that this growth may be an artifact of the pre-existing RN singularity on the ingoing slice NA at (Umax, 0), with U = Umax acting as a Cauchy horizon. Since that caveat is not reflected in the abstract's unconditional implication, the central claim is not yet supported. The authors should either regularize NA (e.g. by adding outgoing radiation) or substantially soften the abstract and conclusion.
  4. [§IV B, Appendix B, Eq. (31)] The Doppler-factor scalings used to explain the 'blueshift focusing instability' are derived for fixed extremal RN in MRT gauge, then applied to fully dynamical near-threshold spacetimes. The qualitative agreement in Figs. 16–17 is suggestive, but no quantitative comparison between the nonlinear Doppler factors and the fixed-background prediction is given. Since this mechanism is used to argue that the interior growth is a property of the threshold solution rather than a numerical artifact, the authors should either provide a quantitative test or present the focusing argument as a heuristic only.
minor comments (4)
  1. [§III B, Fig. 6] In the bottom row, the horizontal-axis labels appear to use '0' where 'A0' is intended; as printed the labels are ambiguous.
  2. [§IV A, Eq. (25)] The quoted uncertainty ±0.029 is only the Fourier bin width. A measured frequency quoted to three significant figures should also include finite-difference and fit uncertainties.
  3. [§II, Eq. (9)] The adaptive mesh parameter C is set to 0.6, but no sensitivity test in C is reported. A brief convergence statement with respect to C would strengthen the numerical claims.
  4. [§IV C] Typo: 'it is difficult see how' should read 'it is difficult to see how.'

Circularity Check

0 steps flagged

No significant circularity: core results are measured from the standard Einstein-Maxwell-Klein-Gordon system and benchmarked against independent analytic scalings; acknowledged caveats are limitations, not circular reductions.

full rationale

The central derivation is not circular. The evolution equations (Eqs. 3-8) are the standard spherically symmetric Einstein-Maxwell-Klein-Gordon system; the matter-sector equations are carried over from the authors' prior work [8], but the new claims—threshold scaling, horizon hair, and interior curvature growth—are extracted from numerical solutions rather than imposed by construction. The critical exponent 1/2 is compared with independently derived analytic results (e.g., [10, 11, 14, 38]) and with the near-extremal surface-gravity relation (Eq. 28), so the tuned value Q* is not itself the source of the scaling law. The horizon charge-density hair is measured on the apparent-horizon proxy and differs among three families (Fig. 8), which is direct evidence that it is not fixed by definition. The interior-singularity conjecture is explicitly caveated in §IV C and §V as possibly an artifact of the super-extremal RN singularity on the ingoing initial-data slice; an acknowledged numerical-extrapolation limitation is not a circularity. The only noticeable self-citations ([8]) supply formalism and NZD-mode properties; they do not define the predicted quantities. No equation in the paper is defined in terms of the quantity it purports to predict.

Axiom & Free-Parameter Ledger

6 free parameters · 4 axioms · 0 invented entities

The central claims rest on a standard GR matter model, but the threshold identification relies on several hand-chosen numerical parameters and on an extrapolation across a finite domain. The most fragile input is the fixed-background Doppler-factor analysis being applied to dynamical spacetimes, which the authors themselves treat as a qualitative heuristic.

free parameters (6)
  • eQ0 (coupling constant times background charge) = 0.6
    Chosen by hand throughout to place the scalar field in the enhanced-instability regime eQ0 ≥ 1/2 (§III). Results are only demonstrated at this coupling.
  • Q* (critical background charge, single pulse) = 1.0033218
    Bisection fine-tuning value for the single-pulse family; defines the threshold used for all Q0 − Q* scaling plots (§III A).
  • Q*,double (critical background charge, double pulse) = 1.0051471
    Threshold value for the double-pulse family (§III B); family-dependent critical parameter.
  • A* (critical amplitude, A0-route) = 0.0179708
    Threshold amplitude in the A0-tuning family at fixed Q0 = 1.01 (§III B); family-dependent critical parameter.
  • Pulse shape parameters = A0 = 0.01, ω̃ = 1, width = 20
    Initial-data family choices that set non-universal prefactors and the horizon hair values (§III, Eq. 13). They are not fitted to the threshold but are free choices.
  • Adaptive mesh constant C = 0.6
    Numerical resolution parameter in Eq. (9). No explicit convergence study is shown in the paper.
axioms (4)
  • standard math The double-null Einstein-Maxwell-Klein-Gordon system (Eqs. 3–8) is the correct spherically symmetric reduction of general relativity.
    Basis of all evolutions; standard GR, but no code or verification artifact is provided.
  • domain assumption Super-extremal RN data on the past null cone, with the singular point at (Umax, 0), can be evolved as a Cauchy problem up to Umax.
    The paper restricts to purely ingoing perturbations and deliberately leaves the RN singularity on NA; the authors note Umax is effectively a Cauchy horizon (§IV C).
  • domain assumption The apparent horizon at the largest V in the domain is a faithful estimate of the event horizon for measuring Q/M and r/M scalings.
    Explicit proxy stated in §III A: 'we simply take to be the apparent horizon on the largest V in our simulation domain'.
  • ad hoc to paper The fixed-extremal-RN Doppler-factor expansion (Appendix B, Eq. B7) carries over to dynamical near-threshold spacetimes.
    The 'blueshift focusing instability' explanation imports fixed-background scaling D ∝ V²Δλ² into the nonlinear context without a rigorous proof, using it to interpret Figures 13–18.

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Cite this review

Pith. "Pith review of A Nonlinear Endpoint of Charged Horizon Instabilities." pith.science (2026). https://pith.science/paper/3Q65DLIO

@misc{pith2026260211256,
  author       = {Pith},
  title        = {Pith review of: A Nonlinear Endpoint of Charged Horizon Instabilities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3Q65DLIO}},
  note         = {Machine review of arXiv:2602.11256}
}
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read the original abstract

We numerically construct asymptotically extremal black holes through the nonlinear evolution of a charged scalar field. Our procedure -- which extends the work of Murata-Reall-Tanahashi to include charged scalar dynamics -- involves the fine-tuned scattering of wave packets within an initially super-extremal Reissner-Nordstrom spacetime. The resulting extremal solution develops an event horizon along which the energy density diverges and the charge density approaches a constant (i.e., the horizon forms with "hair"). We investigate this behavior from the perspective of critical phenomena in gravitational collapse, giving evidence that dynamical extremal black holes act as universal threshold solutions modulo this family-dependent hair. As in the linear instability of fixed extremal backgrounds, the scalar field decays outside the dynamical extremal horizon. But just inside the horizon, the scalar curvature appears to develop unbounded growth. This implies that near-threshold solutions without a black hole could develop correspondingly large curvatures visible from future null infinity.

Figures

Figures reproduced from arXiv: 2602.11256 by Frans Pretorius, Zachary Gelles.

Figure 1
Figure 1. Figure 1: FIG. 1. Sample Penrose diagram of dynamical spacetime [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Construction of a dynamical extremal black hole. Unlike Figure [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Black hole properties as [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Initial conditions for the scalar amplitude along [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Top: Black hole properties as [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Energy density and charge density along the event [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Charge density on the horizon after fine-tuning close [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Radial derivatives of the Ricci scalar along the outer horizon of a black hole as [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Radial derivatives of the Ricci scalar along the “would-be” horizon ( [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Growth of large curvature gradients along a “would [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: , we see that the duration of the instability scales with the universal 1/2 exponent on either side of the critical point: Vdiss ∼ |Q0 − Q∗| −1/2 . (27) Note that the pre-factor of the scaling law differs between the two sides of the critical point. 10 5 |Q0 Q* | 2 × 10 2 3 × 10 2 4 × 10 2 Vdis s Instability Dissipation Timescale |Q0 Q | 1/2 Simulation: Q0 < Q* Simulation: Q0 > Q* FIG. 12. The dissipation… view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Behavior of the energy density along three ingoing [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Energy density plotted as a function of affine time [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Penrose diagram for static sub-extremal (left) and extremal Reissner-Nordstr¨om (right). Contours of constant [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Segment of the (rotated) Penrose diagrams of several [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Behavior of the Ricci scalar in the near-threshold [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗

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Reference graph

Works this paper leans on

49 extracted references · 20 linked inside Pith · cited by 1 Pith paper

  1. [1]

    one-mode unstable

    and successfully construct dynamical extremal black holes by fine tuning charged wave packets incident on super-extremal spacetimes. Like MRT, we similarly find that enhanced Aretakis-like growth persists on the dynamical extremal horizon. We also find that more “generic” charged perturbations of initially extremal spacetimes lead to sub-extremal black ho...

    Pith/arXiv arXiv 2026

  2. [2]

    would-be horizon

    Focusing Instability: Non-Linear Picture Here, we demonstrate that the focusing effects discussed for the fixed RN spacetimes are also present in the dynamical extremal spacetimes, suggesting that one can think of these focusing effects as driving the growth of energy density (cf. Figures 13-14) and curvature gradients. In Figure 16, we plot the Doppler f...

  3. [3]

    Aretakis, in Annales Henri Poincar´ e , Vol

    S. Aretakis, in Annales Henri Poincar´ e , Vol. 12 (Springer, 2011) pp. 1491–1538

    2011

  4. [4]

    Doppler factor

    Focusing Instability: Linear Picture To understand the instability of the dynamical extremal interior, we will take a step back and analyze the behavior of matter perturbations on a fixed background. In particular, we will examine the behavior of outgoing radiation in the fixed RN interior. For static, sub-extremal Reissner-Nordstr¨ om in Kruskal coordina...

  5. [5]

    blueshift instability,

    This result is what ultimately allows Aretakis growth to persist at extremality: With no redshift, energy and charge can accumulate on the horizon without falling in. But even though Doppler effects vanish precisely along the fixed extremal horizon, they re-emerge everywhere else. Again taking ∆ λ ≡ λ − λ∗ to represent the affine time displacement from th...

  6. [6]

    Furthermore, analyzing the charged scalar field is valuable because it serves as a conceptual analog of the vacuum Kerr problem

    showed that on a fixed extremal RN background with a fixed electromagnetic potential, the energy density stored in a charged scalar field actually blows up along the horizon. Furthermore, analyzing the charged scalar field is valuable because it serves as a conceptual analog of the vacuum Kerr problem. We can think of charged scalar field perturbations of...

  7. [7]

    Murata, H

    K. Murata, H. S. Reall, and N. Tanahashi, Classical and Quantum Gravity 30, 235007 (2013), arXiv:1307.6800 [gr-qc, physics:hep-th]

    Pith/arXiv arXiv 2013

  8. [8]

    Aretakis, Communications in mathematical physics 307, 17 (2011)

    S. Aretakis, Communications in mathematical physics 307, 17 (2011)

    2011

  9. [9]

    M. W. Choptuik, Physical review letters 70, 9 (1993)

    1993

  10. [10]

    Aretakis, arXiv e-prints , arXiv:1206.6598 (2012), arXiv:1206.6598 [gr-qc]

    S. Aretakis, arXiv e-prints , arXiv:1206.6598 (2012), arXiv:1206.6598 [gr-qc]

    Pith/arXiv arXiv 2012

  11. [11]

    Angelopoulos, C

    Y. Angelopoulos, C. Kehle, and R. Unger, arXiv preprint arXiv:2410.16234 (2024)

    arXiv 2024

  12. [12]

    Zimmerman, Physical Review D 95, 124032 (2017), publisher: American Physical Society

    P. Zimmerman, Physical Review D 95, 124032 (2017), publisher: American Physical Society

    2017

  13. [13]

    Kehle and R

    C. Kehle and R. Unger, Gravitational collapse to extremal black holes and the third law of black hole thermodynamics (2022), arXiv:2211.15742 [gr-qc, physics:math-ph]

    Pith/arXiv arXiv 2022

  14. [14]

    Gelles and F

    Z. Gelles and F. Pretorius, Phys. Rev. D 112, 064003 (2025), arXiv:2503.04881 [gr-qc]

    Pith/arXiv arXiv 2025

  15. [15]

    double-pulse

    (and likely is also present in a rigorous construction of this scenario presented in [16]). The trends from our simulations suggest that as Vtrap → ∞, the horizon geometry approaches that of an extremal Reissner-Nordstr¨ om black hole. We show this in Figure 4, where the radius r and charge Q of the resultant black hole approach extremality with the relat...

  16. [16]

    S. E. Gralla and P. Zimmerman, Classical and Quantum Gravity 35, 095002 (2018), arXiv:1711.00855 [gr-qc]

    Pith/arXiv arXiv 2018

  17. [17]

    Yang and L

    H. Yang and L. Zou, Phys. Rev. D 109, 104034 (2024), arXiv:2207.04373 [gr-qc]

    Pith/arXiv arXiv 2024

  18. [18]

    Gundlach and L

    C. Gundlach and L. Martel, arXiv e-prints , arXiv:2511.14874 (2025), arXiv:2511.14874 [gr-qc]

    arXiv 2025

  19. [19]

    Poisson and W

    E. Poisson and W. Israel, Phys. Rev. Lett. 63, 1663 (1989)

    1989

  20. [20]

    Hadar, Journal of High Energy Physics 2019, 214 (2019), arXiv:1811.01022 [gr-qc, physics:hep-th]

    S. Hadar, Journal of High Energy Physics 2019, 214 (2019), arXiv:1811.01022 [gr-qc, physics:hep-th]

    Pith/arXiv arXiv 2019

  21. [21]

    W. E. East, arXiv preprint arXiv:2511.20567 (2025)

    Pith/arXiv arXiv 2025

  22. [22]

    Kehle and R

    C. Kehle and R. Unger, arXiv preprint arXiv:2402.10190 (2024)

    Pith/arXiv arXiv 2024

  23. [23]

    T. Hara, T. Koike, and S. Adachi, arXiv e-prints , gr- qc/9607010 (1996), arXiv:gr-qc/9607010 [gr-qc]

    arXiv 1996

  24. [24]

    Hod and T

    S. Hod and T. Piran, Physical Review D 55, 3485–3496 (1997)

    1997

  25. [25]

    Garfinkle and G

    D. Garfinkle and G. C. Duncan, Physical Review D 58, 064024 (1998)

    1998

  26. [26]

    Gundlach and J

    C. Gundlach and J. M. Martin-Garcia, Living Reviews in Relativity 10, 5 (2007)

    2007

  27. [27]

    Gundlach, D

    C. Gundlach, D. Hilditch, and J. M. Mart ´ ın-Garc ´ ıa, Critical phenomena in gravitational collapse (2025), arXiv:2507.07636 [gr-qc]

    Pith/arXiv arXiv 2025

  28. [28]

    Penrose, in Battelle Rencontres (1968) pp

    R. Penrose, in Battelle Rencontres (1968) pp. 121–235

    1968

  29. [29]

    Dafermos, Ann

    M. Dafermos, Ann. Math 158, 875 (2003)

    2003

  30. [30]

    Marolf and A

    D. Marolf and A. Ori, Phys. Rev. D 86, 124026 (2012), arXiv:1109.5139 [gr-qc]

    Pith/arXiv arXiv 2012

  31. [31]

    Luk and S.-J

    J. Luk and S.-J. Oh, Annals of PDE 5, 6 (2019)

    2019

  32. [32]

    Van de Moortel, Communications in Mathematical 19 Physics 360, 103 (2018)

    M. Van de Moortel, Communications in Mathematical 19 Physics 360, 103 (2018)

    2018

  33. [33]

    Marolf, General Relativity and Gravitation 42, 2337 (2010), arXiv:1005.2999 [gr-qc]

    D. Marolf, General Relativity and Gravitation 42, 2337 (2010), arXiv:1005.2999 [gr-qc]

    Pith/arXiv arXiv 2010

  34. [34]

    Garfinkle, Classical and Quantum Gravity 28, 175005 (2011), arXiv:1105.2574 [gr-qc]

    D. Garfinkle, Classical and Quantum Gravity 28, 175005 (2011), arXiv:1105.2574 [gr-qc]

    Pith/arXiv arXiv 2011

  35. [35]

    A. P. Porfyriadis, C. Rosen, and G. Tsaraktsidis, arXiv preprint arXiv:2512.23629 (2025)

    Pith/arXiv arXiv 2025

  36. [36]

    Casals, S

    M. Casals, S. E. Gralla, and P. Zimmerman, Physical Review D 94, 064003 (2016), arXiv:1606.08505 [gr-qc]

    Pith/arXiv arXiv 2016

  37. [37]

    Angelopoulos, S

    Y. Angelopoulos, S. Aretakis, and D. Gajic, Physical Review Letters 121, 131102 (2018), publisher: American Physical Society

    2018

  38. [38]

    S. D. Bishoyi, S. Sabharwal, and G. Khanna, General Relativity and Gravitation 57, 46 (2025), arXiv:2407.06926 [gr-qc]

    Pith/arXiv arXiv 2025

  39. [39]

    Aretakis, G

    S. Aretakis, G. Khanna, and S. Sabharwal, General Relativity and Gravitation 57, 160 (2025), arXiv:2307.03963 [gr-qc]

    arXiv 2025

  40. [40]

    Hod, Physics Letters A 374, 2901 (2010), arXiv:1006.4439 [gr-qc]

    S. Hod, Physics Letters A 374, 2901 (2010), arXiv:1006.4439 [gr-qc]

    Pith/arXiv arXiv 2010

  41. [41]

    Hod, Physics Letters B 710, 349 (2012)

    S. Hod, Physics Letters B 710, 349 (2012)

    2012

  42. [42]

    H. Yang, F. Zhang, A. Zimmerman, D. A. Nichols, E. Berti, and Y. Chen, Physical Review D—Particles, Fields, Gravitation, and Cosmology 87, 041502 (2013)

    2013

  43. [43]

    Zimmerman and Z

    A. Zimmerman and Z. Mark, Phys. Rev. D 93, 044033 (2016)

    2016

  44. [44]

    S. E. Gralla, A. Zimmerman, and P. Zimmerman, Physical Review D 94, 084017 (2016), publisher: American Physical Society

    2016

  45. [45]

    Gajic and J

    D. Gajic and J. Luk, Pure and Applied Analysis 1, 263 (2019)

    2019

  46. [46]

    Gundlach and J

    C. Gundlach and J. M. Martin-Garcia, Phys. Rev. D 54, 7353 (1996), arXiv:gr-qc/9606072

    Pith/arXiv arXiv 1996

  47. [47]

    R. J. W. Petryk, Maxwell-Klein-Gordon Fields in Black Hole Spacetimes , Ph.D. thesis, The University of British Columbia, Vancouver (2005), available at: http://laplace.physics.ubc.ca/People/matt/Doc/Theses/

    2005

  48. [48]

    Choptuik (2004), unpublished

    M. Choptuik (2004), unpublished

    2004

  49. [49]

    Gundlach, T

    C. Gundlach, T. W. Baumgarte, and D. Hilditch, Phys. Rev. D 100, 104010 (2019), arXiv:1908.05971 [gr-qc]

    Pith/arXiv arXiv 2019