Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

This paper derives explicit closed-form JT-gravity solutions for the late-time near-horizon approach to a dynamical extreme Reissner–Nordström black hole, with a singularity-free horizon and a scalar Aretakis instability that persists indef

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 13:34 UTC pith:OZSSQ3GJ

load-bearing objection Explicit JT dilaton profiles for approaching DERN are new and useful, but the central claim overreaches: conditions (40) don't fix the scalar boundary data, and the same dilaton can pair with a non-DERN scalar. the 3 major comments →

arxiv 2512.23629 v3 pith:OZSSQ3GJ submitted 2025-12-29 hep-th gr-qc

Approaching a dynamical extreme black hole horizon

Achilleas P. Porfyriadis , Christopher Rosen , Georgios Tsaraktsidis This is my paper
classification hep-th gr-qc
keywords dynamical extreme Reissner–NordströmJackiw–Teitelboim gravityAretakis instabilityAdS2 × S2 throatnear-extremal black holesblack hole formation thresholdmassless scalar matterexact solutions
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 tries to establish that the late-time near-horizon approach to a dynamical extreme Reissner–Nordström (DERN) black hole can be solved exactly in a two-dimensional Jackiw–Teitelboim (JT) gravity description. Its main claim is that any smooth scalar profile f(U), three constants (a,b,c) obeying b²−4ac=0, and a sufficiently leaky net flux δT(U) satisfying δT(π/2)=δT′(π/2)=0 produce a dilaton field that describes the DERN throat with a singularity-free horizon and scalar Aretakis behavior continuing forever. The result matters because DERN has been proposed as the threshold of black hole formation in Einstein–Maxwell-scalar theory, and until now evidence was numerical or perturbative. If the claim holds, it provides concrete closed-form nonlinear solutions — explicitly worked out in two examples — that encode the whole approach to extremality and make the criticality of DERN a built-in feature of the boundary conditions.

Core claim

The central discovery is that the nonlinear late-time near-horizon dynamics of a DERN black hole are exactly solvable in JT gravity. Imposing linear-Aretakis boundary conditions on the scalar — σ|B, σ′|B, σ″|B → 0 as U→π/2, equivalently δT(π/2)=δT′(π/2)=0 — together with the critical dilaton condition b²−4ac=0, the authors obtain explicit dilaton profiles Φ(U,V) whose late-time boundary value reduces to the static extreme RN vacuum. Two concrete profiles are given: one with nonvanishing Aretakis constant H=1 (f(U)=−2U, δT(U)=−A(π/2−U)², A>2.76) and one with H=0 (f(U)=(π/2−U)², δT(U)=−A(π/2−U)⁴, A>1.73). In both cases the Poincaré horizon U=π/2 is identified with the DERN event horizon and is

What carries the argument

The central object is the JT dilaton Φ(U,V), which encodes the variation of the transverse S² area, evolved against a free massless scalar σ=f(U)+g(V) on a fixed AdS₂ background. The general solution separates into a vacuum part Φvac (the SL(2)-orbit hθθ=ar+b r t+c r(t²−1/r²)), a 'balanced' term, and an integral over the net boundary flux δT(U)=g′²−f′². The mechanism that carries the argument: the boundary conditions δT(π/2)=δT′(π/2)=0 convert the Aretakis instability into simple conditions on the flux, the relation b²−4ac=0 pins the vacuum solution to the critical threshold between sub-extreme and super-extreme RN, and a sufficiently negative δT at early times keeps the singularity locus 1+

Load-bearing premise

The load-bearing assumption is that the two-dimensional JT theory, with the scalar's stress tensor as the only source for the dilaton, captures the full nonlinear s-wave dynamics of the four-dimensional Einstein-Maxwell-scalar theory in the AdS₂×S₂ throat, and that the near-horizon solution can be glued to a static extreme RN exterior; the paper does not construct a global four-dimensional metric.

What would settle it

A four-dimensional numerical simulation of fine-tuned scalar collapse approaching DERN could check whether the near-horizon metric and scalar match the closed-form profiles (42) or (43), and whether the singularity locus stays away from the horizon precisely for A larger than the quoted Amin; finding a global 4D metric built from the JT data that develops a curvature singularity on the horizon would falsify the construction.

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

If this is right

  • The late-time approach to DERN admits explicit closed-form dilaton profiles, giving exact nonlinear templates (Eqs. 42 and 43) that numerical relativity can target.
  • The threshold of black-hole formation, with sub-extreme and super-extreme RN on either side, is enforced by the single condition b²−4ac=0 on the vacuum dilaton.
  • A sufficiently leaky scalar flux δT<0 at early times is necessary for a singularity-free horizon; the minimal amplitudes are Amin≈2.76 for H=1 and Amin≈1.73 for H=0.
  • The Aretakis instability persists indefinitely in the nonlinear description, not as a transient, because the boundary conditions enforce the linear Aretakis behavior at all late times.
  • The approach to DERN is accompanied by a final burst of outgoing scalar flux leaking out of the AdS₂ throat.

Where Pith is reading between the lines

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

  • Editorial inference: If the gluing to a static ERN exterior can be upgraded to a global four-dimensional metric, the same machinery would produce explicit four-dimensional DERN spacetimes, making the threshold condition an exact statement rather than a near-horizon one.
  • Editorial inference: A natural extension, not pursued in the paper, is to classify the full space of admissible (f,δT) choices; the construction suggests DERN may form a large functional family, whose physical reachability from regular Cauchy data remains to be checked.
  • Editorial inference: The same translation of Aretakis conditions into flux conditions might apply to other near-extreme settings, such as charged scalars, higher multipoles, or near-extreme Kerr, where an analogous JT-type boundary-value problem could yield explicit late-time throats.
  • Editorial inference: The predicted minimum leakage amplitudes Amin are concrete, testable numbers; a four-dimensional numerical evolution tuned to the critical family should either reproduce these thresholds or show that the JT reduction misses part of the backreaction.

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 / 4 minor

Summary. The paper proposes an analytic, closed-form description of the late-time near-horizon approach to dynamical extreme Reissner–Nordström (DERN) black holes, using two-dimensional Jackiw-Teitelboim (JT) gravity with a free massless scalar. It reviews the linear Aretakis instability on ERN and the emergence of AdS2×S2 from ERN, sub-ERN, and super-ERN limits. In Section IV it solves the JT equations for the dilaton sourced by the scalar stress tensor, imposes boundary conditions on the dilaton (the ERN threshold b^2-4ac=0) and on the scalar (Aretakis-type conditions), and presents two explicit dilaton profiles (42) and (43) for nonvanishing and vanishing Aretakis constant H. The central claim is that any smooth f(U), constants (a,b,c) satisfying (41), and a sufficiently leaky δT(U) satisfying (40) yield a DERN.

Significance. If correct, the construction would provide explicit, closed-form models of the late-time approach to DERN, complementing numerical [19] and rigorous [21] results and illustrating how JT captures non-linear s-wave dynamics near extremality. The paper is transparent about several assumptions, most notably the validity of the JT reduction and the gluing to an ERN exterior, and it gives concrete examples that are easy to inspect. However, the central sufficiency claim and the explicit examples contain technical gaps that need to be addressed before the main result can be accepted as stated.

major comments (3)
  1. [Section IV, final paragraph (Eqs. 37-40)] The claim that any f(U) and δT(U) satisfying (40) produce a DERN is false as stated. δT = g'^2 - f'^2 constrains only the combination of derivatives; the DERN boundary conditions are the full set (26): f+g, f'+g', f''+g'' → 0 at U=π/2. Conditions (40) are necessary but not sufficient. Concretely, take f(U)=-2U and δT(U)=-A(π/2-U)^2 with A>Amin≈2.76, but choose the opposite sign for g, i.e. g'(π/2)=-2 and g(π/2)=π. Then (40) holds and the dilaton (38) is identical to the H=1 example (42), yet f'(π/2)+g'(π/2)=-4≠0. Equation (24) then gives exterior scalar decay ~1/v instead of the required 1/v^2 of (2), so the solution is not a DERN. The construction must either impose (26) directly or specify the sign of g' and the constant part of g.
  2. [Section IV, Eqs. (42)-(43)] The explicit examples appear inconsistent with the stated vacuum constants a=2, b=c=0. Setting A=0 in (42) gives Φ=2(U-V)cot(U-V)-2, but the vacuum solution (35) for a=2,b=c=0 is Φvac=4cosUcosV/sin(U-V). These differ by an O(1) term; for instance at U=1, V=0.5 the former is ~ -0.17 while the latter is ~ 3.79. Thus (42) does not reduce to Φvac at A=0, nor does it reproduce the boundary behavior Φ≈Φvac on B required for the gluing. The stated condition (41) is therefore not implemented in the written examples. Either the vacuum term is missing from (42)-(43) or the quoted values of (a,b,c) are incorrect; the paper should be corrected and re-checked.
  3. [Section IV and Fig. 4 (JT reduction and gluing)] The physical interpretation of the solutions as describing 4D DERN relies on two assumptions that are acknowledged but not derived: (i) that the JT action (28) with a free scalar accurately captures the non-linear s-wave dynamics of the 4D Einstein-Maxwell-scalar theory, including backreaction; and (ii) that the AdS2 solution can be glued to a static ERN exterior at the boundary B for as long as Φ≪1. The paper does not construct a global 4D metric or provide a quantitative matching. As a result the central claim is conditional on these assumptions. A concrete test would be to compare the late-time behavior of the 4D metric and scalar obtained from (42)-(43) with the numerical DERN solutions of [19] or the rigorous stability results of [21]. I encourage the authors to state this more cautiously, e.g. 'candidate JT descriptions' rather than 'description of DERN' in the abstract.
minor comments (4)
  1. [Section III B, around Eq. (27)] The footnote that the first condition in (27) does not constrain δT because of shift symmetry is correct for the constant part, but the second condition does constrain δT only after choosing a sign of g'. This should be clarified in the text.
  2. [Section IV, Eq. (38)] The rewriting of the general solution in terms of δT with upper limit π/2 is a key step; its derivation from (34) involves a choice of lower integration limits and a sign convention. A short derivation or reference to the intermediate steps would help the reader verify the formula.
  3. [Fig. 5 and text below (42)] The values Amin≈2.76 and ≈1.73 are determined 'by the requirement that the singularity barely grazes the future horizon', but the numerical procedure is not described. Please specify how these values are computed (e.g. root-finding on the singularity locus).
  4. [Throughout] The paper would benefit from an explicit statement of the regime of validity of the JT approximation, including the hierarchy of scales (e.g. λ, M, and the size of Φ) and how the inequality Φ≪1 is checked for the given examples.

Circularity Check

0 steps flagged

No significant circularity: the JT construction is an explicit boundary-value calculation, and the paper openly disclaims deriving Aretakis behavior from scratch.

full rationale

The paper does not pretend to derive DERN from a neutral first-principles input. Its own wording disclaims the only potentially tautological step: 'we are not thinking of the above as a derivation of the Aretakis instability for a scalar on ERN, but rather as a derivation of the appropriate boundary conditions to impose in a purely AdS2 analysis' (Sec. III B). The scalar boundary conditions (27) and the dilaton threshold condition (41) are imposed precisely so that the solutions encode the defining DERN features (Aretakis scalar behavior and the ERN/sub-ERN/super-ERN threshold). That makes the construction reverse-engineered relative to the target, but the paper is explicit about this and does not present it as an independent prediction. The central output, the closed-form dilaton family in Eq. (38) and the examples (42)-(43), follows from genuinely solving the linear dilaton equation (32) with sources (33); it is not a fitted parameter renamed as a prediction, nor is Eq. (38) equivalent to the boundary conditions by construction. Existence and stability of the 4D DERN are imported from external works [19] and [21] (Murata-Reall-Tanahashi and Angelopoulos-Kehle-Unger), which are independent of the present authors. The main self-citation, Ref. [36] (co-authored by Porfyriadis), supplies the anabasis classification used to motivate the condition b^2-4ac=0, but this is a concrete published calculation with stated assumptions and is corroborated by the external DERN literature; it is not a load-bearing circularity. Two caveats are genuine but not circular: the reduction to JT and the gluing to a 4D exterior are assumed rather than derived (footnote 1 notes backreaction aspects not captured by JT; Fig. 4 states 'we can trust the gluing for as long as Φ<<1'), and the final claim that conditions (40) alone suffice is overbroad because (40) does not capture the full scalar conditions f'+g'=0 and f''+g''=0 from (27). These are correctness/sufficiency gaps, not reductions of the output to the input.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The central claim rests on the JT/4D correspondence and on boundary conditions chosen to reproduce the defining DERN features. These are imported from prior work or imposed by hand rather than derived in this paper.

free parameters (4)
  • vacuum dilaton constants a,b,c = a=2, b=0, c=0 in examples (with b^2-4ac=0)
    Integration constants in Φvac; constrained by the ERN threshold condition (41). They set the mass/charge scale of the exterior.
  • leakage amplitude A in δT(U) = A>2.76 (H=1 example), A>1.73 (H=0 example)
    Net flux amplitude chosen large enough to keep the singularity off the horizon; thresholds found numerically in Fig. 5.
  • scalar boundary profile f(U) (and g(U)) = f(U)=-2U (H=1) or f(U)=(π/2-U)^2 (H=0)
    Arbitrary functions defining the scalar field; chosen to realize the Aretakis constant H and boundary conditions (40).
  • Aretakis constant H = 1 in first example; 0 in second
    Determined by f'(π/2); not independent but a free output/input of the construction.
axioms (6)
  • domain assumption JT gravity (28) with dilaton Φ and free scalar σ accurately describes the s-wave sector of the 4D Einstein-Maxwell-scalar theory near an AdS2×S2 throat.
    Invoked in Section IV when using (28)-(30) to describe DERN; supported by [9,11] but with caveats in footnote 1.
  • standard math The general solution for a massless scalar on AdS2 is σ=f(U)+g(V) with arbitrary f,g.
    Used in Eq. (20)/(31); standard d'Alembert solution in 2D null coordinates.
  • domain assumption The singularity locus in JT is 1+Φ=0.
    Cited to [9]; used to determine singularity-free horizon in Section IV and Fig. 5.
  • ad hoc to paper The Poincare horizon U=π/2 can be identified with the DERN event horizon H+ and the AdS2 right boundary B glued to the asymptotically flat ERN exterior.
    Assumed throughout Section IV and Fig. 4; no global 4D matching constructed.
  • ad hoc to paper Boundary conditions (27) and (41) characterize a DERN: scalar Aretakis behavior and dilaton on the ERN threshold b^2-4ac=0.
    Imposed by hand in Section IV based on the desired DERN features; the paper explicitly says it is not deriving them from ERN.
  • domain assumption The anabasis classification hθθ=ar+brt+cr(t^2-1/r^2) with µ=b^2-4ac determines sub-/extreme/super-ERN backreaction.
    Taken from [36] and used to impose (41).

pith-pipeline@v1.3.0-alltime-deepseek · 12978 in / 13823 out tokens · 117929 ms · 2026-08-03T13:34:27.746975+00:00 · methodology

0 comments
read the original abstract

We give an explicit closed form description of the late-time near-horizon approach to dynamical extreme Reissner-Nordstrom (DERN) black holes. These are spherically symmetric dynamical solutions of Einstein-Maxwell theory coupled to a neutral scalar that feature: (i) a spacetime metric which tends to that of a static extreme Reissner-Nordstrom (RN), and (ii) a scalar field which exhibits the linear Aretakis instability ad infinitum in the non-linear theory. We employ the two-dimensional Jackiw-Teitelboim (JT) gravity to solve explicitly for the non-linear s-wave dynamics of the four-dimensional theory near an ${\rm AdS}_2\times {\rm S}^2$ throat. For a teleologically defined black hole horizon, we impose boundary conditions on JT's dilaton field (which encodes the gravitational dynamics) and the scalar matter as follows: (i) the JT dilaton decays at late times on the ${\rm AdS}_2$ boundary to a value that corresponds to a static extreme RN in the exterior, and (ii) the scalar obeys boundary conditions characteristic of linear Aretakis behavior on ${\rm AdS}_2$. We ensure our DERN solutions are singularity-free and we note that our approach to DERN is accompanied by a final burst of outgoing scalar matter flux leaking out of the ${\rm AdS}_2$ throat. The boundary conditions we impose on the JT dilaton place its late-time boundary profile on the threshold of black hole formation with sub-extreme and super-extreme RN on either side of our DERNs.

Figures

Figures reproduced from arXiv: 2512.23629 by Achilleas P. Porfyriadis, Christopher Rosen, Georgios Tsaraktsidis.

Figure 1
Figure 1. Figure 1: FIG. 1. Penrose diagrams showing: (a) DERN arising from maximal development of characteristic [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Penrose diagrams showing the emergence of AdS [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Penrose diagram of AdS [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Penrose diagram of AdS [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of the singularity locus 1 + Φ = 0 on a global AdS [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

discussion (0)

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

Forward citations

Cited by 2 Pith papers

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

  1. A Nonlinear Endpoint of Charged Horizon Instabilities

    gr-qc 2026-02 conditional novelty 7.0

    Charged scalar collapse can form dynamical extremal black holes whose horizons show divergent energy density and constant charge hair, with universal near-threshold scaling.

  2. Taming the Aretakis instability: extremal black holes with multi-degenerate horizons

    gr-qc 2026-04 unverdicted novelty 6.0

    Black holes with infinitely degenerate horizons are proposed to be stable against Aretakis instability, potentially serving as end states.

Reference graph

Works this paper leans on

37 extracted references · 31 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Observational Constraints on Black Hole Spin,

    C. S. Reynolds, “Observational Constraints on Black Hole Spin,”Ann. Rev. Astron. Astrophys.59(2021) 117–154,arXiv:2011.08948 [astro-ph.HE]

    Pith/arXiv arXiv 2021

  2. [2]

    GW231123: A Binary Black Hole Merger with Total Mass 190-265 M ⊙,

    LIGO Scientific, VIRGO, KAGRA Collaboration, A. G. Abacet al., “GW231123: A Binary Black Hole Merger with Total Mass 190-265 M ⊙,”Astrophys. J. Lett.993no. 1, (2025) L25, arXiv:2507.08219 [astro-ph.HE]. 18 0123 -3 -2 -1 0 1 2 3 (a)H= 1 0123 -3 -2 -1 0 1 2 3 (b)H= 0 FIG. 5. Plots of the singularity locus 1 + Φ = 0 on a global AdS 2 strip showing the Poinca...

    Pith/arXiv arXiv 2025

  3. [3]

    Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity,

    T. G. Mertens and G. J. Turiaci, “Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity,”Living Rev. Rel.26no. 1, (2023) 4,arXiv:2210.10846 [hep-th]

    Pith/arXiv arXiv 2023

  4. [4]

    The statistical mechanics of near-extremal black holes,

    L. V. Iliesiu and G. J. Turiaci, “The statistical mechanics of near-extremal black holes,” JHEP05(2021) 145,arXiv:2003.02860 [hep-th]. 19

    Pith/arXiv arXiv 2021

  5. [5]

    Revisiting the logarithmic corrections to the black hole entropy,

    L. V. Iliesiu, S. Murthy, and G. J. Turiaci, “Revisiting the logarithmic corrections to the black hole entropy,”JHEP07(2025) 058,arXiv:2209.13608 [hep-th]

    Pith/arXiv arXiv 2025

  6. [6]

    The evaporation of charged black holes,

    A. R. Brown, L. V. Iliesiu, G. Penington, and M. Usatyuk, “The evaporation of charged black holes,”arXiv:2411.03447 [hep-th]

    Pith/arXiv arXiv

  7. [7]

    Lower Dimensional Gravity,

    R. Jackiw, “Lower Dimensional Gravity,”Nucl. Phys. B252(1985) 343–356

    1985

  8. [8]

    Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,

    C. Teitelboim, “Gravitation and Hamiltonian Structure in Two Space-Time Dimensions,” Phys. Lett. B126(1983) 41–45

    1983

  9. [9]

    Models of AdS 2 backreaction and holography,

    A. Almheiri and J. Polchinski, “Models of AdS 2 backreaction and holography,”JHEP11 (2015) 014,arXiv:1402.6334 [hep-th]

    Pith/arXiv arXiv 2015

  10. [10]

    Conformal Symmetry Breaking and Thermodynamics of Near-Extremal Black Holes,

    A. Almheiri and B. Kang, “Conformal Symmetry Breaking and Thermodynamics of Near-Extremal Black Holes,”JHEP10(2016) 052,arXiv:1606.04108 [hep-th]

    Pith/arXiv arXiv 2016

  11. [11]

    Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,

    J. Maldacena, D. Stanford, and Z. Yang, “Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space,”PTEP2016no. 12, (2016) 12C104, arXiv:1606.01857 [hep-th]

    Pith/arXiv arXiv 2016

  12. [12]

    On the Dynamics of Near-Extremal Black Holes,

    P. Nayak, A. Shukla, R. M. Soni, S. P. Trivedi, and V. Vishal, “On the Dynamics of Near-Extremal Black Holes,”JHEP09(2018) 048,arXiv:1802.09547 [hep-th]

    Pith/arXiv arXiv 2018

  13. [13]

    The Effects of Near-AdS 2 Backreaction on Matter Fields,

    A. Castro, J. Hollander, P. J. Martinez, and E. Verheijden, “The Effects of Near-AdS 2 Backreaction on Matter Fields,”arXiv:2509.08046 [hep-th]

    Pith/arXiv arXiv

  14. [14]

    Gravitational perturbations from NHEK to Kerr,

    A. Castro, V. Godet, J. Sim´ on, W. Song, and B. Yu, “Gravitational perturbations from NHEK to Kerr,”JHEP07(2021) 218,arXiv:2102.08060 [hep-th]

    Pith/arXiv arXiv 2021

  15. [15]

    The stability problem for extremal black holes,

    M. Dafermos, “The stability problem for extremal black holes,”Gen. Rel. Grav.57no. 3, (2025) 60

    2025

  16. [16]

    Gravitational collapse to extremal black holes and the third law of black hole thermodynamics,

    C. Kehle and R. Unger, “Gravitational collapse to extremal black holes and the third law of black hole thermodynamics,”arXiv:2211.15742 [gr-qc]

    Pith/arXiv arXiv

  17. [17]

    Stability and Instability of Extreme Reissner-Nordstr¨ om Black Hole Spacetimes for Linear Scalar Perturbations I,

    S. Aretakis, “Stability and Instability of Extreme Reissner-Nordstr¨ om Black Hole Spacetimes for Linear Scalar Perturbations I,”Commun. Math. Phys.307(2011) 17–63, arXiv:1110.2007 [gr-qc]

    Pith/arXiv arXiv 2011

  18. [18]

    Stability and Instability of Extreme Reissner-Nordstr¨ om Black Hole Spacetimes for Linear Scalar Perturbations II,

    S. Aretakis, “Stability and Instability of Extreme Reissner-Nordstr¨ om Black Hole Spacetimes for Linear Scalar Perturbations II,”Annales Henri Poincare12(2011) 1491–1538,arXiv:1110.2009 [gr-qc]. 20

    Pith/arXiv arXiv 2011

  19. [19]

    What happens at the horizon(s) of an extreme black hole?,

    K. Murata, H. S. Reall, and N. Tanahashi, “What happens at the horizon(s) of an extreme black hole?,”Class. Quant. Grav.30(2013) 235007,arXiv:1307.6800 [gr-qc]

    Pith/arXiv arXiv 2013

  20. [20]

    Frozen Firewall: Generic Singularity Formation on an Extremal Horizon,

    G. T. Horowitz, M. Kolanowski, and J. E. Santos, “Frozen Firewall: Generic Singularity Formation on an Extremal Horizon,”arXiv:2510.18947 [hep-th]

    arXiv

  21. [21]

    Nonlinear stability of extremal Reissner-Nordstr¨ om black holes in spherical symmetry,

    Y. Angelopoulos, C. Kehle, and R. Unger, “Nonlinear stability of extremal Reissner-Nordstr¨ om black holes in spherical symmetry,”arXiv:2410.16234 [gr-qc]

    arXiv

  22. [22]

    Universality and scaling in gravitational collapse of a massless scalar field,

    M. W. Choptuik, “Universality and scaling in gravitational collapse of a massless scalar field,”Phys. Rev. Lett.70(1993) 9–12

    1993

  23. [23]

    On the horizon instability of an extreme Reissner-Nordstr¨ om black hole,

    J. Lucietti, K. Murata, H. S. Reall, and N. Tanahashi, “On the horizon instability of an extreme Reissner-Nordstr¨ om black hole,”JHEP03(2013) 035,arXiv:1212.2557 [gr-qc]

    Pith/arXiv arXiv 2013

  24. [24]

    A note on instabilities of extremal black holes under scalar perturbations from afar,

    S. Aretakis, “A note on instabilities of extremal black holes under scalar perturbations from afar,”Class. Quant. Grav.30(2013) 095010,arXiv:1212.1103 [gr-qc]

    Pith/arXiv arXiv 2013

  25. [25]

    Late-time asymptotics for the wave equation on extremal Reissner–Nordstr¨ om backgrounds,

    Y. Angelopoulos, S. Aretakis, and D. Gajic, “Late-time asymptotics for the wave equation on extremal Reissner–Nordstr¨ om backgrounds,”Adv. Math.375(2020) 107363, arXiv:1807.03802 [gr-qc]

    Pith/arXiv arXiv 2020

  26. [26]

    Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordstr¨ om Spacetime,

    M. A. Apetroaie, “Instability of Gravitational and Electromagnetic Perturbations of Extremal Reissner–Nordstr¨ om Spacetime,”Ann. PDE9no. 2, (2023) 22, arXiv:2211.09182 [gr-qc]

    Pith/arXiv arXiv 2023

  27. [27]

    Gravitational instability of an extreme Kerr black hole,

    J. Lucietti and H. S. Reall, “Gravitational instability of an extreme Kerr black hole,”Phys. Rev. D86(2012) 104030,arXiv:1208.1437 [gr-qc]

    Pith/arXiv arXiv 2012

  28. [28]

    Instability of higher dimensional extreme black holes,

    K. Murata, “Instability of higher dimensional extreme black holes,”Class. Quant. Grav.30 (2013) 075002,arXiv:1211.6903 [gr-qc]

    Pith/arXiv arXiv 2013

  29. [29]

    Horizon Instability of Extremal Black Holes,

    S. Aretakis, “Horizon Instability of Extremal Black Holes,”Adv. Theor. Math. Phys.19 (2015) 507–530,arXiv:1206.6598 [gr-qc]

    Pith/arXiv arXiv 2015

  30. [30]

    Horizon Instability of Extremal Kerr Black Holes: Nonaxisymmetric Modes and Enhanced Growth Rate,

    M. Casals, S. E. Gralla, and P. Zimmerman, “Horizon Instability of Extremal Kerr Black Holes: Nonaxisymmetric Modes and Enhanced Growth Rate,”Phys. Rev. D94no. 6, (2016) 064003,arXiv:1606.08505 [gr-qc]

    Pith/arXiv arXiv 2016

  31. [31]

    Azimuthal instabilities on extremal Kerr,

    D. Gajic, “Azimuthal instabilities on extremal Kerr,”arXiv:2302.06636 [gr-qc]

    Pith/arXiv arXiv

  32. [32]

    Horizon instability of extremal Reissner-Nordstr¨ om black holes to charged perturbations,

    P. Zimmerman, “Horizon instability of extremal Reissner-Nordstr¨ om black holes to charged perturbations,”Phys. Rev. D95no. 12, (2017) 124032,arXiv:1612.03172 [gr-qc]. 21

    Pith/arXiv arXiv 2017

  33. [33]

    Accumulation of charge on an extremal black hole’s event horizon,

    Z. Gelles and F. Pretorius, “Accumulation of charge on an extremal black hole’s event horizon,”Phys. Rev. D112no. 6, (2025) 064003,arXiv:2503.04881 [gr-qc]

    Pith/arXiv arXiv 2025

  34. [34]

    Near-extremal black holes at late times, backreacted,

    S. Hadar, “Near-extremal black holes at late times, backreacted,”JHEP01(2019) 214, arXiv:1811.01022 [hep-th]

    Pith/arXiv arXiv 2019

  35. [35]

    Extremal black hole formation as a critical phenomenon,

    C. Kehle and R. Unger, “Extremal black hole formation as a critical phenomenon,” arXiv:2402.10190 [gr-qc]

    Pith/arXiv arXiv

  36. [36]

    Extreme Black Hole Anabasis,

    S. Hadar, A. Lupsasca, and A. P. Porfyriadis, “Extreme Black Hole Anabasis,”JHEP03 (2021) 223,arXiv:2012.06562 [hep-th]

    Pith/arXiv arXiv 2021

  37. [37]

    Critical Exponents of Extremal Kerr Perturbations,

    S. E. Gralla and P. Zimmerman, “Critical Exponents of Extremal Kerr Perturbations,” Class. Quant. Grav.35no. 9, (2018) 095002,arXiv:1711.00855 [gr-qc]. 22

    Pith/arXiv arXiv 2018