Pith. sign in

REVIEW 4 major objections 4 minor 37 references

Strong scalar-field collapse can destroy the Hayward black hole's regular core, driving the inner horizon to zero with a universal power-law exponent near 0.5.

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 19:35 UTC pith:7HOZI57C

load-bearing objection Plausible and useful numerical study, but the central scaling law needs a coordinate-invariant horizon extraction and convergence tests before it can be trusted. the 4 major comments →

arxiv 2511.23165 v2 pith:7HOZI57C submitted 2025-11-28 gr-qc

Internal structure of Hayward black holes

Caiying Shao , Jun-Qi Guo , Yu Tian , Hongbao Zhang This is my paper
classification gr-qc PACS 04.70.-s04.25.D-
keywords Hayward black holeinner horizonscalar field collapsecritical phenomenaspacelike singularitymass inflationregular black holespower-law scaling
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 evolves a massless scalar field collapsing in a Hayward regular black hole, including full backreaction, and argues that the outcome depends sharply on how strong the initial wave packet is. Weak pulses leave the inner horizon at a finite, slightly contracted radius but drive mass inflation there. Strong pulses shrink the inner horizon to zero, make the metric functions and curvature scalars diverge, and convert the geometry to a Schwarzschild-like one with a spacelike singularity. Near the threshold between these behaviors, the final inner-horizon radius obeys r_- ∝ |p-p*|^γ with γ ≈ 0.5 for all three initial-profile parameters (amplitude, width, position). If correct, the Hayward regular black hole is not dynamically singularity-free, and black-hole interiors exhibit a critical phenomenon with a universal exponent.

Core claim

The central claim is that the Hayward inner horizon is not stable against strong scalar-field collapse. With weak scalar data, the inner horizon settles to a finite radius while the Misner–Sharp mass inflates there; with strong data, the inner horizon contracts to r = 0, σ and φ diverge, the Kretschmann scalar diverges at the core, and the near-singularity field equations reduce to the same asymptotic form as collapse to a Schwarzschild black hole, indicating a spacelike singularity. Approaching the critical amplitude, width, or position of the initial Gaussian packet, the final inner-horizon radius vanishes as a power law with exponent 0.5. The same scaling is reported to persist for Reissn

What carries the argument

The argument runs on a double-null (Kruskal-like) coordinate setup, ds² = 4e^{-2σ} du dv + r² dΩ², in which the Hayward metric is nonsingular and the collapse is evolved as coupled nonlinear PDEs for r, σ, and the scalar field φ, with a Gaussian initial profile parameterized by amplitude A, width B, and center x0. The central diagnostic is the final inner-horizon radius r_- extracted from the evolved r(t,x). The near-critical scaling r_- ∝ |p-p*|^γ is established by numerical fitting over three separate parameter directions, and the spacelike nature of the endpoint is inferred from the asymptotic dominance of the same terms that appear in neutral scalar collapse to a Schwarzschild black hole

Load-bearing premise

Everything rests on the numerical evolution being a faithful solution of the field equations: no convergence or resolution study is given, and the reported r_- is read from a fixed spatial slice, so if discretization error or slice dependence changes the near-threshold values, the scaling law and the spacelike-singularity conclusion are unsupported.

What would settle it

Repeat the strong-field runs with twice and four times the grid resolution and with smaller time steps; if the extracted r_- at fixed p shifts by more than the fitting uncertainty, or if choosing a different spatial slice than x=1.2 changes the fitted γ away from 0.5, the central scaling claim is numerical artifact rather than spacetime physics. Also, if a non-zero r_- is found for any p>p* under better resolution, the claimed threshold is not genuine.

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

If this is right

  • If the critical scaling is real, the Hayward spacetime has a threshold: below it the inner horizon is finite, above it the core collapses to a curvature singularity.
  • The exponent γ ≈ 0.5 being independent of whether one varies amplitude, width, or position points to a true critical phenomenon in the black-hole interior, not an artifact of one family of data.
  • Strong scalar collapse produces a spacelike singularity and Schwarzschild-like geometry, so regular black holes of Hayward type do not by themselves resolve the singularity problem in dynamical collapse.
  • Mass inflation at the inner horizon persists even for weak perturbations, meaning the Cauchy horizon remains a locus of instability.
  • Because the same scaling appears for Reissner–Nordström interiors, the result likely extends to charged black holes and may unify how Cauchy horizons respond to infalling matter.

Where Pith is reading between the lines

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

  • A testable extension is to derive the 1/2 exponent analytically from the near-singularity equations; if the exponent follows solely from those asymptotic equations, it should hold for any regular black hole whose interior approaches the same form.
  • Treating the inner-horizon radius as an order parameter, the collapse of a regular black hole to a singular state resembles a continuous phase transition, and the critical exponent could be measured in independent numerical relativity codes as a consistency check.
  • Observationally, if astrophysical black-hole candidates were Hayward-like, strong infall would destroy the regular core, so the absence of such instability in observed systems would constrain the Hayward scale parameter or the existence of regular cores.
  • A sharper test would run the same collapse with non-Gaussian or asymmetric initial data: if γ changes, the scaling is tied to the profile family, whereas if it stays 0.5 it is universal.

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. The paper studies nonlinear collapse of a massless scalar field on a Hayward regular black hole background using a double-null (t,x) numerical evolution scheme. The authors present equations of motion for r, σ, and ϕ derived from an Einstein-nonlinear-electrodynamics action with a scalar field, and report three main results: (i) weak scalar-field perturbations leave the inner horizon at a finite, though contracted, radius; (ii) sufficiently strong scalar fields drive r to zero at a fixed spatial slice, with divergences in σ, ϕ, the Misner–Sharp mass, and the Kretschmann scalar, which are interpreted as formation of a spacelike singularity; and (iii) near a critical threshold p* for the initial Gaussian profile parameters A, B, and x0, the final inner-horizon radius satisfies r− ∝ |p − p*|^γ with γ ≈ 0.5. The central quantitative claim is Eq. (28) and Fig. 7.

Significance. If established, the result would be significant: it would show that a canonical regular black hole model is not dynamically singularity-free under scalar-field collapse, and it would identify a new critical-scaling law for the black-hole interior. The paper also usefully extends the double-null formalism of Ref. [33] to a regular black hole background, and the derived equations of motion (15)–(17) appear plausible and internally consistent. However, the three headline claims rest entirely on a numerical simulation for which no convergence tests, resolution details, or gauge-invariant horizon extraction are presented. The inner-horizon radius is read off from a fixed-x slice rather than from a geometric trapping-horizon condition, and the claimed spacelike character of the singularity is inferred only by analogy with previous work. These omissions leave the main quantitative conclusion unsupported pending additional validation.

major comments (4)
  1. [§III.B and Fig. 4; Eq. (23)] The inner-horizon radius r− used in the scaling law is never defined geometrically. In §III.B and the Fig. 4 caption the 'inner horizon' is identified with r on the fixed spatial slice x = 1.2, and Fig. 7/Eq. (28) evidently use the same extraction rule. For a spherically symmetric dynamical spacetime the inner horizon is a trapping horizon where the null expansions vanish, equivalently g^{ab} r_{,a} r_{,b} = e^{2σ}(−r_t^2 + r_x^2) = 0; a fixed-x worldline is not such a surface. Since the (t,x) gauge is fixed by the arbitrary initial condition (21), the value r(t, x = const) has no invariant meaning. A coordinate/stopping-time artifact could mimick power-law scaling, especially near criticality where the approach to the final state slows. Please provide an invariant extraction of r−, or at minimum demonstrate slice-independence of the values used in Fig. 7.
  2. [§II and §III.C; Fig. 7] No numerical convergence study is reported. The leapfrog scheme, the extrapolated boundaries, the domain [−5,5], and the time-step initialization by Taylor expansion are only referenced to [33]; no grid resolution, time-step size, domain-size variation, or convergence-factor test is given. Near a critical threshold the relaxation time diverges, so without a resolution and domain study the values of r− used in Eq. (28) may be contaminated by discretization and finite-domain effects. The critical values A*, B*, x0* are quoted to 11–12 digits, yet no error bars, fit residuals, or even the numerical values of r− entering the fit are reported. A resolution/convergence study and an uncertainty estimate for γ are required to support the central claim.
  3. [§III.B and Fig. 5–6] The claim that a spacelike singularity forms is not demonstrated. What is shown is that along the x = 1.2 slice r approaches zero and terms in the equations of motion (15)–(17) simplify to (25)–(27), which are said to resemble the collapse studied in Refs. [33–36]. Striking asymptotic similarity on one coordinate slice is not sufficient to establish the causal character of the resulting singularity. Please provide a geometric diagnosis: e.g., behavior of null geodesics, the location of apparent horizons, or an argument based on the full spacetime structure, rather than an analogy to previous solutions.
  4. [Eq. (28) and §III.C] The universality of the scaling law is not testable from the data presented. The exponents are quoted as 0.49997, 0.500521, and 0.499601 for A, B, and x0 respectively, but no fitting procedure is described, no uncertainties are given, and no tabulated r− versus p data are shown. Moreover, the text states that for p > p* the inner horizon 'vanishes' while for p < p* it is finite, but Fig. 7 plots |p − p*| without indicating whether the two sides behave symmetrically or whether only one side is used. Please provide the fitting details, the data, and a discussion of systematic uncertainties before claiming a clean γ ≈ 1/2 law.
minor comments (4)
  1. [§III.A, Fig. 2 caption] The caption says the evolution is along the x = 0.5 slice, while §III.B and Fig. 4 use x = 1.2. The choice of slice is not justified; if the purpose is to probe the inner horizon, an invariant justification would help.
  2. [§III.A, §III.B] Figures 1 and 3 would benefit from a clearer statement of which panels correspond to which quantity and what the color gradient means; the repeated list of time labels in the captions is unwieldy. Also, Fig. 6 does not label which curves correspond to which term in Eqs. (15)–(17), making the 'negligible' assertion hard to verify.
  3. [Eq. (10)–(13)] The notation is occasionally ambiguous: the parameter s in Eq. (10) is not the same as the action in Eq. (1), and ρ0, ρ1, ρ2 are defined with a semicolon-like notation. Please use distinct symbols for the action and the Hayward parameter, and define all symbols at first use.
  4. [General] The paper cites Refs. [28,29] only in passing as 'the question remains open'; a more substantive discussion of the stability debate and how this work contributes to it would help position the result. Also, the companion paper [37] is mentioned only in the last sentence; if its results are used, they should be stated explicitly in the main text.

Circularity Check

0 steps flagged

No significant circularity: the scaling law (28) is a numerically measured relation, not a consequence of an input that contains it.

full rationale

The central claim, Eq. (28), is obtained by evolving the coupled Einstein-scalar system (15)-(17) from independent Gaussian initial data (20) and then fitting the late-time inner-horizon radius r_- as a function of the initial-profile parameter p. Neither p nor p_* is defined in terms of r_-, and no fitted parameter is renamed as a prediction; the scaling exponent gamma~0.5 is an output of the fitting, not an input. The self-citations are to a numerical implementation [33], a related spherical-collapse study [34], and a companion paper [37] reporting the analogous Reissner-Nordström result; none of them supplies the Hayward scaling law, and the 'spacelike singularity' inference is supported also by independent references [35,36]. The paper's own caveats (extrapolated boundaries, reference to [33] for numerical details, extraction along a fixed slice x=1.2) are legitimate numerical-validation concerns, but they concern whether the measurement is correct and robust, not whether the conclusion is assumed in the premises. No equation in the paper reduces to another by construction, so there is no significant circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The model has few invented entities, but the numerical claims rest on unverified discretization and slicing assumptions. The main free inputs are the chosen Hayward parameters (q=0.5, r+=3) and the critical values of the initial-data parameters, which are fitted thresholds. No new particles or forces are introduced.

free parameters (5)
  • q (magnetic charge) = 0.5
    Hayward background charge; chosen by hand; universality in q not tested.
  • r+ (outer horizon radius) = 3
    Hayward background event-horizon radius; chosen by hand; scaling may depend on r+.
  • A* (critical amplitude) = 0.078606364025
    Critical Gaussian amplitude for initial scalar-field profile (B=1, x0=1); located by tuning until r_- transitions from finite to zero; used as reference point for scaling fit.
  • B* (critical width) = 0.978312460101
    Critical width parameter with A=0.048, x0=1; fitted threshold for scaling fit.
  • x0* (critical position) = 1.008831954973
    Critical central position of the Gaussian with A=0.048, B=1; fitted threshold for scaling fit.
axioms (5)
  • domain assumption Hayward metric (Eq. 10) is a valid regular black hole solution sourced by the nonlinear electrodynamics Lagrangian L(1) from [7].
    The initial background spacetime and the electromagnetic source are taken from Hayward's regular black hole model; the code starts from this metric.
  • domain assumption Massless scalar field minimally coupled to gravity, obeying the wave equation, is a valid matter model for the perturbation.
    The paper postulates this matter content to probe the geometry; no physical derivation of why this scalar field represents a generic perturbation.
  • domain assumption Spherical symmetry and the double-null / Kruskal-like coordinate ansatz cover the spacetime including the interior and near-singularity region.
    The numerical evolution relies on this coordinate system; possible coordinate singularities at the inner horizon are not discussed.
  • ad hoc to paper The leapfrog finite-difference scheme with extrapolated boundary conditions converges to the continuum solution on the domain [-5,5].
    No convergence tests, grid resolution, or boundary-independence checks are reported; the code is not released.
  • ad hoc to paper The final-state inner horizon radius r_- is well-defined and can be extracted from a fixed spatial slice independently of the choice of slice.
    The paper does not describe the extraction algorithm; fixed-x slices may not track a coordinate-independent horizon in a dynamical spacetime.

pith-pipeline@v1.3.0-alltime-deepseek · 10337 in / 21560 out tokens · 197534 ms · 2026-08-03T19:35:56.622797+00:00 · methodology

0 comments
read the original abstract

Regular black holes, free of central singularities, provide an ideal laboratory for probing the geometric structure of spacetime. The global structure of some regular black holes, e.g. Hayward black hole, features an event horizon and a Cauchy horizon, raising fundamental questions about the latter's stability. In this work, we investigate collapse of a scalar field in Hayward spacetime. Under weak scalar perturbations, the inner horizon maintains a stable finite radius. In the circumstance of a strong scalar field, the inner horizon shrinks to zero volume, accompanied by the formation of a spacelike singularity. The Hayward geometry is effectively converted into a Schwarzschild-like geometry. Furthermore, the strength of the scalar field governs the contraction dynamics of the inner horizon. As the parameter $p$ of the initial profile for the scalar field approaches the critical threshold ${p_*}$, the radius of the inner horizon ${r_{-}}$ exhibits a universal scaling behavior: ${r_{-}}\propto{|p - {p_*}|^\gamma}$, with a critical exponent $\gamma\approx 0.5$.

Figures

Figures reproduced from arXiv: 2511.23165 by Caiying Shao, Hongbao Zhang, Jun-Qi Guo, Yu Tian.

Figure 1
Figure 1. Figure 1: FIG. 1. The evolutions for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dynamical evolution of black hole characteristics a [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The evolutions for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dynamical evolution of black hole characteristics a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: illustrates the behavior of the Kretschmann scalar K as r approaches zero in both the original Hayward geometry and the final state. In the initial case, K remains finite at the origin, reflecting its nonsingular central core. In contrast, K diverges at the origin in our model, signaling the formation of a genuine curvature singularity. To probe the spacetime dynamics near the singularity, we systematicall… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Contributions of individual terms in the dynamical e [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The radius of the inner horizon vs. the parameter [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

discussion (0)

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

Reference graph

Works this paper leans on

37 extracted references · 30 linked inside Pith

  1. [1]

    B. P. Abbott et al. (LIGO Scientific, Virgo), Observation of Gravitational Wav es from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    Pith/arXiv arXiv 2016

  2. [2]

    Akiyama et al

    K. Akiyama et al. (Event Horizon Telescope), First M87 Event Horizon Telesco pe Results. I. The Shadow of the Super- massive Black Hole, Astrophys. J. Lett. 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    Pith/arXiv arXiv 2019

  3. [3]

    Murk, Nomen non est omen: Why it is too soon to identify u ltra-compact objects as black holes, Int

    S. Murk, Nomen non est omen: Why it is too soon to identify u ltra-compact objects as black holes, Int. J. Mod. Phys. D 32, 2342012 (2023), arXiv:2210.03750 [gr-qc]

    Pith/arXiv arXiv 2023

  4. [4]

    Rovelli and L

    C. Rovelli and L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys. B 442, 593 (1995), [Erratum: Nucl.Phys.B 456, 753–754 (1995)], arXiv:gr-qc/9411005

    Pith/arXiv arXiv 1995

  5. [5]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Regular black hole in genera l relativity coupled to nonlinear electrodynamics, Phys. R ev. Lett. 80, 5056 (1998), arXiv:gr-qc/9911046

    Pith/arXiv arXiv 1998

  6. [6]

    Borde, Open and closed universes, initial singularit ies and inflation, Phys

    A. Borde, Open and closed universes, initial singularit ies and inflation, Phys. Rev. D 50, 3692 (1994), arXiv:gr-qc/9403049

    Pith/arXiv arXiv 1994

  7. [7]

    S. A. Hayward, Formation and evaporation of regular blac k holes, Phys. Rev. Lett. 96, 031103 (2006), arXiv:gr-qc/0506126

    Pith/arXiv arXiv 2006

  8. [8]

    Balart and E

    L. Balart and E. C. Vagenas, Regular black holes with a non linear electrodynamics source, Phys. Rev. D 90, 124045 (2014), arXiv:1408.0306 [gr-qc]

    Pith/arXiv arXiv 2014

  9. [9]

    Culetu, On a regular charged black hole with a nonlinea r electric source, Int

    H. Culetu, On a regular charged black hole with a nonlinea r electric source, Int. J. Theor. Phys. 54, 2855 (2015), arXiv:1408.3334 [gr-qc]

    Pith/arXiv arXiv 2015

  10. [10]

    Fan and X

    Z.-Y. Fan and X. Wang, Construction of Regular Black Hol es in General Relativity, Phys. Rev. D 94, 124027 (2016), arXiv:1610.02636 [gr-qc]

    Pith/arXiv arXiv 2016

  11. [11]

    Construction of regular black holes in general relativity

    B. Toshmatov, Z. Stuchl ´ ık, and B. Ahmedov, Comment on “ Construction of regular black holes in general relativity” , Phys. Rev. D 98, 028501 (2018), arXiv:1807.09502 [gr-qc]

    Pith/arXiv arXiv 2018

  12. [12]

    Bardeen, Non-singular general relativistic gravit ational collapse, in Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity (1968) p

    J. Bardeen, Non-singular general relativistic gravit ational collapse, in Proceedings of the 5th International Conference on Gravitation and the Theory of Relativity (1968) p. 87

    1968

  13. [13]

    Uchikata, S

    N. Uchikata, S. Yoshida, and T. Futamase, New solutions of charged regular black holes and their stability, Phys. Re v. D 86, 084025 (2012), arXiv:1209.3567 [gr-qc]

    Pith/arXiv arXiv 2012

  14. [14]

    S. W. Hawking, Breakdown of Predictability in Gravitat ional Collapse, Phys. Rev. D 14, 2460 (1976)

    1976

  15. [15]

    Bonanno and M

    A. Bonanno and M. Reuter, Renormalization group improv ed black hole space-times, Phys. Rev. D 62, 043008 (2000), arXiv:hep-th/0002196

    Pith/arXiv arXiv 2000

  16. [16]

    Platania, Dynamical renormalization of black-hole spacetimes, Eur

    A. Platania, Dynamical renormalization of black-hole spacetimes, Eur. Phys. J. C 79, 470 (2019), arXiv:1903.10411 [gr-qc]

    Pith/arXiv arXiv 2019

  17. [17]

    Fan, Critical phenomena of regular black holes in anti-de Sitter space-time, Eur

    Z.-Y. Fan, Critical phenomena of regular black holes in anti-de Sitter space-time, Eur. Phys. J. C 77, 266 (2017), arXiv:1609.04489 [hep-th]

    Pith/arXiv arXiv 2017

  18. [18]

    Guo and Y.-G

    Y. Guo and Y.-G. Miao, Weinhold geometry and thermodyna mics of Bardeen AdS black holes, Nucl. Phys. B 980, 115839 (2022), arXiv:2107.01866 [gr-qc]

    Pith/arXiv arXiv 2022

  19. [19]

    K. A. Bronnikov, R. A. Konoplya, and A. Zhidenko, Instab ilities of wormholes and regular black holes supported by a phantom scalar field, Phys. Rev. D 86, 024028 (2012), arXiv:1205.2224 [gr-qc]

    Pith/arXiv arXiv 2012

  20. [20]

    J. Li, M. Hong, and K. Lin, Dirac quasinormal modes in sph erically symmetric regular black holes, Phys. Rev. D 88, 064001 (2013), arXiv:1308.6499 [gr-qc]

    Pith/arXiv arXiv 2013

  21. [21]

    Fernando and J

    S. Fernando and J. Correa, Quasinormal Modes of Bardeen Black Hole: Scalar Perturbations, Phys. Rev. D 86, 064039 (2012), arXiv:1208.5442 [gr-qc]

    Pith/arXiv arXiv 2012

  22. [22]

    Yue and Y.-Q

    Y. Yue and Y.-Q. Wang, Frozen Hayward-boson stars, JCAP 05, 066, arXiv:2312.07224 [gr-qc]

    Pith/arXiv arXiv

  23. [23]

    Li and C

    Z. Li and C. Bambi, Measuring the Kerr spin parameter of r egular black holes from their shadow, JCAP 01, 041, arXiv:1309.1606 [gr-qc]

    Pith/arXiv arXiv

  24. [24]

    Abdujabbarov, M

    A. Abdujabbarov, M. Amir, B. Ahmedov, and S. G. Ghosh, Sh adow of rotating regular black holes, Phys. Rev. D 93, 104004 (2016), arXiv:1604.03809 [gr-qc]

    Pith/arXiv arXiv 2016

  25. [25]

    Tsukamoto, Black hole shadow in an asymptotically-fl at, stationary, and axisymmetric spacetime: The Kerr-Newm an and rotating regular black holes, Phys

    N. Tsukamoto, Black hole shadow in an asymptotically-fl at, stationary, and axisymmetric spacetime: The Kerr-Newm an and rotating regular black holes, Phys. Rev. D 97, 064021 (2018), arXiv:1708.07427 [gr-qc]

    Pith/arXiv arXiv 2018

  26. [26]

    C. Lan, H. Yang, Y. Guo, and Y.-G. Miao, Regular Black Hol es: A Short Topic Review, Int. J. Theor. Phys. 62, 202 (2023), arXiv:2303.11696 [gr-qc]. 10

    Pith/arXiv arXiv 2023

  27. [27]

    Penrose, Structure of space-time, in Battelle Rencontres (1968) pp

    R. Penrose, Structure of space-time, in Battelle Rencontres (1968) pp. 121–235

    1968

  28. [28]

    Regular evaporating black hol es with stable cores

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacil io, and M. Visser, Comment on “Regular evaporating black hol es with stable cores”, Phys. Rev. D 108, 128501 (2023), arXiv:2212.07458 [gr-qc]

    Pith/arXiv arXiv 2023

  29. [29]

    Comment on ‘Regular evaporating black holes with stable cores’

    A. Bonanno, A.-P. Khosravi, and F. Saueressig, Reply to “Comment on ‘Regular evaporating black holes with stable cores’ ”, Phys. Rev. D 108, 128502 (2023), arXiv:2402.07416 [gr-qc]

    Pith/arXiv arXiv 2023

  30. [30]

    Ori, Inner structure of a charged black hole: An exact mass-inflation solution, Phys

    A. Ori, Inner structure of a charged black hole: An exact mass-inflation solution, Phys. Rev. Lett. 67, 789 (1991)

    1991

  31. [31]

    Bonanno, A.-P

    A. Bonanno, A.-P. Khosravi, and F. Saueressig, Regular black holes with stable cores, Phys. Rev. D 103, 124027 (2021), arXiv:2010.04226 [gr-qc]

    Pith/arXiv arXiv 2021

  32. [32]

    Carballo-Rubio, F

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, C. Pacil io, and M. Visser, Inner horizon instability and the unstabl e cores of regular black holes, JHEP 05, 132, arXiv:2101.05006 [gr-qc]

    Pith/arXiv arXiv

  33. [33]

    Guo and P

    J.-Q. Guo and P. S. Joshi, Interior dynamics of neutral a nd charged black holes, Phys. Rev. D 92, 064013 (2015), arXiv:1507.01806 [gr-qc]

    Pith/arXiv arXiv 2015

  34. [34]

    J.-Q. Guo, D. Wang, and A. V. Frolov, Spherical collapse in f (R) gravity and the Belinskii-Khalatnikov-Lifshitz conje cture, Phys. Rev. D 90, 024017 (2014), arXiv:1312.4625 [gr-qc]

    Pith/arXiv arXiv 2014

  35. [35]

    P. R. Brady and J. D. Smith, Black hole singularities: A N umerical approach, Phys. Rev. Lett. 75, 1256 (1995), arXiv:gr- qc/9506067

    arXiv 1995

  36. [36]

    M. L. Gnedin and N. Y. Gnedin, Destruction of the Cauchy h orizon in the Reissner-Nordstrom black hole, Class. Quant. Grav. 10, 1083 (1993)

    1993

  37. [37]

    Shao, J.-Q

    C.-Y. Shao, J.-Q. Guo, Y. Tian, and H.-B. Zhang, Emergen ce of critical phenomena from the black hole interior, (2025 ), arXiv:2511.17193 [gr-qc]

    arXiv 2025