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Cauchy Horizon (In)Stability of Regular Black Holes

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For regular black holes, the paper claims, the Misner-Sharp mass at the Cauchy horizon has only three possible late-time growth laws — exponential, polynomial, or logarithmic — with the logarithmic case reached by a…

desk verdict Solid phase-space analysis for Bardeen, Hayward, and Dymnikova, but the headline AS-collapse log-attractor rests on a flawed asymptotic derivation in Appendix B. read the letter →

arxiv 2507.03581 v1 pith:ZOBTY34V submitted 2025-07-04 gr-qc hep-th

Alfio Bonanno , Antonio Panassiti , Frank Saueressig This is my paper
classification gr-qchep-th
keywords CauchyhorizonmassinflationregularblackholesOrimodelMisner-Sharpquasi-fixedpointssingularitystrengthholestability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether the Cauchy horizon inside a regular black hole is unstable, as it is in Reissner-Nordström. Working with the Ori model, in which outgoing radiation is a thin null shell colliding with the ingoing Price tail, it claims that the late-time growth of the Misner-Sharp mass can be only one of three things: exponential, polynomial, or logarithmic. In particular, for the AS-collapse geometry, a non-singular collapse solution proposed in the context of quantum gravity, the mass grows as $r_{-}^{3}/(12\xi)\,\left[\log(y_0 v^{-p})\right]^{2}$, so it stays of the same order as the black-hole mass and is reached with essentially no transient mass-inflation episode. The conclusion is that exponential mass inflation is not generic for regular black holes, and that polynomial or logarithmic growth produces only a weak singularity in the standard classification.

What carries the argument

The load-bearing object is the Ori model: outgoing radiation is collapsed into a spherically symmetric thin null shell, while the ingoing flux is fixed by Price's late-time tail $m_-(v)=m_0-\beta/v^{p-1}$; the two meet at the shell and are linked by the junction condition. The dynamics reduces to two first-order equations, one for the shell radius $R(v)$ and one for the mass $m_+(v)$ in the region between the shell and the Cauchy horizon. The central concept is a quasi-fixed point: a value $m_{+,\ast}$ where the right-hand side of the mass equation vanishes when $R=r_-$, so the flow is governed by an attractive or repulsive fixed point, and the scaling of the Misner-Sharp mass is read off by a Frobenius expansion around it. For the AS-collapse attractor, the paper introduces $x(v)=1+6\xi m_+/R^{3}$ and analyzes the late-time equation (B6) through the ansatz $y(v)=y_0\left[1+\sum_{n\geq 1}y_n\log^{n}v\right]$, whose coefficients are fixed recursively by a generating function; because the authors state that they are not aware of an analytic solution of (B6), the logarithmic law is carried by this log-polynomial ansatz itself.

What would settle it

Integrate the reduced equation (B6), or the full Ori system, to very late times with several initial conditions and compare the reconstructed Misner-Sharp mass with the logarithmic law: if the ratio $M_+(v)/\bigl[(r_{-}^{3}/(12\xi))\log^{2}(y_0 v^{-p})\bigr]$ does not approach a universal constant, or if residuals require terms not captured by the recursively fixed coefficients, the logarithmic attractor fails. A more direct check is to find a non-analytic contribution, such as a $\log\log v$ correction, in the solution of (B6).

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Extended reading notes

Core claim

The central claim, stated within the Ori model, is that regular black holes fall into exactly three late-time behaviors for the Misner-Sharp mass at the Cauchy horizon. The Bardeen geometry and the Reissner-Nordström geometry share a repulsive quasi-fixed point and exhibit eternal exponential mass inflation, $M_+(v)\propto v^{-p}e^{\kappa_- v}$. The Hayward geometry has a global attractive quasi-fixed point whose basin of attraction is the whole phase space, giving universal power-law growth $M_+(v)\propto v^{p}$. The Dymnikova geometry has a one-sided attractor at $m_+=0$ and generically still ends in exponential mass inflation, apart from trajectories that terminate at finite $v$. The new case is the AS-collapse regular black hole: its attractive quasi-fixed point gives $M_+^{\rm att}(v)\simeq \frac{r_{-}^{3}}{12\xi}\,\left[\log(y_0 v^{-p})\right]^{2}$, reached from a wedge of phase space and with no appreciable transient mass-inflation phase. The paper reads this as showing that the Reissner-Nordström mass-inflation instability does not automatically transfer to regular black holes.

Load-bearing premise

The logarithmic attractor rests on the assumption that the late-time solution of the differential equation (B6) is exactly described by a log-polynomial series, while the authors state that they do not know an analytic solution of that equation; if the true attractor contains any non-analytic late-time term, the logarithmic law is not established.

Editorial extensions

If this is right

  • If the central claim is correct, exponential mass inflation is not the generic fate for regular black holes: only Bardeen-type flows (and Dymnikova flows reaching $v=\infty$) suffer it.
  • The Hayward attractor is global: every initial condition, even after a finite transient mass-inflation episode, ends at the universal power law $M_+(v)\propto v^{p}$, with no free parameter.
  • The AS-collapse attractor keeps the Misner-Sharp mass within the same order of magnitude as the black-hole mass, and the Weyl-squared curvature invariant asymptotes to a constant rather than diverging exponentially.
  • The three mass behaviors reduce to two singularity-strength classes: exponential growth gives a strong singularity, while polynomial and logarithmic growth give weak singularities, so geodesics may admit a $C^{1}$ continuation.
  • The phase-space structure shows that the tamed growth in the Hayward case is generic due to its cylindrical topology, whereas the AS-collapse attractor is confined to a wedge between two repulsive quasi-fixed points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the completeness of the log-polynomial ansatz for (B6) is the obvious place to test; if the true late-time solution contains non-analytic terms such as $\log\log v$, the logarithmic law would need revision.
  • Editorial inference: the classical taming of mass inflation in the AS-collapse geometry does not by itself settle the full stability question, since quantum field theory in curved spacetime can still produce divergent stress-energy near the horizon; that comparison is not made in the paper.
  • Editorial inference: the dichotomy between linear and nonlinear dependence of the Misner-Sharp mass on $m_+$ suggests a practical criterion — geometries whose mass function is linear in $m_+$ fall into the exponential class, while nonlinear mass functions can admit polynomial or logarithmic attractors — which could be used to scan other regular black hole proposals.
  • Editorial inference: the results give a target for numerical relativity: a full dynamical collapse simulation of the AS-collapse geometry should show whether the thin-shell idealization substantially changes the late-time logarithmic growth found here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the Ori model of Cauchy-horizon mass inflation for four families of regular black holes: Bardeen, Hayward, Dymnikova, and a model arising from asymptotically safe collapse (AS-collapse). It proposes a phase-space classification of the late-time dynamics of the Misner-Sharp mass along an ingoing null shell: eternal exponential mass inflation (Bardeen and one Dymnikova branch), a polynomial attractor (Hayward), and a logarithmic attractor (AS-collapse). The Hayward result is presented as generic through a global basin-of-attraction argument, and the AS-collapse logarithmic growth is the paper's central new claim. The analytic argument for the logarithmic attractor is relegated to Appendix B and is supplemented by numerical integrations shown in Fig. 4.

Significance. If the results are correct, the paper substantially changes the expected stability picture of regular black holes: the standard Reissner-Nordstrom mass-inflation instability would not be generic, and the AS-collapse model would exhibit an extremely tame, logarithmic growth of the mass at the Cauchy horizon. The paper's organization is clear, and the phase-space treatment of the Bardeen, Hayward, and Dymnikova cases is a useful systematic contribution. The claim of a global Hayward attractor and the explicit basin analysis are valuable, as is the recognition that non-analytic mass functions require a case-by-case treatment. However, the central new result for AS-collapse depends entirely on the asymptotic analysis of Appendix B, and that analysis as written is not internally consistent. No code or data is shipped, so the numerical confirmation cannot currently be checked independently.

major comments (3)
  1. [Appendix B, Eqs. (B8)-(B13) and Eq. (82)] The derivation of the logarithmic attractor is internally inconsistent. For y(v) of the assumed form (B11), log y(v) = O(log log v), so the right-hand side of (B8) is dominated by the term -t kappa_- v log v; neither the left-hand side nor any term in (B11) can cancel it. Eq. (B13) would require log tilde-y ~ 12 log v, i.e. tilde-y ~ v^12, which contradicts x(v) -> 0 and is not representable by a polynomial in log v. Equivalently, substituting x(v) = y0 v^{-p} into (B6) leaves a residual of order v^{-p} log v, larger than the derivative of the purported solution. In addition, the transcendental equation for y0 appears as a + b y0 + ... in (B10) but as 3a + b y0 + ... in (82); both cannot follow from the same balancing, and neither matches the coefficient obtained by substituting x = y0 v^{-p} into (B6). Thus Eq. (81), the parameter-free attractor claim, and the absence of transient mass inflation are not established by the presented analysis.
  2. [Section V C, Fig. 4 (bottom row)] The numerical evidence for the log attractor consists of two trajectories with a single constant y0 fixed by solving (82), displayed over v up to roughly 100. No residuals, convergence in the upper limit of integration, or machine-readable code/data are given. A slowly varying prefactor such as x(v) approx y0 v^{-p} / log^k v produces a |M_+(v)| that is indistinguishable from the claimed form over this range with a one-constant fit. Please provide the numerical data or code, and a residual analysis demonstrating that the log form is selected at asymptotically large v.
  3. [Section V B, Eqs. (78)-(80) and Table II] The reconstruction of the Misner-Sharp mass for the repulsive AS branch is sensitive to the meaning of the notation in (3e). Under the standard reading M(r) = (r^3/(12 xi)) [log(1 + 6 xi m / r^3)]^2, substituting m_+ = exp(c v^{-p} e^{kappa_- v}) gives M_+ ~ v^{-2p} e^{2 kappa_- v}, not the v^{-p} e^{kappa_- v} stated in Eq. (80) and Table II. Under the alternative reading log[(...)^2], Eq. (81) and the text's log-squared discussion need to be reworded consistently. Please state the convention unambiguously and correct all affected entries in the table and the main text.
minor comments (5)
  1. [Table I] The header of Table I lists 'mrep_+,* matt_+,* mrep_+,*' with an apparently duplicated repulsive column; if the two repulsive points for AS-collapse are intended, they should be labeled distinctly (e.g. mrep,1 and mrep,2).
  2. [Section V A, Bardeen paragraph] The text says 'phase diagram shown in the top-right corner of Fig. 3' when referring to the Bardeen case; the Bardeen panel is the top-left diagram, so the pointer should be corrected.
  3. [Eqs. (81) and Table II] The notation log(y0 v^{-p})^2 is ambiguous between [log(y0 v^{-p})]^2 and log[(y0 v^{-p})^2]; this ambiguity propagates into the abstract's claim of 'logarithmic' growth and should be clarified at first use.
  4. [Footnote 2 and Eq. (3e)] The analytic continuation of the AS-collapse mass function to real values for all m should be spelled out, including the branch choice for the logarithm when the argument becomes negative.
  5. [References] Reference [31] appears to conflate two Ayon-Beato and Garcia papers; the two works should be cited separately to allow the reader to identify the relevant result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the logarithmic AS-collapse attractor is derived from the Ori equations via an explicit asymptotic analysis, not fitted or assumed as its own conclusion.

full rationale

I walked the paper's derivation chain. The Ori-model equations (22) and (28) are obtained from the junction condition (24) and combined with the externally stated Price law (29). The late-time attractors in Sect. V B are obtained by substituting the explicit mass profiles (3) into these equations and solving the resulting ordinary differential equations asymptotically. The Hayward power-law attractor, though attributed to refs. [43,44], is re-derived here through the Frobenius expansions (67), (73), and (74); hence those self-citations are not load-bearing for the present derivation. The central new claim, the AS-collapse logarithmic attractor (81), is derived in Appendix B from the differential equation (B6). The log-polynomial ansatz (B11) is an assumed functional form, not a fitted parameter, and the resulting log-squared growth of the Misner-Sharp mass is not inserted as an input but emerges from the v^{-p} power-law factor selected by the dominant-balance argument. In the numerical section, the integration constants c are calibrated by matching the already-derived asymptotics at one value of v; this is a consistency check, not a fitted parameter renamed as a prediction. The AS-collapse mass function itself is imported from the self-cited prior work [30], but it is an external input from a collapse model and is not constructed to reproduce the logarithmic law. I also note the paper's own caveat in Appendix B that 'we are not aware of an analytic solution of (B6)', and the completeness of the log-polynomial ansatz is not proven; the dominant balance leading to (B13) also appears questionable. Those are correctness risks, not circular reductions of the conclusion to its inputs. Overall, the derivation is self-contained with respect to the orienting dynamical equations, and the central claim does not reduce by construction to any fitted or self-defined quantity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper does not fit free parameters to its target claim; Price-law inputs and normalization choices set the constant position of the attractors but not the scaling exponents. The main liabilities are the Ori-model idealization, the assumed asymptotic ansatz for Eq. (B6), and the analytic continuation of the AS-collapse mass function.

free parameters (4)
  • p (Price law exponent) = 12
    Chosen as the standard quadrupole radiative-tail value; the late-time exponents in Table II depend on p, but p is an input from Price's law, not fitted to the target result.
  • m_0 (asymptotic mass in Price law) = 1
    Set to 1 in the numerical phase-space analysis; a normalization that does not change the scaling exponents, though it enters y0 in Eq. (82).
  • β (Price-law amplitude) = 1
    Set to 1 in Table I; it affects the attractor constant y0 but is not fitted to the claimed log or power laws.
  • geometry parameters (a, l, γ, ξ) = a=0.586, l=0.483, γ=0.892, ξ=0.167
    Chosen to make κ−=1 in the representative numerical examples; the asymptotic scaling claims are argued to be independent of these values, but the 'no transient mass inflation' claim for AS-collapse is only demonstrated at ξ=0.167.
assumptions (5)
  • domain assumption The Ori thin-shell model: outgoing backreaction is a single null shell and ingoing radiation a continuous null flux, with junction conditions from GR determining the mass jump.
    The entire analysis lives inside this simplified model; real Cauchy horizon backreaction may be more complex. Introduced in Sect. III.
  • domain assumption Price's law for the late-time ingoing tail, m_-(v)=m0 - β/v^{p-1} with p≥12.
    Standard radiative-decay input from [58,59]; not derived in this paper. Entered at Eq. (29).
  • domain assumption Generic smoothness and finiteness of f_+, F(v) near the attractor for the universal power-law derivation.
    Assumed in Sect. IV around Eq. (40); the authors note Dymnikova and AS-collapse violate these, motivating the case-by-case analysis.
  • standard math The late-time dynamics admit Frobenius-type asymptotic series, R(v)-r_- ~ v^{-(p-1)} and m_+(v) with powers and log corrections.
    Standard asymptotic method, but it assumes the attractor is reached in a power-law or log-corrected power-law manner. Used in Eqs. (30), (41), (B7)-(B11).
  • ad hoc to paper Analytic continuation of the AS-collapse mass function so that M(r) is real for all m, as stated in footnote 2.
    The original model [30] is extended by hand; the log attractor sits at the boundary of the original logarithm's domain, so the continuation is load-bearing for the model's global phase space.

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Cite this review

Pith. "Pith review of Cauchy Horizon (In)Stability of Regular Black Holes." pith.science (2026). https://pith.science/paper/ZOBTY34V

@misc{pith2026250703581,
  author       = {Pith},
  title        = {Pith review of: Cauchy Horizon (In)Stability of Regular Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOBTY34V}},
  note         = {Machine review of arXiv:2507.03581}
}
read the original abstract

A common feature of regular black hole spacetimes is the presence of an inner Cauchy horizon. The analogy to the Reissner-Nordstr\"om solution then suggests that these geometries suffer from a mass-inflation effect, rendering the Cauchy horizon unstable. Recently, it was shown that this analogy fails for certain classes of regular black holes, including the Hayward solution, where the late-time behavior of the mass function no longer grows exponentially but follows a power law. In this work, we extend these results in a two-fold way. First, we determine the basin-of-attraction for the power-law attractor, showing that the tamed growth of the mass function is generic. Second, we extend the systematic analysis to the Bardeen geometry, the Dymnikova black hole, and a spacetime arising from a non-singular collapse model newly proposed in the context of asymptotically safe quantum gravity. Remarkably, in the latter solution, the Misner-Sharp mass at the Cauchy horizon remains of the same order of magnitude of the mass of the black hole, since its growth is just logarithmic.

Figures

Figures reproduced from arXiv: 2507.03581 by the authors.

Figure 1
Figure 1. FIG. 1: Radiation falling in to a Reissner-Nordstr¨om black [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Penrose diagram illustrating the Ori-model of the in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Illustration of the dynamics arising from the Ori model for [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Illustration of the dynamics found for the Ori-model for va [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison, in a log-plot, between different dynam [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

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Forward citations

Cited by 2 Pith papers

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

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.