Numerical study on the robustness of the stability for stable black holes

Run-Qiu Yang; Zhan-Feng Mai

REVIEW 4 major objections 4 minor 42 references

Numerical study on the robustness of the stability for stable black holes

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A near-horizon bump of any size can drive a stable black hole unstable.

desk verdict A genuinely new numerical claim that infinitesimal near-horizon potential bumps can destabilize Schwarzschild, but the load-bearing extrapolation to arbitrarily small amplitude rests on numerical evidence that is not yet documented well enough. read the letter →

arxiv 2506.07562 v4 pith:3R3NYWYN submitted 2025-06-09 gr-qc hep-th

Zhan-Feng Mai , Run-Qiu Yang This is my paper

classification gr-qchep-th

keywords blackholestabilityRegge-Wheelerpotentialnear-horizonperturbationsspectruminstabilityscalarwavestime-domainnumericalrelativitycriticaldistancescalingtortoisecoordinate

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

The paper asks whether the stability of a stable black hole survives arbitrarily small changes to the geometry near its horizon. Using the Schwarzschild solution as the background and encoding the change as a small local deformation of the Regge-Wheeler potential, the authors solve the scalar wave equation in the time domain and find a critical distance: when the deformation sits closer to the horizon than that distance, the perturbation stops decaying and instead grows exponentially at late time. The critical distance grows as the amplitude shrinks, but without bound, so there is no nonzero amplitude below which the black hole is guaranteed to remain stable. The same qualitative behavior holds for zero-mean stochastic and low-frequency time-dependent local potentials. If this is right, two black holes that are nearly identical in their exterior geometry can have completely different late-time fates.

What carries the argument

The central object is the deformed Regge-Wheeler potential $V_{\mathrm{eff}} = V_{\mathrm{RW}}(r_*) + \epsilon V_p(r_* - a, t)$ for s-wave scalar perturbations of Schwarzschild, with $V_{\mathrm{RW}} = h h'/r$ and tortoise coordinate mapping the horizon to $r_* \to -\infty$. The analysis solves the one-dimensional wave equation with no-reflect boundary conditions using fourth-order finite differences and a fourth-order Runge-Kutta time integrator, then locates the critical distance $a_c$ by bisection from the appearance of late-time exponential growth. The horizon's infinite tortoise-coordinate range is the load-bearing geometric feature: it allows a small local potential placed arbitrarily far down the throat to act over an unbounded interval, unlike the bounded $r_*$ range of a horizonless star.

What would settle it

Run the same time-domain evolutions at a sequence of decreasing grid spacings and time steps while fixing $\epsilon$ and $a$: if the exponential growth rate shrinks toward zero or $a_c$ stops growing as $\epsilon \to 0$, the infinitesimal-instability claim fails. A complementary check is a frequency-domain search for a quasinormal mode with positive imaginary part that appears only when $a < a_c$; without such a mode there is no spectral instability to match the time-domain growth.

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Extended reading notes

Core claim

The central claim is that the stability of the Schwarzschild black hole against s-wave scalar perturbations is not robust: for a local deformation $V_{\mathrm{eff}} = V_{\mathrm{RW}}(r_*) + \epsilon V_p(r_* - a, t)$, with $V_p$ compactly supported (or random, or time-dependent), there is a critical distance $a_c$ such that for $a < a_c$ the perturbation grows exponentially instead of decaying. Numerically, for static negative bumps $a_c = 1/(\bar{V}_p \epsilon)$ where $\bar{V}_p$ is the spatial integral of $V_p$; for stochastic zero-mean potentials $a_c \propto 1/\epsilon^2$; for low-frequency oscillatory bumps with $w \to 0^+$, $a_c \propto 1/\epsilon$. Since $a_c$ grows without bound as $\epsilon \to 0$, there is no threshold amplitude below which the black hole is guaranteed stable: any positive $\epsilon$ can destabilize if placed close enough. The paper also finds that potentials that stabilize Minkowski spacetime can destabilize a black hole, and that a regular star with no horizon does not show this infinitesimal-bump instability, so the effect is tied to the horizon's infinite tortoise-coordinate range.

Load-bearing premise

The results hinge on the assumption that the late-time exponential growth seen in the numerical integrations is a genuine instability of the deformed wave equation and that the fitted power laws for $a_c$ keep holding as $\epsilon$ goes to zero; if the growth is a numerical artifact or the scaling bends at smaller amplitudes, the infinitesimal-bump conclusion does not follow.

Editorial extensions

If this is right

A black hole surrounded by any matter that produces a negative, stochastic, or low-frequency oscillatory near-horizon potential deformation will, if the deformation lies closer than $a_c$, develop exponentially growing scalar perturbations rather than settling down.
The energy-flux calculation in the paper implies that during this instability the scalar field transfers energy into the horizon and out to infinity, so the end state is not a static hairy configuration but a vacuum Schwarzschild black hole with larger horizon area.
Near-horizon quantum fluctuations, redshifted to $\sim 1/r_h$ scales in tortoise coordinates, act as small stochastic deformations; the paper concludes that an isolated black hole is not guaranteed to remain stable over extended timescales if such fluctuations persist.
The comparison with a Hayward star shows the effect is horizon-specific: an infinitesimal negative bump placed as close as possible to a regular star does not trigger instability, so the phenomenon is not just the negative bump acting in flat space.
The authors note that analogous instability may occur near cosmological horizons in asymptotically de Sitter black holes.

Reading between the lines

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

Editorial extension: the power-law scalings predict a sharp quantitative signature in the frequency domain; a spectral analysis should show a quasinormal mode crossing the real axis exactly when $a$ drops below $a_c$, with growth rate tied to the distance past the threshold.
Editorial extension: because the critical distance diverges only polynomially in $1/\epsilon$, one can test the claim by computing $a_c$ at two or three smaller amplitudes with higher-resolution grids and checking whether $\epsilon a_c$ (or $\epsilon^2 a_c$) truly stays constant; a bend in the curve would signal a breakdown of the infinitesimal-bump conclusion.
Editorial extension: for rotating black holes, the tortoise coordinate still runs to $-\infty$ but angular momentum modifies the effective potential; spin may change the critical distance and the scaling exponents, and the connection to turbulent black holes mentioned in the paper becomes testable.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: this is a genuinely new observation, not just a numerical confirmation of something in the literature. Prior spectrum-instability work showed QNM spectra are sensitive to positive bumps while time-domain ringdown stays robust. This paper instead reports genuine late-time exponential growth from arbitrarily small negative or stochastic local potentials placed near the horizon, with scaling ac ~ 1/epsilon and 1/epsilon^2. If true, that overturns the usual robustness assumption for linearized BH stability. The star comparison in Appendix B and the DEC example in Appendix A are nice supporting touches; the energy-flux appendix adds physical interpretation. The paper is also honest about being a toy model.

The soft spot is the numerics, and it is load-bearing. Section II describes fourth-order finite differences and RK4 but gives no domain size, grid spacing, time step, or boundary implementation, and there are no convergence tests or error bars. The central 'infinitesimal' claim is an extrapolation of power laws fitted to six or seven bisection points. The stress-test worry is real: the measured ac ~ 1/epsilon and 1/epsilon^2 are exactly what you'd expect if the instability is a bound-state-like mode that only fits inside a finite domain once the well is close enough to the inner boundary. With no domain-size study, the apparent exponential growth at late times could be a finite-domain artifact rather than a genuine mode of the infinite-line problem. The stochastic case is also shown for one realization of the noise, so the universality claim is not yet established.

I don't think the claim is wrong; the qualitative picture—negative near-horizon wells competing with horizon absorption—is plausible, and the authors may well be right. But the paper as written does not yet provide the evidence needed to distinguish a real effect from an artifact. This is a fixable problem: a grid-convergence study, a domain-size scan, and a demonstration that the growth rate and critical distance converge as the domain grows would settle it. The scaling laws should also be tested over a wider range of epsilon.

Bottom line: this deserves a serious referee—the question is important and the result, if confirmed, matters. I would send it to review, expecting major revisions on the numerics. I wouldn't cite the infinitesimal claim in my own work until that's done.

Recommendation: engage, but condition acceptance on the numerical robustness being documented.

Referee Report

4 major / 4 minor

Summary. The manuscript studies the linear stability of Schwarzschild black holes against small local deformations of the Regge-Wheeler potential. The authors numerically solve the (1+1)-dimensional wave equation for an s-wave scalar field with three types of potential perturbations: a static negative Gaussian bump, a stochastic local potential, and a low-frequency time-dependent Gaussian bump. For each case they report that, when the perturbation is placed sufficiently close to the horizon, the scalar field grows exponentially at late times, and they extract scaling laws for the critical distance as a function of the perturbation amplitude: a_c ~ 1/(epsilon) for static negative bumps and low-frequency bumps, and a_c ~ 1/epsilon^2 for stochastic bumps. The paper concludes that infinitesimal near-horizon deformations can overturn the stability of a stable black hole, and it speculates that horizon-induced redshift of quantum fluctuations could destabilize isolated black holes.

Significance. If substantiated, the central claim would be significant: it would show that the linear stability of Schwarzschild is not structurally robust against small localized potential perturbations, contradicting the common expectation that sufficiently small metric deformations cannot change the stability of an initially stable black hole. The manuscript is clearly written, addresses a timely question, and includes useful supplementary analyses (a DEC-compatible construction of a negative bump in Appendix A, a comparison with a horizonless star in Appendix B, and an energy-flux argument in Appendix C). However, the significance is entirely conditional on the numerical reliability of the time-domain simulations and on the validity of the extrapolated scaling laws; neither is established in the present version.

major comments (4)

[II] The numerical method is described only as a fourth-order finite-difference spatial discretization with RK4 time integration and 'no reflect' boundary conditions. The paper does not state the computational domain size in r*, the grid spacing, the time step, the duration of the simulations, or the precise implementation of the boundary conditions. This is load-bearing because the instability is diagnosed from visual exponential growth in Figures 1-3 and the critical distances are located by bisection; without grid-convergence and domain-size studies, the observed growth could be a finite-domain artifact, a boundary reflection, or a transient effect. In particular, the reported scaling a_c ~ 1/epsilon and 1/epsilon^2 coincides with the tail length of a would-be bound state, so the threshold may be set by the numerical boundary rather than by the continuum problem. A convergence study and a domain-size dependence test are essential before the central claim can be accepted.
[III, Eq. (6)] The scaling law a_c = 1/(V_p epsilon) is presented as an asymptotic relation, but it is based on a small number of bisection points (visible in the right panel of Fig. 1) and is then extrapolated to arbitrarily small epsilon. No criterion is given for what constitutes 'instability' in the bisection procedure, no growth-rate threshold is specified, and no independent spectral (frequency-domain) confirmation is provided to show that a genuine exponentially growing mode crosses the real axis. The conclusion that 'infinitesimal' negative bumps always destabilize the black hole therefore exceeds the evidence presented, since the fitted power law could bend or break down at smaller epsilon.
[IV and V, Eqs. (11) and (13)] The stochastic and low-frequency scaling laws a_c ~ 1/epsilon^2 and 1/epsilon are obtained by fitting the 'most left 6-7 points' in Figures 2 and 3, and the text claims universality after 'testing many different functions' without showing any of those tests. For the stochastic case, the spectral content of the random potential is not quantified, and no ensemble-averaging or statistical convergence is described. The 1/epsilon^2 scaling in particular is not explained by a simple bound-state estimate and would need independent confirmation, for example by solving for quasinormal modes or using a spectral method, before it can be treated as a physical result.
[Appendix B] The comparison with a Hayward star is invoked to argue that the instability is specific to black hole geometry, but the argument is only that the tortoise coordinate has finite range for a star and infinite range for a black hole. This is a heuristic distinction, not a controlled numerical comparison: the stellar background also has a different potential profile and a different inner boundary condition. To support the claim that the effect is due to the near-horizon r* -> -infinity structure, the authors would need to compare black-hole and horizonless backgrounds with the same type of localized perturbation and the same numerical setup, or provide an analytic argument for the role of the semi-infinite domain.

minor comments (4)

[II] The text says 'forth-order Rugge-Kutta method'; this should be 'fourth-order Runge-Kutta method'.
[III, Fig. 1] The red line in the right panel of Fig. 1 is said to be 'not by fitting', but the definition of V_p is only given in the text; it would be clearer to explicitly state the integrated value of V_p for the chosen Gaussian profile and to show the relation between the fitted line and the data points with residual errors.
[IV, Eq. (8)] The notation is confusing because the same symbol W is used for the spatial random function and for its zero-average integral condition, and U is used both for the temporal random function and for its time average; please use distinct symbols for the function and the mean.
[V, Fig. 3] The left panel of Fig. 3 states w = 0.02 and sigma = 4, while the right panel states w = 0.001 and sigma = 4; the text discusses both cases but does not clearly state which parameters correspond to which data set in the bisection plot. Clarify the parameter choices and the number of points used for each fit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical instability claims and scaling laws are empirical outputs of time-domain simulations, not consequences of fitted inputs by construction.

full rationale

The paper's central derivation chain is numerical: it solves the perturbed wave equation (2) with potential (3) for three classes of Vp, locates a critical distance a_c by bisection, and reports the scaling laws (6), (11), and (13). These scaling laws are outputs of the simulations, not inputs fitted to reproduce them. The 'negative static bump' law a_c = 1/(Vp epsilon) is an inferred empirical relation; the paper explicitly labels the plotted line as 'not by fitting' only in the sense that the line is drawn from the formula after the exponent is read off the bisection data. No equation in the paper defines stability or instability in terms of a_c, and no parameter fitted to the late-time growth data is renamed as a prediction. Self-references are limited to background material (e.g., Ref. [26] on spectral instability, Ref. [39] announcing future frequency-domain work) and are not load-bearing for the stated numerical conclusion. The heuristic redshift argument in Sec. IV is an independent scaling estimate, not a derivation of the numerical results. The main concerns with the paper are numerical-convergence and extrapolation risks (no stated grid spacing or domain size, no convergence study, and power laws inferred from a finite range of epsilon), but those are correctness/robustness issues, not circularity. There is no step in which a claimed prediction is equivalent by construction to an input.

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

The central numerical result rests on four explicit modeling assumptions: the additive-potential encoding, the numerical boundary and convergence assumptions, the extrapolation of fitted scaling laws to arbitrarily small amplitudes, and the representativeness of the single stochastic realization. The Planck-redshift estimate is an additional heuristic used only for physical interpretation. The fitted quantities are the slopes and intercepts of the scaling laws; the critical distances themselves are measured outputs, not free inputs.

free parameters (4)

Stochastic scaling-law exponent = -2 (slope of ln|a_c| vs ln epsilon, Fig. 2)
The line ln|a_c| = -2 ln epsilon + 0.02 is a best fit to the leftmost 7 bisection points; used to claim a_c proportional to 1/epsilon^2.
Stochastic scaling-law intercept = 0.02
Intercept from the same straight-line fit; a free constant of the empirical relation.
Low-frequency scaling-law exponent = -1 (slope of ln|a_c| vs ln epsilon, Fig. 3)
The line ln|a_c| = -ln epsilon - 0.93 is a best fit to the leftmost 6 bisection points; used to claim a_c proportional to 1/epsilon.
Low-frequency scaling-law intercept = -0.93
Intercept from the same straight-line fit; a free constant of the empirical relation.

assumptions (5)

ad hoc to paper All near-horizon environmental or quantum effects can be encoded as an additive local perturbation epsilon Vp(r*, t) to the Regge-Wheeler potential of a single s-wave scalar test field.
Eq. (3); no derivation from a matter action or metric perturbation, explicitly acknowledged as a toy model.
domain assumption The numerical evolution uses no-reflection boundary conditions at both horizon and infinity and is stable enough that observed exponential growth is physical.
Sec. II imposes no-reflect boundaries; no grid-convergence or independent spectral check is presented.
ad hoc to paper The fitted power laws extend to arbitrarily small epsilon, so that a critical distance always exists for any infinitesimal amplitude.
Secs. III-V extrapolate the bisection-based fits; this is the key step behind the infinitesimal-bump claim.
ad hoc to paper Planck-scale quantum fluctuations near the horizon are redshifted to O(r_h) stochastic potential variations in tortoise coordinates.
Sec. IV heuristic estimate; used to connect the stochastic toy model to quantum fluctuations, but not needed for the static negative bump result.
domain assumption A single pseudo-random realization of W and U represents the generic stochastic behavior.
Sec. IV uses one explicit realization and asserts universality after testing many functions without showing the ensemble spread.

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Pith. "Pith review of Numerical study on the robustness of the stability for stable black holes." pith.science (2026). https://pith.science/paper/3R3NYWYN

@misc{pith2026250607562,
  author       = {Pith},
  title        = {Pith review of: Numerical study on the robustness of the stability for stable black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3R3NYWYN}},
  note         = {Machine review of arXiv:2506.07562}
}

read the original abstract

This paper numerically studies if the stability of a stable black hole is robust against the small perturbation on geometry near its event horizon. In an other word, we numerically study if two nearly identical black holes may exhibit completely different stabilities at late time. As a toy model, it encodes the such perturbation into deformations of Regge-Wheeler potential. It considers three different types of local deformations-the negative static bump potential, the stochastic potential and bump potential modulated by time function in low frequency limit. Our numerical results show that infinitesimal local deformations on Regge-Wheeler potential near the horizon can overturn stability of a stable black hole, implying that late-time behavior of a stable black hole is extremely sensitive to geometry near horizon. Specially, certain deformations that stabilize systems in flat backgrounds can destabilize otherwise stable black holes. It also shows that horizon-induced redshift transforms near-horizon quantum fluctuations into classical-scale stochastic deformations capable of triggering instability, implying that even an isolated black hole cannot keep stable if the near-horizon quantum noise could be hold in extended timescales.

Figures

Figures reproduced from arXiv: 2506.07562 by the authors.

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

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**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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Adrian Ka-Wai Chung, Joseph Gais, Mark Ho-Yeuk Che- ung, and Tjonnie G. F. Li, “Searching for ultralight bosons with supermassive black hole ringdown,” Phys. Rev. D 104, 084028 (2021)

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Perturbations of the Vaidya metric in the frequency do- main: Quasinormal modes and tidal response,

Lodovico Capuano, Luca Santoni, and Enrico Barausse, “Perturbations of the Vaidya metric in the frequency do- main: Quasinormal modes and tidal response,” Phys. Rev. D 110, 084081 (2024) , arXiv:2407.06009 [gr-qc]

2024 arXiv
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Instability of reissner-nordström black holes in de sitter backgrounds,

Vitor Cardoso, Madalena Lemos, and Miguel Marques, “Instability of reissner-nordström black holes in de sitter backgrounds,” Phys. Rev. D 80, 127502 (2009)

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Agnostic black hole spectroscopy: Quasinormal mode content of numer- ical relativity waveforms and limits of validity of linear perturbation theory,

Vishal Baibhav, Mark Ho-Yeuk Cheung, Emanuele Berti, Vitor Cardoso, Gregorio Carullo, Roberto Cotesta, Wal- ter Del Pozzo, and Francisco Duque, “Agnostic black hole spectroscopy: Quasinormal mode content of numer- ical relativity waveforms and limits of validity of linear pert...

2023
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The (in)stability of quasinormal modes of Boulware-Deser- Wheeler black hole in the hyperboloidal framework,

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Tur- bulent black holes,

Huan Yang, Aaron Zimmerman, and Luis Lehner, “Tur- bulent black holes,” Phys. Rev. Lett. 114, 081101 (2015)

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The analysis of frequency domain for static negative bump and static stochastic potential will be addressed in an upcoming publication
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Formation and evaporation of reg- ular black holes,

Sean A. Hayward, “Formation and evaporation of reg- ular black holes,” Phys. Rev. Lett. 96, 031103 (2006) , arXiv:gr-qc/0506126

2006 arXiv
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When the negative bump is suﬃciently far way from the star, the instability can still be triggered but this is the eﬀect of negative bump in ﬂat space

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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negative bump

− sin(t/π) and W (x) = w0 [1 − tanh(4(x − 2)) tanh(4(x + 2))]Q(x). (9) with Q(x) = sin x + cos √ 2x + cosπx − sinπx +c0. (10) Herec0 andw0 are chosen so thatW has zero average and normalization condition max |W | = 1 is satisﬁed. The detailed forms of Eq. ( 9) is not important. We choose functions ( 9) and ( 10) just as a concrete example. The results are...

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arXiv 2018