Analytical Criteria for Black Hole Instability in Non-Minimally Coupled Scalar-Tensor Theories

Davood Mahdavian Yekta; Majid Karimabadi; S. A. Alavi

Analytical Criteria for Black Hole Instability in Non-Minimally Coupled Scalar-Tensor Theories

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper derives closed-form critical coupling constants beyond which regular black holes become unstable under non-minimal scalar-field perturbations, identifying the threshold with an extremum of the near-horizon effective potential loca

desk verdict Useful analytic formulas for instability thresholds in regular black holes, but the central threshold criterion is asserted rather than proven, and the QNM/area-quantization section contains a factor-of-two error. read the letter →

arxiv 2607.19755 v2 pith:4OAL2KH7 submitted 2026-07-22 gr-qc

Majid Karimabadi , Davood Mahdavian Yekta , S. A. Alavi This is my paper · ORCID

classification gr-qc

keywords regularblackholesnon-minimalscalar-tensorcouplingtachyonicinstabilityRegge-Wheelerequationnear-horizonapproximationquasinormalmodesareaquantizationcriticalconstant

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 tries to establish that regularity does not protect black holes from instability once a scalar field is allowed to couple to the spacetime curvature. The authors study four families of regular black holes — non-commutative Schwarzschild, Bardeen, Hayward, and ABG — under two non-minimal couplings: the field coupled directly to the Ricci scalar, and its derivatives coupled to the Einstein tensor. They derive closed-form expressions for the critical coupling at which the black hole becomes unstable, essentially by demanding that the effective potential of the Regge-Wheeler equation have an extremum exactly at the horizon. If the derivation holds, the threshold is computable from horizon data alone: horizon radius, surface gravity, and the second derivative of the metric function. At the threshold, the quasinormal ringing gives way to a purely imaginary mode, and the area spectrum A = 8πn follows without the usual highly-damped approximation.

What carries the argument

The machinery is the near-horizon expansion of the effective potential in the Regge-Wheeler equation for a static spherically symmetric metric with f(r_h) = 0. Writing f(r) ≈ 2κ x + (1/2) f''(r_h) x² with x = r − r_h, and expanding the potential as V(x) ≈ a x² + b x, the paper identifies the instability threshold with b = 0, i.e. V'(r_h) = 0. Solving b = 0 for the coupling ζ gives the critical values. In the Ricci-coupled model this is a direct equation; in the Einstein-tensor-coupled model the coefficients a and b themselves come from the near-horizon expansion of the more complicated kinetic coupling, which is an additional approximation the paper notes.

What would settle it

For the NC Schwarzschild tensor model with θ = 0.2, ℓ = 2, μ = 0.5, the paper predicts ζ_c = 5.673. Numerically integrate the exact Regge-Wheeler equation for ζ = 5.6 (just below) and search for a negative well outside r_h or late-time growth in the time-domain profile; finding either would contradict the claim that the threshold sits exactly at b = 0. Alternatively, check directly whether the exact first derivative V'(r_h) vanishes at the predicted ζ_c for the tensor model using the full potential, not the near-horizon approximation.

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Extended reading notes

Core claim

At the critical value of the non-minimal coupling, the near-horizon effective potential of the Regge-Wheeler equation, V(x) ≈ a x² + b x, has its extremum exactly at the event horizon; the condition is b = 0. For the Ricci-coupled scalar model this condition is exact and yields ζ_c(Ricci) = [r_h² μ² + 2 r_h κ + ℓ(ℓ+1)] / [r_h² f''(r_h) + 8 r_h κ − 2]. For the Einstein-tensor-coupled model, after a near-horizon expansion, it yields ζ_c(Einstein) = 2 r_h [r_h(r_h μ² + 2κ) + ℓ(ℓ+1)] / [(2 r_h κ + ℓ(ℓ+1))(r_h f''(r_h) + 4κ)]. At this threshold the perturbations neither ring nor decay — the late-time tail is a straight line — and the real part of the near-horizon quasi-normal frequency vanishes,

Load-bearing premise

The load-bearing premise, asserted and supported by one plotted numerical example rather than proved, is that the onset of instability is exactly the condition b = 0, i.e. an extremum of the near-horizon effective potential located on the horizon; if a negative potential well could already exist for smaller coupling without touching the horizon, the derived critical values would be wrong, and the tensor-model formula further inherits the near-horizon approximation the paper i

Editorial extensions

If this is right

For any spherically symmetric black hole whose metric function has a Taylor expansion near the horizon, Eqs. (3.8) and (3.9) give a direct algebraic prediction of the coupling at which scalar perturbations become unstable, without solving the full perturbation equations.
In the Schwarzschild limit both formulas diverge, recovering the known linear stability of Schwarzschild against these scalar perturbations for any finite coupling.
In the tensor model, the critical coupling becomes independent of the multipole ℓ in both the massless (μ = 0) and eikonal (ℓ → ∞) limits, while in the Ricci model ℓ → ∞ removes the instability entirely; these are testable predictions.
At ζ = ζ_c the near-horizon QNM frequencies become purely imaginary, so the onset of instability is accompanied by a mode that neither oscillates nor decays.
The spacing of near-horizon QNM frequencies at criticality is Δω = κ = 2πT_H, reproducing the area spectrum A = 8πn without invoking highly damped modes.

Reading between the lines

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

Inference: Because the two closed-form formulas depend only on r_h, κ, f''(r_h), μ, and ℓ, they should be directly testable against full numerical integration of the Regge-Wheeler equation for any metric in the same class, including the singular Reissner-Nordström geometry; the paper does not carry out that check.
Inference: The paper's b = 0 criterion, if true, implies a geometric picture of the onset of instability: the negative well is born exactly on the horizon and then migrates outward; this could be checked by tracking the location of the minimum of V(r) as ζ is swept through ζ_c for all four families.
Inference: The area-quantization result at ζ_c suggests that critical coupling could serve as a proxy for highly damped modes in other contexts, such as computing grey-body factors or entropy spectra, though the authors do not explore those.
Inference: A gap in the argument is that absence of instability for ζ < ζ_c is not proven analytically; a full proof would need to show the exact potential has no negative well outside the horizon whenever ζ is below the b = 0 value.

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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

Dave,

The useful thing here is the pair of closed-form critical coupling expressions, Eqs. (3.8) and (3.9), for the onset of scalar-field instability on a class of regular black holes. If you take the near-horizon quadratic expansion at face value, the Ricci-coupled formula follows cleanly from V'(r_h)=0, and the tensor formula is at least plausible. That turns a numerical threshold search into a one-line algebraic check. The paper checks the Schwarzschild limit, the eikonal limit, and the dependence on scalar mass; those checks are consistent, and the time-domain profiles in Fig. 1 support the claimed critical values for the examples shown.

The soft spots are real. First, the central identification - instability starts exactly when the near-horizon potential has an extremum on the horizon, i.e., b=0 - is asserted, not proved. In the pure quadratic model V(x)=a x^2+b x on the half-line x>0, a zero-energy bound state requires b = -2κ sqrt(a) for the lowest mode, not b=0. So the paper's criterion is not self-evident, and one plotted example is weak evidence. The formulas may be good approximations, but calling them exact overstates what is shown. Second, the tensor formula is not derived in the text; the reader is asked to trust the algebra. Third, the QNM section has a clear factor-of-two error: applying the paper's own Gamma-pole condition gives ω = -b/(2 sqrt(-a)) - iκ(2n+1), not -iκ(n+1/2). That changes the frequency spacing from κ to 2κ and undermines the area-quantization claim advertised in the abstract. That section should be corrected or retracted.

The core instability analysis is a useful heuristic, but it is not a rigorous threshold theorem. The paper would be credible if it presented the criterion as an empirical or approximate condition and supplied a derivation of the tensor formula plus a systematic numerical test across the parameter range. As it stands, the gap between the claim and the evidence is too large for unconditional acceptance.

Send it to a referee who knows black hole perturbation theory; it is not a desk reject. The referee should ask for the tensor derivation, a correction of the QNM factor, and either a proof or an honest downgrading of the threshold criterion. If those are supplied, the formulas are worth publishing.

Best,

Referee Report

3 major / 5 minor

Summary. The paper analyses linear scalar perturbations of spherically symmetric regular black holes (non-commutative Schwarzschild and Bardeen/Hayward/ABG) in two non-minimally coupled models: Ricci coupling and Einstein-tensor coupling. Expanding the metric near the horizon as f ≈ 2κx + ..., the effective potential is approximated by V(x) ≈ a x² + b x. The instability threshold is identified with V'(r_h) = 0, which yields closed-form critical couplings, Eqs. (3.8) and (3.9). The authors further derive near-horizon QNM frequencies, argue that at the critical coupling the modes become purely imaginary, and use the spacing of these modes to reproduce an equidistant area spectrum A = 8πn without the highly-damped approximation. Numerical verification is based on time-domain profiles reported in the authors' earlier work.

Significance. If correct, Eqs. (3.8)-(3.9) would be remarkably simple horizon-data formulas for the onset of instability, with no fitted parameters. The paper performs several sensible consistency checks (Schwarzschild limit, eikonal limits, mass dependence) and the qualitative trends in ℓ, θ, q match the authors' earlier numerics. The derivation of the threshold, however, rests on an unproved equivalence between V'(r_h) = 0 and the appearance of an unstable mode, and the QNM section contains a factor-of-two algebraic error. The manuscript is therefore a promising but incomplete treatment; its main claims need additional spectral justification.

major comments (3)

[Sec. 3.1, Eq. (3.3)] The identification of the instability threshold with b = 0 is the load-bearing step, but it is not proved. b = 0 is only the condition that the near-horizon effective potential has an extremum at the horizon. For a Schrödinger operator with potential a x² + b x (or any generic potential with a negative well), the onset of a zero-energy bound state is controlled by the lowest eigenvalue crossing zero, which generally occurs at a finite negative b, not at b = 0. The evidence offered, Fig. 1 for a single NC-Schwarzschild configuration and a sentence that other plots behave similarly, is heuristic. Please prove the equivalence or provide a controlled numerical test (e.g., compute the fundamental QNM frequency as a function of ζ and show that Im ω → 0 exactly at b = 0), or Eqs. (3.8)-(3.9) remain unjustified.
[Sec. 3.2, Eq. (3.14)] The Gamma-pole algebra is incorrect. Solving the displayed condition gives ω = -b/(2√-a) - iκ(2n+1), not ω = -b/(2√-a) - iκ(n+1/2). Consequently the spacing of near-horizon modes is Δω = 2κ rather than κ, so the area quantization in Sec. 3.3 becomes A = 4πn, not A = 8πn. The qualitative result Re ω = 0 at b = 0 survives, but Eq. (3.17), the area-spectrum claim, and the comparison with the highly-damped literature must be revised.
[Abstract, Sec. 3.1, Eqs. (3.8)-(3.9)] The abstract describes Eqs. (3.8) and (3.9) as 'exact expressions', but the text states that the near-horizon recipe gives the exact result only for the scalar model and that the tensor case requires the near-horizon expansion. Eq. (3.9) is therefore approximate, and no error estimate is given. Please either limit the exactness claim to the scalar-model formula or quantify the accuracy of Eq. (3.9) by comparing with higher-order terms or with direct numerical solution of the full potential.

minor comments (5)

[Sec. 1] The term 'the so-called RGB' should be 'RBH'.
[Sec. 2.1, footnote 1] The d-dimensional redefinition A = ψ/r^{(d-2)/2} is stated only for the Ricci-coupled model; clarify whether the tensor-model reduction in Sec. 2.2 has an analogous form.
[Eqs. (3.12)-(3.13)] The generalized Laguerre function notation is incomplete (missing script L) and parentheses are unbalanced; please rewrite these equations for readability.
[Fig. 4 caption] 'Comparying' should be 'Comparing'.
[Sec. 3.3] 'General area quantization' overstates the result; the derivation applies to the near-horizon sector of spherically symmetric backgrounds and should be phrased accordingly.

Circularity Check

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No circularity found: the critical-coupling formulas follow from the stated near-horizon condition b=0, and the threshold identification is an external numerical premise, not a fitted input or self-referential reduction.

full rationale

The central derivation is self-contained. The paper starts from the full effective potentials (2.6) and (2.11), expands f(r) near the horizon as f(x)=2κx+..., substitutes into those potentials to obtain the near-horizon form V(x)≈ax²+bx with the explicit coefficients (3.4)-(3.7), and then imposes V'(0)=0, i.e. b=0, to solve for ζ_c in Eqs. (3.8) and (3.9). No constant in these formulas is fitted to instability data; the formulas are algebraic solutions of a stated horizon-locality condition. The identification of b=0 with the onset of tachyonic instability is an assumption supported by the authors' earlier numerical time-domain work [31,32] and by the plotted example in Fig. 1, but that is an evidential/conjectural link rather than a circular definition: the instability threshold is not inserted into the derivation, and the derivation does not presuppose the value of ζ_c. The purely imaginary mode result follows from the derived formula ω=-b/(2√-a)-iκ(n+1/2) evaluated at b=0, and the area-quantization result follows from Δω=κ; both are consequences, not inputs. The self-citations [31,32] provide prior numerical support and do not constitute a load-bearing self-citation chain, since the analytic expressions would stand or fall on the independent numerical check of the threshold criterion. Any deficiency in the threshold criterion is a correctness risk, not circularity.

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

No free parameters are fitted to data; μ, ℓ, θ, q, and M are model inputs. The load-bearing assumptions are the horizon-extremum threshold criterion and the correctness of the tensor-model effective potential. No new particles or entities are introduced.

assumptions (5)

domain assumption Test-field approximation: the non-minimally coupled scalar field does not backreact on the RBH background.
The actions (2.1) and (2.7) are solved on fixed black-hole metrics; no backreaction or self-consistency condition is imposed.
ad hoc to paper The instability threshold coincides with an extremum of the effective potential located exactly on the event horizon.
This is the central identification used to set b=0 in Eq. (3.3). It is supported by one numerical example but not proven as an exact condition for all cases.
domain assumption The near-horizon expansion f(x) = 2κx + (1/2)f''(r_h)x² + ... is sufficient to determine the threshold.
Used in Eq. (3.2) for all RBH; assumes the leading terms capture the relevant instability dynamics.
domain assumption The Regge-Wheeler effective potentials (2.6) and (2.11) are correct reductions of the scalar field equations.
Long calculations are omitted, especially for the tensor model, so the reader must accept the potential and auxiliary functions as correct.
domain assumption Maggiore's quantization relation Δω = |ω_I|_n - |ω_I|_{n-1} applies to the near-horizon modes.
Used in Sec. 3.3 to convert the QNM imaginary spacing into an area spectrum; this is an external result being imported.

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Cite this review

Pith. "Pith review of Analytical Criteria for Black Hole Instability in Non-Minimally Coupled Scalar-Tensor Theories." pith.science (2026). https://pith.science/paper/4OAL2KH7

@misc{pith2026260719755,
  author       = {Pith},
  title        = {Pith review of: Analytical Criteria for Black Hole Instability in Non-Minimally Coupled Scalar-Tensor Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OAL2KH7}},
  note         = {Machine review of arXiv:2607.19755}
}

read the original abstract

In this paper, we show that the robustness of black hole stability is not preserved when perturbations are subjected to critical values of the coupling constant in two non-minimally coupled scalar-tensor models, particularly for several regular black holes. Our main result is the derivation of a general near-horizon analytical criterion for the onset of instability, which yields analytical expressions for the critical coupling constants in both coupling models. Numerical time-domain analysis shows that these critical values are coincident with the threshold points of the field perturbation profiles. At the critical coupling, the effective potential of the Regge-Wheeler equation develops an extremum exactly at the location of event horizon. We further show that, in the near-horizon regime, the real parts of the quasi-normal frequencies vanish, giving rise to purely imaginary modes. Finally, we recover the universal area quantization of spherically symmetric black holes at the critical coupling without resorting to the highly damped-mode approximation, and show that the result is independent of the particular non-minimal coupling model.