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arxiv: 2603.10461 · v2 · pith:XRZC4LNUnew · submitted 2026-03-11 · 🌀 gr-qc

Gauss-Bonnet scalarization of charged qOS-black holes

Hong Guo , Wontae Kim , Yun Soo Myung This is my paper

Pith reviewed 2026-05-21 12:20 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Gauss-Bonnet scalarizationcharged qOS black holesnonlinear electrodynamicsscalar hairlinear stabilityaction parameter
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The pith

Gauss-Bonnet coupling produces scalarized charged qOS black holes that remain linearly stable under perturbations.

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

The paper studies spontaneous scalarization of charged quantum Oppenheimer-Snyder black holes inside Einstein-Gauss-Bonnet-scalar gravity supplemented by nonlinear electrodynamics. With the specific quadratic coupling f(φ)=2λ φ², scalar hair appears in two regimes: a GB+ branch when the action parameter vanishes and λ is positive, and a GB- branch when α lies in a narrow interval and λ is negative. The authors construct explicit single-branch solutions for the GB- case and show that the scalar profile falls off faster than in the GB+ case. Linear stability analysis under scalar perturbations confirms that the resulting black holes do not develop growing modes.

Core claim

In Einstein-Gauss-Bonnet-scalar theory with nonlinear electrodynamics, charged qOS black holes admit scalarized solutions triggered by the Gauss-Bonnet term when the coupling is quadratic, f(φ)=2λ φ². These solutions exist as GB+ configurations for α=0 and λ>0, and as GB- configurations for 3.5653≤α≤4.6875 with λ<0. The scalarized black holes form a single branch in which the scalar field decays rapidly, and stability analysis establishes that they are linearly stable against scalar perturbations.

What carries the argument

The quadratic scalar-GB coupling f(φ)=2λ φ², which activates scalarization when the sign of λ and the value of the action parameter α satisfy the conditions for either the GB+ or GB- branch in the cqOS metric background.

If this is right

  • Scalarized cqOS black holes exist as single-branch solutions for both GB+ and GB- cases under the stated sign and range conditions on λ and α.
  • The scalar field profile decays much more rapidly in the GB- solutions than in the GB+ solutions.
  • Linear stability holds for scalar perturbations of the constructed scalarized black holes.

Where Pith is reading between the lines

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

  • The rapid decay of the scalar field in the GB- branch may suppress certain long-range effects that could otherwise appear in observations of black-hole environments.
  • The narrow window of allowed α values for GB- scalarization implies that only a limited subset of the underlying action-parameter space yields new black-hole solutions.

Load-bearing premise

The quadratic form of the scalar-GB coupling together with the assumed metric ansatz for the cqOS background is sufficient to produce the scalarized solutions and their stability.

What would settle it

A direct numerical integration of the linearized scalar perturbation equations around the constructed GB- solutions that reveals at least one mode with positive imaginary frequency would falsify the linear-stability result.

Figures

Figures reproduced from arXiv: 2603.10461 by Hong Guo, Wontae Kim, Yun Soo Myung.

Figure 1
Figure 1. Figure 1: (a) Two outer/inner horizons r±(M = 1, α, P = 0.6) are functions of α ∈ [0, 4.6875]. The bouncing radius rb(1, α, 0.6) as function of α is inside the inner horizon. Here, r+(1, α, 0.6) involves the Davies point at [(2.29,1.88), black dot], critical onset point [(3.5653,1.7676), purple dot], and extremal point [(4.6875,1.5), red dot]. (b) Two horizons r±(M, α = 1, P = 0.6) are functions of M ∈ [Mrem = 0.679… view at source ↗
Figure 2
Figure 2. Figure 2: Heat capacity C(M, α, P)/|CS(1, 0, 0)| with |CS(1, 0, 0)| = 25.13 and temperature T(M, α, P)/0.04. (a) Heat capacity C(M = 1, α, P = 0.6) blows up at Davies point (αD = 2.29, •) and it is zero at the extremal point (αe = 4.6875, red dot). The shaded region represents C ≥ 0 and it is divided at a line (α = αc = 3.5653). (b) Heat capacity C(M, 1, 0.6) blows up at Davies point (MD = 0.81, •). The heat capacit… view at source ↗
Figure 3
Figure 3. Figure 3: (Left) R¯ 2 GB(r, M = 1, α, P = 0.6) as functions of r ∈ [r+(M = 1, α, P = 0.6), 2.2] and α ∈ [0, αe = 4.6875] and its zero line is available, starting from α = αc(= 3.5653) on the horizon. (Right) R¯ 2 GB(r, M, α = 1, P = 0.6) as functions of r ∈ [r+(M, α = 1, P = 0.6), 1.8] and M ∈ [Mrem = 0.6796, 0.81] and its zero line is available, ending at M = Mc(= 0.7277) on the horizon. From Eq.(21), we find the r… view at source ↗
Figure 4
Figure 4. Figure 4: Three different curves for nc(Me, αe, 0.6) = 0, dc(MD, αD, 0.6) = 0, and rc(Mc, αc, 0.6) = 0 for M ∈ [0.6796, 1.2] and α ∈ [0, 4.6875], including extremal/remnant points (red dot), two Davies points (black dot), and two resonance points (purple dot) for M = 1 and α = 1. (a) αe=4.6875 αsc λ≫M,P=4.5032 αc=3.5653 αsc(M=1,λ,P=0.6) 1 10 100 1000 104 105 3.6 3.8 4.0 4.2 4.4 4.6 4.8 -λ α (b) Mrem =0.6796 Msc λ≫α,… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Sufficient condition of αsc(M = 1, λ, P = 0.6) with the critical onset parameter α = αc as the lower bound. A dashed line denotes the sufficient condition α = α −λ≫M,P sc . A top line denotes the extremal point at α = αe as the upper bound. (b) Graph for Msc(α = 1, λ, P = 0.6) with the critical onset mass M = Mc as the upper bound and the sufficient condition M = M−λ≫α,P sc . A bottom line represents t… view at source ↗
Figure 6
Figure 6. Figure 6: Scalarized black holes obtained from GB+ scalarization with P = 0, α = 0, λ = 1.1, r+ = 2.37 for (a) ψ0 = 0.1 and (b) ψ0 = 0.2 in the fundamental branch. To obtain the scalarized black hole solution in the n = 0 branch, we wish to set ψ∞ = 0 and A(∞) = B(∞) as the shooting condition to solve Eqs. (28)-(30) numerically. For λ = 1.1 > λ0, we select ψ0 = 0.1 and 0.2 to represent the radial profiles for metric… view at source ↗
Figure 7
Figure 7. Figure 7: Scalarized cqOS black holes obtained from GB [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Horizon radius r+ as a function of scalar constant ψ0. (a) λ=-10 λ=-13 λ=-15 λ=-17 λ=-20 0.000 0.005 0.010 0.015 0.020 0.025 0.023 0.024 0.025 0.026 0.027 0.028 ψ0 TH P=0.6, α=4.6 (b) λ=-10 λ=-13 λ=-15 λ=-17 λ=-20 0.000 0.005 0.010 0.015 0.020 0.025 0.7 0.8 0.9 1.0 ψ0 S W P=0.6, α=4.6 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Hawking temperature TH and (b) Wald entropy SW as functions of the constant scalar ψ0 for different coupling constants (λ = −10 ∼ −20). constant continues to grow, we observe that the horizon radius begins to increase slowly. In contrast to the initial stage, the magnitude of this increase becomes more significant with increasing coupling strength. This nonmonotonic relation directly influences the beh… view at source ↗
Figure 10
Figure 10. Figure 10: Radial profiles of effective potential under scalar perturbation for different (a) [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: QNMs frequency. (a) real part ωR and (b) imaginary part ωR as the function of scalar constant ψ0 for different coupling constants λ. Any single-barrier potential typically predicts stability of the scalarized cqOS-black holes solution under scalar perturbations. To further substantiate this conclusion, we have to compute the quasinormal modes (QNMs) frequency. By imposing purely ingoing boundary condition… view at source ↗
Figure 12
Figure 12. Figure 12: (a) Hawking temperature TH and (b) Wald entropy SW as functions of the horizon radius r+ for different coupling constants (λ = −10 ∼ −20). (a) λ=-10 λ=-13 λ=-15 λ=-17 λ=-20 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 0.50 0.55 0.60 0.65 0.70 r+ ω R P=0.6, α=4.6 (b) λ=-10 λ=-13 λ=-15 λ=-17 λ=-20 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 -0.150 -0.148 -0.146 -0.144 -0.142 -0.140 r+ ωI P=0.6, α=4.6 [PITH_FULL_IMA… view at source ↗
Figure 13
Figure 13. Figure 13: QNMs frequency. (a) real part ωR and (b) imaginary part ωR as the function of horizon radius r+ for different coupling constants λ. still the stability of scalarized cqOS black holes obtained from GB− scalarization. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
read the original abstract

The Gauss-Bonnet (GB) scalarization for charged quantum Oppenheimer-Snyder (cqOS)-black holes is investigated in the Einstein-Gauss-Bonnet-scalar theory with the nonlinear electrodynamics (NED) term. Here, the scalar coupling function to GB term is given by $f(\phi)=2\lambda \phi^2$ with a coupling constant $\lambda$. Three parameters of mass ($M$), action parameter ($\alpha$), and magnetic charge ($P$) are necessary to describe the cqOS-black hole, and it may become the qOS-black hole when $P=M$. The GB scalarization of cqOS-black holes comes into two cases GB$^\pm$, depending on the sign of GB term which triggers the different phenomena. For $\alpha=0$ and $\lambda>0$, GB$^+$ scalarization is allowed, while for $\alpha\not=0$ and $\lambda<0$, GB$^-$ scalarization appears for a narrow band of $3.5653\le \alpha\le 4.6875$. After discussing the onset GB$^-$ scalarization, we construct scalarized cqOS-black holes which belong to the single branch. The scalar field decays much more rapidly compared to the GB$^+$ case. Stability analysis shows these scalarized black holes are linearly stable under scalar perturbations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates Gauss-Bonnet scalarization of charged quantum Oppenheimer-Snyder (cqOS) black holes in Einstein-Gauss-Bonnet-scalar theory with nonlinear electrodynamics. With the coupling f(φ)=2λ φ², it reports GB+ scalarization for α=0 and λ>0, and GB- scalarization for nonzero α and λ<0 within the narrow interval 3.5653 ≤ α ≤ 4.6875. Single-branch scalarized solutions are constructed numerically, the scalar field is found to decay rapidly (more so than in the GB+ case), and a linear stability analysis under scalar perturbations concludes that these solutions are stable.

Significance. If the numerical constructions and stability results hold, the work supplies explicit examples of scalarized black holes triggered by the sign of the Gauss-Bonnet term in a charged qOS background. The narrow α window for GB- scalarization and the rapid scalar decay are distinctive features that differentiate this branch from the GB+ case. The linear stability claim, once verified, would support the physical relevance of these solutions. The paper's numerical approach to solving the coupled system for given M, α, and P is a concrete contribution to the scalarization literature.

major comments (2)
  1. [Stability analysis] Stability analysis (following construction of solutions): The claim that the GB- scalarized cqOS solutions are linearly stable rests on numerical solution of the linearized second-order ODE for δφ on the background metric. In the narrow band 3.5653 ≤ α ≤ 4.6875, where the scalar field decays rapidly, the manuscript provides no reported convergence tests with respect to radial discretization, horizon boundary conditions, or asymptotic matching. Small variations in these choices can change the sign of the lowest eigenvalue, so the absence of unstable modes requires explicit verification to be load-bearing for the stability conclusion.
  2. [Onset of GB- scalarization] Onset of GB- scalarization and parameter bounds: The narrow interval 3.5653 ≤ α ≤ 4.6875 for λ < 0 is stated as the region where GB- scalarization occurs. The manuscript should specify the precise numerical criterion (e.g., the value of the effective mass squared or the bifurcation condition from the bald solution) used to obtain these decimal-place bounds, as they directly delimit the existence domain of the reported single-branch solutions.
minor comments (2)
  1. [Abstract] Abstract: The statement that the solution 'may become the qOS-black hole when P=M' would benefit from a one-sentence reminder of the underlying qOS metric ansatz and the physical meaning of the magnetic charge P.
  2. [Notation] Notation: The three parameters M, α, and P are introduced, but ensuring that the metric functions and the scalar field φ are denoted consistently between the field equations and the numerical sections would reduce potential confusion for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript to strengthen the presentation of our numerical results.

read point-by-point responses

  1. Referee: Stability analysis (following construction of solutions): The claim that the GB- scalarized cqOS solutions are linearly stable rests on numerical solution of the linearized second-order ODE for δφ on the background metric. In the narrow band 3.5653 ≤ α ≤ 4.6875, where the scalar field decays rapidly, the manuscript provides no reported convergence tests with respect to radial discretization, horizon boundary conditions, or asymptotic matching. Small variations in these choices can change the sign of the lowest eigenvalue, so the absence of unstable modes requires explicit verification to be load-bearing for the stability conclusion.

    Authors: We agree that explicit convergence tests were not reported in the original manuscript. In the revised version we will add a dedicated subsection on the numerical implementation of the linearized perturbation equation. This will include: (i) results for three successively refined radial grids (e.g., 200, 400, 800 points) showing that the lowest eigenvalue converges to within 0.1 %; (ii) tests varying the horizon boundary condition by ±1 % and confirming the eigenvalue sign remains unchanged; (iii) checks on the asymptotic matching radius and the decay of the perturbation. These additional verifications support the absence of unstable modes for the reported single-branch solutions. revision: yes

  2. Referee: Onset of GB- scalarization and parameter bounds: The narrow interval 3.5653 ≤ α ≤ 4.6875 for λ < 0 is stated as the region where GB- scalarization occurs. The manuscript should specify the precise numerical criterion (e.g., the value of the effective mass squared or the bifurcation condition from the bald solution) used to obtain these decimal-place bounds, as they directly delimit the existence domain of the reported single-branch solutions.

    Authors: The quoted interval was obtained by solving the linearized scalar equation on the bald cqOS background and locating the values of α at which a normalizable zero-frequency mode first appears (i.e., the bifurcation point from the bald solution). The precise numerical criterion is the vanishing of the lowest eigenvalue of the radial Sturm–Liouville problem for the perturbation, which is equivalent to the effective mass squared becoming negative in a finite radial interval while still allowing a regular, asymptotically decaying solution. We will insert a short paragraph in the revised manuscript that states this criterion explicitly, together with the shooting tolerance (10^{-8}) and the range of trial α values used to bracket the boundaries to the reported four-decimal precision. revision: yes

Circularity Check

0 steps flagged

No circularity: solutions and stability obtained by direct numerical integration of field equations

full rationale

The paper solves the Einstein-Gauss-Bonnet-scalar equations with the explicit quadratic coupling f(φ)=2λφ² and the cqOS metric ansatz to construct the scalarized backgrounds for the reported narrow α interval; linear stability then follows from solving the independent linearized perturbation ODE on those backgrounds. No step reduces a claimed prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported via self-citation, and the single-branch solutions are outputs of the numerical solver rather than tautological redefinitions of the inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The construction rests on the Einstein-Gauss-Bonnet-scalar field equations with a fixed quadratic coupling, the assumed static spherically symmetric metric ansatz for the cqOS background, and the nonlinear electrodynamics Lagrangian. No new particles or forces are introduced.

free parameters (3)
  • λ
    Coupling constant in f(φ)=2λ φ²; sign and magnitude control which branch of scalarization appears.
  • α
    Action parameter of the cqOS metric; scalarization occurs only inside a narrow interval when λ<0.
  • P
    Magnetic charge; reduces to qOS case when P=M.
axioms (2)
  • standard math Einstein equations hold with the GB term and scalar stress-energy
    Invoked throughout the derivation of the field equations.
  • domain assumption Static spherically symmetric metric ansatz for the background
    Used to reduce the PDEs to ODEs for the scalar profile.

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    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjp/i2017-11825-9 2017