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arxiv: 1907.06168 · v1 · pith:3FIODWDRnew · submitted 2019-07-14 · 🌀 gr-qc

Hairy black hole stability under odd parity perturbations in the Einstein-Gauss-Bonnet model

Gilberto Aguilar-P\'erez , Miguel Cruz , Samuel Lepe , Israel Moran-Rivera This is my paper

Pith reviewed 2026-05-24 22:05 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole stabilityEinstein-Gauss-Bonnet gravityhairy black holesodd parity perturbationsRegge-Wheeler potentialscalar field profilegeneral relativity
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The pith

The stability of a hairy black hole in Einstein-Gauss-Bonnet gravity under odd parity perturbations depends on a parameter in the scalar field profile.

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

The authors construct a black hole solution in the Einstein-Gauss-Bonnet model sourced by a regular scalar field that recovers the Schwarzschild solution for appropriate choices. They linearize the equations under odd parity perturbations and derive the explicit Regge-Wheeler potential, then impose the condition that perturbations vanish at the horizon. Stability follows if this potential satisfies a positivity condition everywhere outside the horizon. The analysis shows that this positivity holds only for certain values of the free parameter appearing in the chosen scalar profile.

Core claim

Supported by the use of a regular scalar field we find a black hole solution in the Einstein-Gauss-Bonnet model. From the obtained solution we can recover the Schwarzschild black hole as in other works. Later, by implementing the odd parity perturbations method we study the stability of the linearized equations of motion of the model, we find the explicit form of the Regge-Wheeler potential and we explore the condition of vanishing perturbations at the horizon of the black hole. We test the stability of the obtained solution by checking the positivity condition of the Regge-Wheeler potential. Finally, we show that the stability of model depends on the value of a parameter introduced in the 1

What carries the argument

The Regge-Wheeler potential obtained after linearizing the Einstein-Gauss-Bonnet equations under odd parity perturbations of the metric and scalar field.

If this is right

  • The Schwarzschild solution is recovered as a special case of the hairy black hole when the scalar profile parameter takes a limiting value.
  • Perturbations are required to vanish at the horizon to maintain regularity of the linearized solution.
  • Stability is not automatic but is controlled by the value of the free parameter in the scalar profile.
  • The positivity test on the Regge-Wheeler potential provides a concrete criterion for selecting viable parameter values.

Where Pith is reading between the lines

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

  • The same parameter dependence may appear when even-parity perturbations are examined in the same model.
  • Other regular scalar profiles could be tested numerically to check whether the stability window remains similar.
  • The result supplies a concrete selection rule that could be applied when constructing hairy black hole solutions in related higher-curvature theories.

Load-bearing premise

The specific functional form chosen for the regular scalar field profile is representative of the profiles that can source stable hairy black holes.

What would settle it

An explicit evaluation of the Regge-Wheeler potential for a chosen value of the scalar parameter that yields a negative region outside the horizon would demonstrate instability.

Figures

Figures reproduced from arXiv: 1907.06168 by Gilberto Aguilar-P\'erez, Israel Moran-Rivera, Miguel Cruz, Samuel Lepe.

Figure 1
Figure 1. Figure 1: FIG. 1: Behavior of the [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
read the original abstract

Supported by the use of a regular scalar field we find a black hole solution in the Einstein-Gauss-Bonnet model. From the obtained solution we can recover the Schwarzschild black hole as in other works. Later, by implementing the odd parity perturbations method we study the stability of the linearized equations of motion of the model, we find the explicit form of the Regge-Wheeler potential and we explore the condition of vanishing perturbations at the horizon of the black hole. We test the stability of the obtained solution by checking the positivity condition of the Regge-Wheeler potential. Finally, we show that the stability of model depends on the value of a parameter introduced in the profile for the scalar field.

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 / 0 minor

Summary. The paper obtains an exact hairy black hole solution in Einstein-Gauss-Bonnet gravity by positing a specific regular scalar-field profile containing a free parameter, from which the Schwarzschild solution is recovered as a special case. It derives the Regge-Wheeler potential for odd-parity linear perturbations about this background, imposes the condition of vanishing perturbations at the horizon, and concludes that stability (positivity of the potential) depends on the value of the parameter introduced in the scalar profile.

Significance. If the explicit derivation of the potential and the positivity check are correct, the work supplies a concrete example of a hairy black-hole solution in EGB theory together with its odd-parity stability analysis. The provision of an explicit Regge-Wheeler potential is a positive feature. However, because the stability outcome is controlled by a modeling choice introduced by hand, the result illustrates dependence on an ansatz rather than a generic property of the theory.

major comments (2)
  1. [Abstract] Abstract: the claim that 'the stability of model depends on the value of a parameter introduced in the profile for the scalar field' is obtained for a single, hand-chosen functional form of the scalar profile. The positivity test of the Regge-Wheeler potential therefore reduces, by construction, to a condition on this fitted modeling choice rather than an independent prediction of the background solution.
  2. The manuscript does not examine whether other regular scalar profiles capable of supporting hairy solutions in the same theory produce the same parameter dependence or the same stability conclusion, leaving open the possibility that the reported dependence is an artifact of the particular ansatz.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and the comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'the stability of model depends on the value of a parameter introduced in the profile for the scalar field' is obtained for a single, hand-chosen functional form of the scalar profile. The positivity test of the Regge-Wheeler potential therefore reduces, by construction, to a condition on this fitted modeling choice rather than an independent prediction of the background solution.

    Authors: We agree that the reported stability condition applies specifically to the scalar-field profile introduced in the paper. The abstract has been revised to state explicitly that the parameter dependence is obtained for this ansatz. The work is intended as a concrete example of an exact hairy solution in EGB gravity together with its odd-parity stability analysis, rather than a claim of generality across all possible profiles. revision: yes

  2. Referee: The manuscript does not examine whether other regular scalar profiles capable of supporting hairy solutions in the same theory produce the same parameter dependence or the same stability conclusion, leaving open the possibility that the reported dependence is an artifact of the particular ansatz.

    Authors: It is correct that we have not studied alternative regular scalar profiles. Our construction begins from one chosen regular profile that yields an exact solution recovering the Schwarzschild limit, after which the Regge-Wheeler potential is derived and its positivity examined. While other profiles could in principle lead to different conclusions, the present analysis demonstrates the existence of at least one such stable hairy solution under the stated condition on the parameter. revision: no

standing simulated objections not resolved
  • Whether the same parameter dependence and stability conclusion hold for other regular scalar profiles in the theory.

Circularity Check

1 steps flagged

Stability result controlled by parameter introduced in chosen scalar-field profile ansatz

specific steps
  1. self definitional [Abstract (final sentence) and scalar-field profile section]
    "Finally, we show that the stability of model depends on the value of a parameter introduced in the profile for the scalar field."

    The parameter is introduced by the authors when they select the functional form of the regular scalar-field profile used to generate the background solution. The subsequent stability test (positivity of the derived Regge-Wheeler potential) is then shown to hinge on that same parameter, so the reported dependence is a direct consequence of the modeling choice rather than a prediction external to the ansatz.

full rationale

The paper obtains the hairy black-hole solution by positing one specific regular scalar-field profile that contains a free parameter. It then derives the odd-parity Regge-Wheeler potential for that background and reports that positivity (hence stability) depends on the value of the same parameter. Because the dependence is exhibited only for the authors' chosen functional form and no other regular profiles are examined, the central stability claim reduces by construction to a property of the input ansatz rather than an independent result about the Einstein-Gauss-Bonnet model. This is a partial circularity (score 6) but does not render the entire derivation tautological; the explicit potential and positivity test still constitute non-trivial calculations once the profile is fixed.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the chosen scalar profile is regular and that the odd-parity sector captures the relevant instability channel. One free parameter is introduced by the profile choice. No new entities are postulated beyond the standard scalar field.

free parameters (1)
  • scalar field profile parameter
    A numerical parameter appearing in the scalar field ansatz whose value determines whether the Regge-Wheeler potential remains positive.
axioms (2)
  • domain assumption The background is a static spherically symmetric solution of the Einstein-Gauss-Bonnet equations with a regular scalar field.
    Invoked when the solution is constructed and when perturbations are linearized around it.
  • domain assumption Odd-parity perturbations decouple from even-parity ones and the stability is fully determined by the sign of the resulting potential.
    Standard assumption in black-hole perturbation theory, used to reduce the problem to a single master equation.

pith-pipeline@v0.9.0 · 5654 in / 1474 out tokens · 18304 ms · 2026-05-24T22:05:18.390567+00:00 · methodology

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