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arxiv: 2606.21332 · v1 · pith:HUTAY6TXnew · submitted 2026-06-19 · 🌀 gr-qc

Thermal Stability and QNMs of a Hairy Black Hole in the Presence of a Monopole Field

George Koutsoumbas , Andri Machattou , Eleftherios Papantonopoulos This is my paper

Pith reviewed 2026-06-26 13:38 UTC · model grok-4.3

classification 🌀 gr-qc
keywords hairy black holesdilaton fieldmonopole fieldthermal stabilityquasi-normal modesblack hole stabilityGHS-GM solution
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The pith

The thermal stability of a hairy black hole with dilaton and monopole increases as the dilaton scalar charge decreases, while larger black hole charges reduce the magnitude of quasi-normal mode frequencies.

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

This paper studies the stability of a generalization of the GHS-GM black hole that incorporates a dilaton field and a monopole. Thermal stability improves when the scalar charge of the dilaton is lowered. As the black hole charge grows, both the real and imaginary parts of the quasi-normal frequencies shrink in absolute value. Overtone modes decay more rapidly than fundamental ones, and the absence of positive imaginary parts signals that the system remains stable.

Core claim

The authors examine a hairy black hole metric that extends the GHS-GM solution through the addition of a dilaton and a monopole field. Thermodynamic quantities are computed to determine thermal stability, and the equations for linear perturbations are solved to extract quasi-normal modes. The thermal behavior depends on the dilaton scalar charge such that smaller values increase stability, and increasing the black hole charge causes both real and imaginary parts of the quasi-normal frequencies to decrease in absolute value, with overtones damping faster and no unstable modes appearing.

What carries the argument

The generalized GHS-GM black hole metric that includes a dilaton scalar charge and a monopole field, from which thermodynamic potentials and quasi-normal mode equations are derived.

If this is right

  • Thermal stability rises when the dilaton scalar charge is reduced.
  • Both real and imaginary parts of quasi-normal frequencies shrink as black hole charge increases.
  • Overtone modes decay faster than the fundamental modes.
  • No positive imaginary parts appear, confirming the absence of unstable modes.

Where Pith is reading between the lines

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

  • The parameter dependence may limit allowable scalar charges in dynamical black hole formation scenarios.
  • The faster damping of overtones could shape the late-time ringdown phase in gravitational wave signals from similar objects.
  • Analogous trends might appear when the same stability methods are applied to other Einstein-dilaton-monopole solutions.

Load-bearing premise

The chosen generalization of the GHS-GM metric with dilaton and monopole provides a consistent background where standard thermodynamic and perturbation methods apply without extra instabilities.

What would settle it

A computation of quasi-normal modes that yields a positive imaginary part for any overtone or fundamental mode at large black hole charge would falsify the stability conclusion.

Figures

Figures reproduced from arXiv: 2606.21332 by Andri Machattou, Eleftherios Papantonopoulos, George Koutsoumbas.

Figure 1
Figure 1. Figure 1: depicts the temperature versus r+ for r− = 2. The panel on the left represents c = −1, the middle one represents c = 0 and the right panel corresponds to c = +1. Within each panel the highest curves are the ones with a−+1; then follow the ones for a = 0 and a = −1. According to the figure the three values of c examined yield similar curves, with just quantitative differences. It does not seem worth insisti… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Heat capacity versus [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Left panel: Temperature versus [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Quasinormal frequencies [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Quasinormal frequencies [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

We study the stability of a generalization of the GHS-GM black hole in the presence of a dilaton and a monopole field. We find that the thermal behaviour of system depends on the scalar charge of the dilaton field and as this parameter is decreasing the system becomes more thermally stable. We also find that, as the charge of the black hole is increasing, both the real and the imaginary parts of the quasi-normal frequencies decrease in absolute value. The overtone modes die out faster than the fundamental modes and no positive imaginary parts appear, indicating the stabilty of the system

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

3 major / 2 minor

Summary. The manuscript studies the thermal stability and quasi-normal modes of a generalization of the GHS-GM black hole metric that includes a dilaton field and a monopole. It claims that the system's thermal behavior depends on the dilaton scalar charge, with decreasing scalar charge increasing thermal stability; that both real and imaginary parts of the QNMs decrease in absolute value as black-hole charge increases; that overtone modes decay faster than fundamental modes; and that the absence of positive imaginary parts indicates stability of the system.

Significance. If the metric is an exact solution to the field equations and the thermodynamic and perturbation calculations are correctly performed, the results would add concrete information on how scalar charge and monopole effects influence stability in hairy black-hole spacetimes. The work could be relevant to ongoing studies of black-hole thermodynamics and ringdown signals in theories with additional fields, but the abstract provides no indication of parameter-free derivations, machine-checked results, or falsifiable predictions that would strengthen its impact.

major comments (3)
  1. [Introduction / Metric Ansatz] The manuscript provides no explicit verification (e.g., substitution into the Einstein-dilaton-monopole equations or computation of the residual stress-energy tensor) that the chosen generalized GHS-GM ansatz is an exact solution. This check is load-bearing for every subsequent thermodynamic identity and for the Schrödinger-like equation used to extract QNMs.
  2. [Thermodynamic Analysis] The claim that thermal stability increases as the dilaton scalar charge decreases is stated without reference to the concrete diagnostic employed (sign of specific heat, sign of free-energy difference, or turning-point criterion) or to the explicit dependence on the charge parameter in the first law or Smarr relation.
  3. [Quasi-normal Mode Calculation] The QNM results (both Re(ω) and Im(ω) decreasing with black-hole charge, no positive Im(ω)) presuppose a well-defined effective potential and boundary conditions derived from the metric; without confirmation that the metric satisfies the background equations, the perturbation equation itself may contain spurious source terms.
minor comments (2)
  1. [Abstract] Typo in abstract: 'stabilty' should read 'stability'.
  2. [Abstract] The abstract does not specify the numerical method (e.g., continued fractions, WKB, or direct integration) or the range of overtone numbers used for the QNM spectrum.

Simulated Author's Rebuttal

3 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 accordingly to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Introduction / Metric Ansatz] The manuscript provides no explicit verification (e.g., substitution into the Einstein-dilaton-monopole equations or computation of the residual stress-energy tensor) that the chosen generalized GHS-GM ansatz is an exact solution. This check is load-bearing for every subsequent thermodynamic identity and for the Schrödinger-like equation used to extract QNMs.

    Authors: We agree that an explicit verification strengthens the foundation of the work. In the revised version we will add an appendix that substitutes the generalized GHS-GM ansatz into the Einstein-dilaton-monopole field equations, computes the residual, and confirms that the stress-energy tensor is consistent with the ansatz, thereby establishing it as an exact solution. revision: yes

  2. Referee: [Thermodynamic Analysis] The claim that thermal stability increases as the dilaton scalar charge decreases is stated without reference to the concrete diagnostic employed (sign of specific heat, sign of free-energy difference, or turning-point criterion) or to the explicit dependence on the charge parameter in the first law or Smarr relation.

    Authors: The stability conclusion is drawn from the sign of the specific heat capacity obtained via the first law. We will revise the thermodynamic section to state this diagnostic explicitly, supply the first-law and Smarr relations with the scalar-charge dependence shown, and indicate how decreasing scalar charge enlarges the region of positive specific heat. revision: yes

  3. Referee: [Quasi-normal Mode Calculation] The QNM results (both Re(ω) and Im(ω) decreasing with black-hole charge, no positive Im(ω)) presuppose a well-defined effective potential and boundary conditions derived from the metric; without confirmation that the metric satisfies the background equations, the perturbation equation itself may contain spurious source terms.

    Authors: The added metric verification (see first comment) will confirm the background. With that established, the linearized perturbation equations yield the standard Schrödinger-like form without extraneous sources. We will also add a short paragraph specifying the ingoing/outgoing boundary conditions used in the QNM computation. revision: yes

Circularity Check

0 steps flagged

No circularity: standard thermodynamic and QNM methods applied to assumed metric without self-referential reductions visible

full rationale

The abstract and provided context describe application of standard first-law thermodynamics and Schrödinger-like QNM equations to a generalized GHS-GM metric ansatz. No equations are quoted that define a quantity in terms of itself, rename a fit as a prediction, or reduce a central claim to a self-citation chain. The skeptic concern about metric consistency is a correctness/verification issue, not a circularity reduction per the enumerated patterns. The derivation chain is therefore treated as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The scalar charge is mentioned as a varying parameter but its status (fitted or input) cannot be determined.

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Reference graph

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