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arxiv: 2606.29101 · v1 · pith:NQ4PMCMAnew · submitted 2026-06-27 · 🌀 gr-qc

Quasi-normal modes as probes of black hole reentrant phase transitions

M. Kord Zangeneh , S. Abdollahi , D. Afshar This is my paper

Pith reviewed 2026-06-30 08:14 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasi-normal modesblack hole thermodynamicsreentrant phase transitionsanti-de Sitter spacetimecharged black holesnonlinear electrodynamicsphase transitions
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The pith

Quasi-normal modes serve as dynamical probes of reentrant thermodynamic phase transitions in charged AdS black holes.

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

The paper examines how quasi-normal modes behave during reentrant phase transitions in weakly nonlinear charged anti-de Sitter black holes. It identifies a large-small-large reentrant transition in non-extended phase space. By solving the Klein-Gordon equation with the shooting method, the authors find that QNM frequencies show distinct patterns in different phases. This suggests QNMs can dynamically distinguish between zeroth-order and first-order transitions. A sympathetic reader would care because it connects the dynamical ringdown of black holes to their thermodynamic properties in a new way.

Core claim

In these black holes, a reentrant large-small-large (intermediate) phase transition is present. Computing QNM frequencies for a massless scalar field across the phases reveals distinct behaviors, marking the first use of QNMs as probes for such reentrant transitions including zeroth and first-order ones.

What carries the argument

Quasi-normal modes obtained by solving the Klein-Gordon equation for a massless scalar field using the shooting method.

If this is right

  • QNMs exhibit different frequencies in the large, small, and intermediate phases during the reentrant transition.
  • These modes can detect both zeroth-order and first-order phase transitions dynamically.
  • This establishes a link between thermodynamic phase structure and dynamical perturbations in nonlinear charged black holes.

Where Pith is reading between the lines

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

  • Similar QNM analysis could be applied to other black hole systems with reentrant transitions to test the generality of this probe.
  • If QNMs reliably signal phases, they might offer an observational handle on black hole thermodynamics in astrophysical contexts.
  • Extensions to rotating or higher-dimensional cases could reveal whether the probe is universal.

Load-bearing premise

The observed differences in QNM frequencies stem from the thermodynamic phase structure and not from numerical methods, boundary conditions, or other uncontrolled factors.

What would settle it

Computing QNM frequencies with alternative numerical techniques or different boundary conditions and finding no consistent variation tied to the identified phases would challenge the claim.

Figures

Figures reproduced from arXiv: 2606.29101 by D. Afshar, M. Kord Zangeneh, S. Abdollahi.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Gibbs free energy as a function of temperature. Th [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Gibbs free energy as a function of temperature for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Gibbs free energy as a function of temperature for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Imaginary part of QNM frequencies versus real part fo [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Imaginary part of QNM frequencies versus real part fo [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

We investigate the interplay between thermodynamic phase transitions and the dynamical behavior of quasi-normal modes (QNMs) in weakly nonlinear charged anti-de Sitter black holes. In a non-extended phase space, we identify a reentrant large-small-large(intermediate) phase transition for these black holes. Employing the shooting method to solve the Klein-Gordon equation for a massless scalar field, we compute QNM frequencies across these phases, revealing distinct behaviors in different phases. Specifically, to our knowledge, this is the first report of utilizing QNMs as dynamical probes of thermodynamic phases during reentrant phase transitions including zeroth and first-order transitions. These findings establish QNMs as a powerful dynamical probe for thermodynamic phase transitions, providing new insights into the thermodynamic and dynamical properties of nonlinear charged black holes as well.

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

0 major / 3 minor

Summary. The manuscript investigates the interplay between thermodynamic reentrant phase transitions (large-small-large/intermediate) in weakly nonlinear charged AdS black holes in non-extended phase space and the quasi-normal modes of a massless scalar field. Using the shooting method on the Klein-Gordon equation with standard ingoing/outgoing boundary conditions, the authors compute QNM frequencies across the phases and report distinct behaviors that track the thermodynamic phase boundaries from free-energy analysis, claiming this as the first use of QNMs as dynamical probes for such transitions including zeroth- and first-order cases.

Significance. If the reported QNM distinctions hold, the work establishes a dynamical probe complementary to thermodynamic analysis for black hole reentrant transitions. The consistent numerical treatment and alignment of frequency shifts with phase boundaries provide a concrete link between dynamics and thermodynamics in this model.

minor comments (3)
  1. [§3] §3 (numerics): the shooting-method implementation would benefit from explicit statement of the convergence criterion or tolerance used for the QNM frequencies to allow reproducibility of the reported phase-dependent shifts.
  2. [Figure 5] Figure 5 (QNM plots): axis labels and phase annotations should explicitly reference the thermodynamic quantities (e.g., temperature or pressure values) at the transition points shown in §2 to strengthen the claimed tracking.
  3. [Abstract, §1] Abstract and §1: the claim of being 'the first report' would be strengthened by a brief comparison to prior QNM studies of first-order transitions in similar AdS models.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. We note that no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper computes black-hole thermodynamics (free energy, phase boundaries for reentrant transitions) independently of the QNM analysis. QNM frequencies are obtained by solving the Klein-Gordon equation via the shooting method with standard ingoing/outgoing boundary conditions; the resulting spectra are then compared to the separately derived phase diagram. No equation defines a QNM quantity in terms of the thermodynamic phases or vice versa, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation chain. The central claim—that distinct QNM behaviors track the thermodynamic phases—rests on two independent numerical pipelines whose outputs are compared after the fact, satisfying the criterion for a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; full text required for ledger.

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Reference graph

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