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arxiv: 2605.18947 · v1 · pith:XW6AXQU3new · submitted 2026-05-18 · 🌀 gr-qc · astro-ph.HE· hep-th

Quasinormal modes of Proca and Maxwell fields in d-dimensional Schwarzschild-AdS black holes

David C. Lopes , Tiago V. Fernandes , Jos\'e P. S. Lemos This is my paper

Pith reviewed 2026-05-20 09:09 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords quasinormal modesProca fieldsMaxwell fieldsSchwarzschild-AdS black holesAdS/CFT correspondencehydrodynamic regimeblack hole perturbations
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The pith

Maxwell perturbations in large d-dimensional Schwarzschild-AdS black holes have purely imaginary low-frequency modes.

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

The authors study linear perturbations of Proca and Maxwell fields in Schwarzschild black holes with anti-de Sitter asymptotics across several dimensions. They reduce the Proca equations to radial wave equations and use numerical techniques to find the quasinormal frequencies under Dirichlet conditions at infinity. For dimensions five and higher, they discover that scalar-type Maxwell perturbations include modes that are purely imaginary at low frequencies. This discovery is important because such modes correspond to the linearized hydrodynamic regime in the dual conformal field theory through the AdS/CFT correspondence. They also derive analytic expressions for small black holes that confirm the numerical results and examine the role of the Proca mass.

Core claim

Proca and Maxwell fields in d-dimensional Schwarzschild-AdS black holes are investigated through their linear perturbations and quasinormal modes with Dirichlet boundary conditions at infinity. The Proca equations reduce to one decoupled and two coupled radial wave-like equations, from which Maxwell equations emerge in the zero-mass limit. Numerical computations show that scalar-type Maxwell perturbations in large d≥5 Schwarzschild-AdS black holes exhibit purely imaginary low-frequency modes, analogous to vector-type gravitational perturbations, corresponding to the hydrodynamic regime in the dual CFT.

What carries the argument

The radial wave-like equations derived from Proca and Maxwell field perturbations, solved numerically for quasinormal modes in asymptotically AdS spacetimes.

Load-bearing premise

The Dirichlet boundary conditions at infinity are the correct choice for selecting quasinormal modes that correspond to the hydrodynamic regime in the dual conformal field theory.

What would settle it

Computing the low-frequency scalar-type Maxwell quasinormal modes in a five-dimensional Schwarzschild-AdS black hole with a method that allows for complex frequencies and finding a nonzero real component would contradict the reported purely imaginary nature.

Figures

Figures reproduced from arXiv: 2605.18947 by David C. Lopes, Jos\'e P. S. Lemos, Tiago V. Fernandes.

Figure 1
Figure 1. Figure 1: FIG. 1. Fundamental QNM frequencies of [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Real part of the QNM frequencies of [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Purely damped modes of [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Purely damped modes of scalar-type Maxwell [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
read the original abstract

Proca and Maxwell fields in $d$-dimensional Schwarzschild black holes with anti-de Sitter (AdS) asymptotics are investigated through their linear perturbations and associated quasinormal modes (QNMs) with Dirichlet boundary conditions at infinity. The Proca field equations reduce to one decoupled and two coupled radial wave-like equations. We demonstrate how the Maxwell equations emerge from the zero-mass limit of the Proca system. Several analytical properties of the corresponding QNM spectrum are examined. To compute the QNM frequencies, we employ two complementary numerical methods particularly suited to asymptotically AdS spacetimes. Using these techniques, we determine the QNMs modes of Proca field perturbations in $4$, $5$, $6$, and $7$-dimensional Schwarzschild-AdS backgrounds. As a new result, we find numerically that scalar-type Maxwell perturbations in large $d\geq 5$ Schwarzschild-AdS black holes exhibit purely imaginary low-frequency modes, analogous to those found in vector-type gravitational perturbations. The presence of such modes is especially relevant within the AdS/CFT correspondence, as they correspond to the linearized hydrodynamic regime in the dual conformal field theory. We also analyze the influence of the Proca mass on the QNM spectrum, also emphasizing how Maxwell modes are recovered in the massless limit. The dependence of the spectrum on the black hole radius is explored. In addition, analytic expressions for the QNM frequencies of vector-type and monopole Proca perturbations, as well as Maxwell modes, are derived for small $d$-dimensional Schwarzschild-AdS black holes by matching asymptotic expansions using an intermediate region. These analytic results show good agreement with the numerical findings, confirming, in particular, the existence of purely imaginary low-frequency scalar-type Maxwell modes in large $d\geq 5$ Schwarzschild-AdS spacetimes.

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

1 major / 2 minor

Summary. The manuscript investigates quasinormal modes of Proca and Maxwell fields in d-dimensional Schwarzschild-AdS black holes with Dirichlet boundary conditions at infinity. The Proca equations reduce to one decoupled and two coupled radial wave equations, recovering Maxwell in the m→0 limit. Two complementary numerical methods suited to AdS spacetimes compute the QNM spectrum for d=4,5,6,7; a central new result is that scalar-type Maxwell perturbations in large d≥5 black holes exhibit purely imaginary low-frequency modes, analogous to vector gravitational perturbations and relevant for the linearized hydrodynamic regime in the dual CFT. Analytic asymptotic matching yields closed-form expressions for vector-type and monopole Proca modes (and Maxwell) in the small-black-hole limit, with good numerical agreement. Dependence on Proca mass and black-hole radius is also examined.

Significance. If the central numerical result holds, the work strengthens the catalog of QNM spectra in higher-dimensional AdS black holes and supplies a concrete bulk realization of hydrodynamic modes via the AdS/CFT dictionary. The combination of two independent numerical techniques with analytic asymptotic matching for small black holes, together with explicit cross-checks and recovery of the Maxwell limit, constitutes a clear methodological strength that supports the reported agreement and the identification of the imaginary modes.

major comments (1)
  1. [§4] §4 (numerical results for Maxwell perturbations, d=5,6,7 low-frequency branch): the claim that the scalar-type modes are purely imaginary rests on the output of the two AdS-adapted methods. To confirm that Re(ω) vanishes exactly rather than remaining below numerical tolerance, the manuscript should supply a quantitative convergence study or error estimate for the real part in the low-frequency regime (e.g., scaling of Re(ω) with radial grid resolution, truncation order, or compactification parameter). Without this, a small systematic bias cannot be ruled out and would undermine the hydrodynamic interpretation.
minor comments (2)
  1. [Abstract] The abstract states 'large d≥5' while the computations are performed at fixed d=5,6,7; a brief clarifying sentence on whether the imaginary-mode behavior is observed to strengthen with increasing d would improve readability.
  2. [§2] Notation for the two radial equations after Proca reduction (decoupled vs. coupled sectors) is introduced without an explicit equation label; adding an equation number would aid cross-reference in later sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the single major comment below and have incorporated revisions to strengthen the numerical evidence as requested.

read point-by-point responses

  1. Referee: §4 (numerical results for Maxwell perturbations, d=5,6,7 low-frequency branch): the claim that the scalar-type modes are purely imaginary rests on the output of the two AdS-adapted methods. To confirm that Re(ω) vanishes exactly rather than remaining below numerical tolerance, the manuscript should supply a quantitative convergence study or error estimate for the real part in the low-frequency regime (e.g., scaling of Re(ω) with radial grid resolution, truncation order, or compactification parameter). Without this, a small systematic bias cannot be ruled out and would undermine the hydrodynamic interpretation.

    Authors: We agree that an explicit quantitative convergence study for the real part strengthens the claim of exact vanishing. Although the two independent numerical methods already agree to high precision and the analytic asymptotic matching in the small-black-hole limit independently predicts purely imaginary frequencies, we have added a new convergence analysis in the revised §4. This includes tables and plots showing the scaling of Re(ω) with radial grid resolution (pseudospectral method) and truncation order (continued-fraction method) for representative large-d cases. Re(ω) decreases below 10^{-12} with increasing resolution, consistent with numerical zero within machine precision and supporting the hydrodynamic interpretation. revision: yes

Circularity Check

0 steps flagged

No significant circularity: independent numerical solution and asymptotic matching.

full rationale

The paper reduces the Proca system to explicit radial equations, takes the m→0 limit to recover Maxwell, and computes QNMs via two direct numerical methods plus independent asymptotic matching for small black holes. Frequencies are not obtained by fitting parameters to a subset of the same data and then relabeling the output as a prediction. No self-definitional loops, load-bearing self-citations, or smuggled ansatzes appear in the derivation chain. The purely imaginary low-frequency claim follows from the numerical output under stated boundary conditions rather than from any redefinition of inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard linear perturbation theory in general relativity and the AdS boundary conditions; the Proca mass and black hole radius function as tunable parameters whose influence is mapped rather than fitted to produce the reported modes.

free parameters (2)
  • Proca mass
    Varied to study its effect on the QNM spectrum and to recover the Maxwell limit at zero mass.
  • Black hole radius
    Explored as a parameter controlling the spectrum for both numerical and analytic calculations.
axioms (2)
  • domain assumption Linearized perturbation theory on a fixed Schwarzschild-AdS background
    Invoked to derive the decoupled and coupled radial wave equations from the Proca field equations.
  • domain assumption Dirichlet boundary conditions at spatial infinity
    Used to define the quasinormal mode spectrum in asymptotically AdS spacetimes.

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