arxiv: 2604.05988 · v2 · submitted 2026-04-07 · 🌀 gr-qc · hep-th

Recognition: 2 theorem links

· Lean Theorem

Quasinormal modes of coupled metric-dilaton perturbations in two-dimensional stringy black holes

Wen-Hao Bian , Zhu-Fang Cui

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:22 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords quasinormal modestwo-dimensional black holesstring theorymetric-dilaton couplinglinear stabilitySchrodinger equationsblack hole perturbations

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The pith

The two-dimensional MSW black hole is linearly stable against coupled metric-dilaton perturbations.

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

This paper studies the ringing modes triggered by small coupled changes to both the metric and the dilaton inside the Mandal-Sengupta-Wadia black hole of two-dimensional string theory. It rewrites the perturbation equations as coupled Schrödinger problems in tortoise coordinates and computes the complex frequencies numerically. Every frequency found has a negative imaginary part, which means the perturbations always die away and the black hole returns to equilibrium. These intrinsic modes generally oscillate at a real frequency as they damp, in contrast to external scalar fields that produce only decaying non-oscillatory modes. The damping becomes slower when the string central-charge parameter is larger.

Core claim

The linear perturbation equations for the coupled metric and dilaton fields are reduced, after field redefinitions, to a pair of coupled Schrödinger-type equations. Numerical integration with purely ingoing waves at the horizon and purely outgoing waves at infinity produces a discrete spectrum of complex frequencies. All computed modes satisfy Im(ω) < 0, establishing linear stability. The modes generically possess nonzero real parts, corresponding to oscillatory behavior in the gravity-dilaton sector, with the real part showing nonmonotonic dependence on overtone number and the damping rate decreasing as √k increases.

What carries the argument

Reduction of the coupled metric-dilaton perturbation equations to Schrödinger-type eigenvalue problems via field redefinitions of the conformal factor and dilaton.

If this is right

The black hole is stable against small intrinsic perturbations in its geometry and dilaton.
Relaxation involves oscillatory ringing with non-zero real frequencies.
Increasing the central charge parameter prolongs the relaxation time by reducing the damping rate.
The real frequency component varies nonmonotonically with the overtone number.

Where Pith is reading between the lines

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

These oscillatory modes may directly reflect the internal string-theoretic degrees of freedom rather than external fields.
Similar stability conclusions might hold for other two-dimensional dilaton-gravity models if analogous reductions are possible.
The nonmonotonic overtone behavior could be tested in related higher-dimensional string black holes to see if it is a general feature.

Load-bearing premise

Field redefinitions can reduce the coupled linear equations exactly to Schrödinger-type problems whose numerical solutions with standard boundary conditions give the complete spectrum.

What would settle it

Discovery of any quasinormal mode with positive imaginary part in the numerical spectrum would demonstrate instability.

Figures

Figures reproduced from arXiv: 2604.05988 by Wen-Hao Bian, Zhu-Fang Cui.

**Figure 2.** Figure 2: FIG. 2. (Color online) Evolution of QNMs for parameters [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (Color online) Time evolution of the two branches [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

read the original abstract

We investigate the quasinormal modes (QNMs) associated with intrinsic metric-dilaton coupled perturbations of the Mandal-Sengupta-Wadia (MSW) black hole in two-dimensional string theory. Through suitable field redefinitions, the gravity-dilaton system is expressed in terms of the conformal factor and a redefined dilaton field, allowing the linear perturbation equations to be reduced to coupled Schrodinger-type eigenvalue equations in the tortoise coordinate. By imposing the standard QNMs' boundary conditions of purely ingoing waves at the horizon and purely outgoing waves at spatial infinity, we numerically determine the complex frequency spectrum. All modes satisfy Im$(\omega)<0$, confirming the linear stability of the MSW black hole under intrinsic coupled perturbations. Unlike external scalar-field perturbations, which yield purely imaginary frequencies, the intrinsic perturbations generically exhibit nonvanishing real parts, corresponding to oscillatory modes of the gravity-dilaton sector. The real part of the frequency displays a nonmonotonic dependence on the overtone number, while increasing the central-charge parameter $\sqrt{k}$ systematically decreases the damping rate and prolongs the relaxation time. These results indicate that intrinsic perturbations probe internal dynamical degrees of freedom and reveal characteristic features of the relaxation dynamics of two-dimensional stringy black holes.

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.

Desk Editor's Note private letter to a colleague

The paper computes QNMs for coupled metric-dilaton perturbations on the MSW 2D black hole, finds stability plus non-monotonic real parts, but the numerical spectrum extraction is the part that still needs checking.

read the letter

The paper computes the quasinormal mode spectrum for coupled metric and dilaton perturbations around the Mandal-Sengupta-Wadia black hole in two-dimensional string theory. It finds that all modes are stable with negative imaginary parts, and reports some new features like non-monotonic real parts depending on overtone number and the central charge parameter. They do a reasonable job reducing the perturbation equations to a pair of coupled Schrödinger-type problems using field redefinitions, then applying standard ingoing and outgoing boundary conditions to get the complex frequencies numerically. This is new because prior work mostly looked at external test fields, which give purely imaginary frequencies, whereas here the intrinsic perturbations have real parts and thus oscillate while damping. The dependence on √k is also tracked, showing slower damping for larger k. The soft spots are mostly in the numerical side. For coupled systems, it's easy to miss modes or get numerical artifacts if the discretization or solver isn't checked thoroughly, especially with non-monotonic potentials or varying parameters. The abstract doesn't provide the explicit equations, the specific numerical scheme used, any convergence tests, or comparisons to analytic limits like the external scalar case. Without those, the claim that the spectrum is complete and the non-monotonic behavior is physical rather than numerical remains a bit tentative. The stability conclusion is plausible but rests on that numerical completeness. This paper is aimed at people working on lower-dimensional gravity models and quasinormal modes in string theory contexts. A specialist in 2D black holes would find the specific results useful for comparison, but it doesn't open up broad new directions. I would bring it to a reading group for the niche audience, but it's not something I'd cite broadly. It deserves peer review because the calculation is concrete and the stability result is worth having checked by referees who can look at the full numerics.

Referee Report

1 major / 2 minor

Summary. The manuscript studies quasinormal modes of intrinsic coupled metric-dilaton perturbations around the Mandal-Sengupta-Wadia black hole in two-dimensional string theory. Suitable field redefinitions reduce the linearized equations to a system of coupled Schrödinger-type eigenvalue problems in tortoise coordinates. Standard ingoing (horizon) and outgoing (infinity) boundary conditions are imposed, and the complex frequencies are extracted numerically. The central results are that every computed mode satisfies Im(ω) < 0 (linear stability), that the real parts are generically nonzero (oscillatory behavior unlike external scalars), that Re(ω) varies non-monotonically with overtone number, and that increasing the central-charge parameter √k decreases the damping rate.

Significance. If the numerical spectrum is shown to be complete and free of discretization artifacts, the work would usefully characterize the relaxation dynamics of two-dimensional stringy black holes under their intrinsic gravitational-dilaton degrees of freedom. The explicit contrast with external scalar perturbations and the systematic dependence on √k supply concrete, falsifiable features of the linear response in this exactly solvable model. The reduction to coupled Schrödinger form is a clear technical strength that could enable further analytic or semi-analytic checks.

major comments (1)

[Numerical implementation and results] The stability conclusion (all Im(ω) < 0) and the claim of a complete discrete spectrum rest on the numerical solution of the coupled Schrödinger system. The manuscript should supply explicit verification that the chosen scheme (shooting, continued fractions, or matrix discretization) recovers known analytic limits, such as the purely imaginary frequencies of external scalar perturbations, and that results converge under grid refinement or basis truncation for the range of √k considered. Without these checks, branches may be missed when the effective potentials are non-monotonic.

minor comments (2)

[Perturbation equations] The abstract and text should clarify whether the field redefinitions are invertible and preserve the full physical spectrum, including any possible zero modes or constraints from the two-dimensional diffeomorphism invariance.
[Results] A table or plot summarizing the lowest few modes for at least two values of √k would make the non-monotonic Re(ω) behavior and the damping-rate trend quantitatively accessible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below and will revise the paper to incorporate the requested numerical verifications.

read point-by-point responses

Referee: [Numerical implementation and results] The stability conclusion (all Im(ω) < 0) and the claim of a complete discrete spectrum rest on the numerical solution of the coupled Schrödinger system. The manuscript should supply explicit verification that the chosen scheme (shooting, continued fractions, or matrix discretization) recovers known analytic limits, such as the purely imaginary frequencies of external scalar perturbations, and that results converge under grid refinement or basis truncation for the range of √k considered. Without these checks, branches may be missed when the effective potentials are non-monotonic.

Authors: We agree that explicit numerical validation is essential to substantiate the stability results and the discrete nature of the spectrum. In the revised manuscript we will add a dedicated subsection describing our numerical implementation, which uses a matrix discretization method with finite differences on a uniform grid in tortoise coordinates. We will demonstrate that the code reproduces the known purely imaginary quasinormal frequencies for external scalar perturbations in the appropriate decoupling limit. We will also present convergence tests by successively refining the grid size and showing that the extracted frequencies stabilize to within a specified tolerance for all values of √k considered. These additions will directly address the possibility of missed modes arising from non-monotonic potentials. revision: yes

Circularity Check

0 steps flagged

No circularity in the derivation chain

full rationale

The paper derives the coupled perturbation equations via field redefinitions to Schrödinger-type form, then numerically solves the eigenvalue problem under standard ingoing/outgoing boundary conditions to extract the QNM frequencies. The stability result (all Im(ω)<0) and the distinction from external scalar modes follow from these computed values rather than any algebraic identity or fitted parameter that presupposes the outcome. The background parameter √k enters as an independent input from the MSW solution, and no self-citation chain or ansatz is invoked to force the spectrum. The analysis is therefore self-contained against the numerical benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard linearization of 2D dilaton gravity around a known black-hole background and on the conventional definition of quasinormal modes. No new particles or forces are introduced.

free parameters (1)

central-charge parameter √k
Model parameter inherited from the MSW background solution; its value is varied to observe trends in damping rates.

axioms (2)

domain assumption Linear perturbation theory is valid for small metric-dilaton fluctuations around the MSW background
Invoked to obtain the coupled wave equations from the action.
domain assumption Quasinormal-mode boundary conditions consist of purely ingoing waves at the horizon and purely outgoing waves at spatial infinity
Standard choice used to select the discrete complex-frequency spectrum.

pith-pipeline@v0.9.0 · 5525 in / 1578 out tokens · 57299 ms · 2026-05-10T19:22:36.032349+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Through suitable field redefinitions, the gravity-dilaton system is expressed in terms of the conformal factor and a redefined dilaton field, allowing the linear perturbation equations to be reduced to coupled Schrödinger-type eigenvalue equations in the tortoise coordinate.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

All modes satisfy Im(ω)<0, confirming the linear stability of the MSW black hole under intrinsic coupled perturbations.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

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Reference graph

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