arxiv: 2605.03730 · v1 · submitted 2026-05-05 · 🌀 gr-qc

Recognition: unknown

Total transmission modes in draining bathtub model with vorticity

Liang-Bi Wu, Zhe Yu

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Pith reviewed 2026-05-07 14:00 UTC · model grok-4.3

classification 🌀 gr-qc

keywords total transmission modesdraining bathtub modelvorticitypseudospectral methodanalog gravityblack hole analogswave propagation

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The pith

Total transmission modes in the draining bathtub model with vorticity have spectra whose imaginary parts can be positive or negative depending on parameters.

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

The paper numerically computes total transmission modes for scalar waves propagating in a draining bathtub vortex that includes vorticity. It applies the Chebyshev-Lobatto pseudospectral method to the radial wave equation with ingoing boundary conditions imposed at the horizon and at infinity. The resulting spectra for the right modes show imaginary parts that are positive for some parameter choices and negative for others. Higher overtones display pronounced spectral mobility, meaning small parameter shifts produce large frequency changes. These properties matter for determining whether waves in such analog systems amplify or damp over time.

Core claim

Numerical results show that the right TTM spectra can possess positive imaginary parts, while for certain parameters they acquire negative imaginary parts. The extreme sensitivity of the higher overtones is manifested as their pronounced spectral mobility.

What carries the argument

Chebyshev-Lobatto pseudospectral discretization of the radial master equation for the draining bathtub model with vorticity, subject to ingoing boundary conditions at both the horizon and spatial infinity.

Load-bearing premise

The chosen boundary conditions are exactly ingoing at the horizon and at infinity, and the pseudospectral method resolves the discrete spectra without artifacts or unresolved convergence issues.

What would settle it

A high-resolution time-domain evolution of the wave equation for a parameter set reported to have positive imaginary frequency that instead shows only decay or no exponential growth would falsify the reported spectra.

Figures

Figures reproduced from arXiv: 2605.03730 by Liang-Bi Wu, Zhe Yu.

**Figure 1.** Figure 1: FIG. 1: Eigenfunctions of the fundamental modes for view at source ↗

**Figure 2.** Figure 2: FIG. 2: The left panels display the TTM trajectories as the parameter view at source ↗

read the original abstract

We investigate the total transmission modes (TTMs) in the draining bathtub model (DBM) with vorticity using the Chebyshev-Lobatto pseudospectral method, where the boundary conditions of the total transmission modes are both ingoing at the event horizon and infinity. Numerical results show that the (right) TTM spectra can possess positive imaginary parts, while for certain parameters they acquire negative imaginary parts. The extreme sensitivity of the higher overtones is manifested as their pronounced spectral mobility.

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 numerically computes TTM spectra in the vorticity-extended draining bathtub and reports positive imaginary parts for some parameters, but provides no convergence or validation data for the sensitive higher overtones.

read the letter

The main result is a set of numerical spectra for total transmission modes in the draining bathtub model once vorticity is included. Using the Chebyshev-Lobatto pseudospectral method on the wave equation with ingoing conditions at both the horizon and infinity, the authors find that the frequencies can have positive imaginary parts for certain parameter choices and negative ones for others, with the higher overtones showing strong shifts under small changes in the setup. This is a direct numerical extension of earlier DBM work, and the concrete spectra are new for the vortical case. The method choice is standard for this class of radial problems, and the reported spectral mobility follows naturally from the numerics once the model is set up. The limitation is that the abstract and description give no residual norms, grid-refinement tests, or checks against the zero-vorticity limit. Because the paper itself flags the extreme sensitivity of the higher modes, the absence of those diagnostics leaves open whether the positive imaginary parts are physical or arise from truncation or boundary enforcement. The dual ingoing boundary conditions add another layer where small implementation details can matter. This is a narrow numerical study aimed at analog-gravity researchers who already work with fluid models and quasinormal-mode techniques. Someone in that subfield could use the spectra as a reference point or starting guess for further runs. It is worth sending for peer review so that experts can ask for the missing convergence tables and limiting-case comparisons; the underlying calculation is straightforward enough that those additions should be feasible.

Referee Report

2 major / 0 minor

Summary. The manuscript numerically studies total transmission modes (TTMs) in the draining bathtub model with vorticity. It employs the Chebyshev-Lobatto pseudospectral method to discretize the wave equation subject to ingoing boundary conditions at both the event horizon and spatial infinity. The central results are that the (right) TTM spectra can exhibit positive imaginary parts, acquire negative imaginary parts for certain parameter choices, and that higher overtones display extreme sensitivity manifested as pronounced spectral mobility.

Significance. If the reported spectra are numerically robust, the finding that vorticity permits positive imaginary parts in TTM spectra would be of interest for analog gravity and scattering problems, as it suggests possible amplification channels distinct from standard quasinormal-mode behavior. The documented sensitivity of higher overtones could also inform the design of numerical or experimental probes of such modes.

major comments (2)

[Abstract and Numerical Results] The abstract and numerical results section present spectra with positive and negative imaginary parts without error bars, convergence tests with respect to polynomial degree, residual norms, or validation against known limits (e.g., zero vorticity). Given the explicit statement that higher overtones are extremely sensitive, this omission directly affects the reliability of the claim that positive imaginary parts are physical rather than numerical artifacts.
[Numerical Method] The Chebyshev-Lobatto pseudospectral implementation with simultaneous ingoing conditions at the horizon and infinity is not accompanied by any demonstration of spectral convergence, grid-refinement studies, or checks for spurious modes. This is load-bearing for the central claim of spectral mobility in the higher overtones.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their insightful comments, which highlight important aspects of numerical reliability in our study of total transmission modes. We respond to each major comment in detail below and commit to enhancing the manuscript with additional numerical evidence.

read point-by-point responses

Referee: [Abstract and Numerical Results] The abstract and numerical results section present spectra with positive and negative imaginary parts without error bars, convergence tests with respect to polynomial degree, residual norms, or validation against known limits (e.g., zero vorticity). Given the explicit statement that higher overtones are extremely sensitive, this omission directly affects the reliability of the claim that positive imaginary parts are physical rather than numerical artifacts.

Authors: We agree that the absence of explicit error bars and convergence tests in the original submission weakens the presentation, especially given the noted sensitivity of higher overtones. Although our internal checks confirmed the robustness of the positive imaginary parts for the reported parameters, we did not include them in the manuscript. In the revision, we will add a dedicated subsection on numerical validation, including plots of eigenvalue convergence versus polynomial degree N, residual norm estimates for the discretized operator, and a direct comparison to the zero-vorticity limit, where our spectra match previously published results for the standard draining bathtub model. Error bars will be estimated from the difference between consecutive N values. revision: yes
Referee: [Numerical Method] The Chebyshev-Lobatto pseudospectral implementation with simultaneous ingoing conditions at the horizon and infinity is not accompanied by any demonstration of spectral convergence, grid-refinement studies, or checks for spurious modes. This is load-bearing for the central claim of spectral mobility in the higher overtones.

Authors: The referee correctly identifies that demonstrations of convergence were missing. The Chebyshev-Lobatto method was chosen because it naturally accommodates the boundary conditions through the choice of collocation points and the mapping to the compactified domain. To address this, the revised manuscript will include grid-refinement studies showing that the TTM spectra, including the spectral mobility of higher overtones, remain stable under increases in the number of points. We will also present checks for spurious modes by examining the condition number of the discretized matrix and comparing results from different mappings. These additions will substantiate that the observed positive imaginary parts and mobility are physical features rather than numerical artifacts. revision: yes

Circularity Check

0 steps flagged

Direct numerical solution of wave equation shows no circularity

full rationale

The paper performs a direct numerical computation of total transmission modes by discretizing the wave equation in the draining bathtub model with vorticity using the Chebyshev-Lobatto pseudospectral method, subject to ingoing boundary conditions at both the event horizon and infinity. No analytical derivation chain, parameter fitting, or self-citation is invoked to obtain the spectra; the reported positive/negative imaginary parts and higher-overtone mobility are outputs of solving the discretized eigenvalue problem from the model equations. The computation is self-contained against the input PDE and BCs with no reduction by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce or specify any free parameters, axioms, or invented entities beyond the standard draining bathtub model with added vorticity; all content is numerical evaluation of an existing framework.

pith-pipeline@v0.9.0 · 5361 in / 1112 out tokens · 94151 ms · 2026-05-07T14:00:55.668291+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

51 extracted references · 46 canonical work pages · 6 internal anchors

[1]

Nollert, Class

H.-P. Nollert, Class. Quant. Grav.16, R159 (1999)

1999
[2]

K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel.2, 2 (1999), arXiv:gr-qc/9909058

work page internal anchor Pith review arXiv 1999
[3]

Black hole spectroscopy: from theory to experiment

E. Berti et al., (2025), arXiv:2505.23895 [gr-qc]

work page internal anchor Pith review arXiv 2025
[4]

Quasinormal modes of black holes and black branes

E. Berti, V . Cardoso, and A. O. Starinets, Class. Quant. Grav.26, 163001 (2009), arXiv:0905.2975 [gr-qc]

work page internal anchor Pith review arXiv 2009
[5]

Trivedi, A

O. Trivedi, A. Gurrola, and R. J. Scherrer, (2026), arXiv:2603.04375 [gr-qc]

work page arXiv 2026
[6]

B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. Lett.116, 221101 (2016), [Erratum: Phys.Rev.Lett. 121, 129902 (2018)], arXiv:1602.03841 [gr-qc]

work page Pith review arXiv 2016
[7]

Abbottet al

R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. D103, 122002 (2021), arXiv:2010.14529 [gr-qc]

work page arXiv 2021
[8]

Tests of General Relativity with GWTC-3

R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), (2021), arXiv:2112.06861 [gr-qc]

work page internal anchor Pith review arXiv 2021
[9]

Chandrasekhar, The mathematical theory of black holes (1985)

S. Chandrasekhar, The mathematical theory of black holes (1985)

1985
[10]

Andersson, Class

N. Andersson, Class. Quant. Grav.11, L39 (1994)

1994
[11]

Maassen van den Brink, Phys

A. Maassen van den Brink, Phys. Rev. D62, 064009 (2000), arXiv:gr-qc/0001032

work page arXiv 2000
[12]

Berti, gr-qc/0411025

E. Berti, Conf. Proc. C0405132, 145 (2004), arXiv:gr-qc/0411025

work page internal anchor Pith review arXiv 2004
[13]

Keshet and A

U. Keshet and A. Neitzke, Phys. Rev. D78, 044006 (2008), arXiv:0709.1532 [hep-th]

work page arXiv 2008
[14]

G. B. Cook and M. Zalutskiy, Class. Quant. Grav.33, 245008 (2016), arXiv:1603.09710 [gr-qc]

work page arXiv 2016
[15]

G. B. Cook and M. Zalutskiy, Phys. Rev. D94, 104074 (2016), arXiv:1607.07406 [gr-qc]

work page arXiv 2016
[16]

G. B. Cook, L. S. Annichiarico, and D. J. Vickers, Phys. Rev. D99, 024008 (2019), arXiv:1808.00987 [gr-qc]

work page arXiv 2019
[17]

G. B. Cook and S. Lu, Phys. Rev. D107, 044043 (2023), arXiv:2211.14955 [gr-qc]

work page arXiv 2023
[18]

Longhi, Phys

S. Longhi, Phys. Rev. A82, 031801 (2010), arXiv:1008.5298 [quant-ph]

work page arXiv 2010
[19]

Total absorption of tailored incoming signals by black holes

F. Tuncer, V . Cardoso, R. Panosso Macedo, and T. F. M. Spieksma, (2025), arXiv:2509.19451 [gr-qc]

work page internal anchor Pith review arXiv 2025
[20]

Wu, Y .-S

L.-B. Wu, Y .-S. Zhou, Z. Yu, M.-F. Jia, and L.-M. Cao, (2026), arXiv:2603.22897 [gr-qc]

work page arXiv 2026
[21]

Kwon and S

Y . Kwon and S. Nam, Class. Quant. Grav.28, 035007 (2011), arXiv:1102.3512 [hep-th]

work page arXiv 2011
[22]

Zhou, M.-F

Y .-S. Zhou, M.-F. Ji, L.-B. Wu, and L.-M. Cao, (2025), arXiv:2512.16372 [gr-qc] . 11

work page arXiv 2025
[23]

J. L. Jaramillo, R. Panosso Macedo, and L. Al Sheikh, Phys. Rev. X11, 031003 (2021), arXiv:2004.06434 [gr-qc]

work page arXiv 2021
[24]

Cao, L.-B

L.-M. Cao, L.-B. Wu, and Y .-S. Zhou, Sci. China Phys. Mech. Astron.68, 100411 (2025), arXiv:2412.21092 [gr-qc]

work page arXiv 2025
[25]

Cao, J.-N

L.-M. Cao, J.-N. Chen, L.-B. Wu, L. Xie, and Y .-S. Zhou, Sci. China Phys. Mech. Astron.67, 100412 (2024), arXiv:2401.09907 [gr-qc]

work page arXiv 2024
[26]

Destounis, R

K. Destounis, R. P. Macedo, E. Berti, V . Cardoso, and J. L. Jaramillo, Phys. Rev. D104, 084091 (2021), arXiv:2107.09673 [gr-qc]

work page arXiv 2021
[27]

Chen, L.-B

J.-N. Chen, L.-B. Wu, and Z.-K. Guo, Class. Quant. Grav.41, 235015 (2024), arXiv:2407.03907 [gr-qc]

work page arXiv 2024
[28]

Cai, L.-M

R.-G. Cai, L.-M. Cao, J.-N. Chen, Z.-K. Guo, L.-B. Wu, and Y .-S. Zhou, Phys. Rev. D111, 084011 (2025), arXiv:2501.02522 [gr-qc]

work page arXiv 2025
[29]

Cao, M.-F

L.-M. Cao, M.-F. Ji, L.-B. Wu, and Y .-S. Zhou, Phys. Rev. D112, 124022 (2025), arXiv:2508.13894 [gr-qc]

work page arXiv 2025
[30]

Barcelo, S

C. Barcelo, S. Liberati, and M. Visser, Living Rev. Rel.8, 12 (2005), arXiv:gr-qc/0505065

work page arXiv 2005
[31]

W. G. Unruh, Phys. Rev. Lett.46, 1351 (1981)

1981
[32]

Visser, Class

M. Visser, Class. Quant. Grav.15, 1767 (1998), arXiv:gr-qc/9712010

work page arXiv 1998
[33]

Schutzhold and W

R. Schutzhold and W. G. Unruh, Phys. Rev. D66, 044019 (2002), arXiv:gr-qc/0205099

work page arXiv 2002
[34]

Weinfurtner, E

S. Weinfurtner, E. W. Tedford, M. C. J. Penrice, W. G. Unruh, and G. A. Lawrence, Phys. Rev. Lett.106, 021302 (2011), arXiv:1008.1911 [gr-qc]

work page arXiv 2011
[35]

L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett.85, 4643 (2000), arXiv:gr-qc/0002015

work page Pith review arXiv 2000
[36]

Barcelo, S

C. Barcelo, S. Liberati, and M. Visser, Class. Quant. Grav.18, 1137 (2001), arXiv:gr-qc/0011026

work page arXiv 2001
[37]

T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, Science319, 1367 (2008), arXiv:0711.4796 [gr-qc]

work page Pith review arXiv 2008
[38]

Belgiorno, S

F. Belgiorno, S. L. Cacciatori, M. Clerici, V . Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V . G. Sala, and D. Faccio, Phys. Rev. Lett.105, 203901 (2010), arXiv:1009.4634 [gr-qc]

work page arXiv 2010
[39]

Observation of quantum Hawking radiation and its entanglement in an analogue black hole

J. Steinhauer, Nature Phys.12, 959 (2016), arXiv:1510.00621 [gr-qc]

work page Pith review arXiv 2016
[40]

Torres, S

T. Torres, S. Patrick, A. Coutant, M. Richartz, E. W. Tedford, and S. Weinfurtner, Nature Phys.13, 833 (2017), arXiv:1612.06180 [gr-qc]

work page arXiv 2017
[41]

Berti, V

E. Berti, V . Cardoso, and J. P. S. Lemos, Phys. Rev. D70, 124006 (2004), arXiv:gr-qc/0408099

work page arXiv 2004
[42]

S. R. Dolan, E. S. Oliveira, and L. C. B. Crispino, Phys. Rev. D79, 064014 (2009), arXiv:0904.0010 [gr-qc]

work page arXiv 2009
[43]

Patrick, A

S. Patrick, A. Coutant, M. Richartz, and S. Weinfurtner, Phys. Rev. Lett.121, 061101 (2018), arXiv:1801.08473 [gr-qc]

work page arXiv 2018
[44]

Patrick, On the analogy between black holes and bathtub vortices, Ph.D

S. Patrick, On the analogy between black holes and bathtub vortices, Ph.D. thesis, Nottingham U. (2019), arXiv:2009.02133 [gr-qc]

work page arXiv 2019
[45]

Malato Corr ˆea, C

M. Malato Corr ˆea, C. F. B. Macedo, R. Panosso Macedo, and L. A. Oliveira, Phys. Rev. D112, 024036 (2025), arXiv:2504.00107 [gr-qc]

work page arXiv 2025
[46]

L. T. de Paula, P. H. C. Siqueira, R. Panosso Macedo, and M. Richartz, Phys. Rev. D111, 104064 (2025), arXiv:2504.00106 [gr-qc]

work page arXiv 2025
[47]

Wang and D.-R

Z.-X. Wang and D.-R. Guo, Special Functions (World Scientific Publishing Company, 1989)

1989
[48]

Jansen, Eur

A. Jansen, Eur. Phys. J. Plus132, 546 (2017), arXiv:1709.09178 [gr-qc]

work page arXiv 2017
[49]

Li, W.-L

G.-R. Li, W.-L. Qian, and R. G. Daghigh, Phys. Rev. D110, 064076 (2024), arXiv:2406.10782 [gr-qc]

work page arXiv 2024
[50]

Li, W.-L

G.-R. Li, W.-L. Qian, Q. Pan, R. G. Daghigh, J. C. Morey, and R.-H. Yue, Phys. Rev. D112, 064071 (2025), arXiv:2504.13265 [gr-qc]

work page arXiv 2025
[51]

Xie, L.-B

L. Xie, L.-B. Wu, and Z.-K. Guo, Phys. Rev. D112, 024054 (2025), arXiv:2505.21303 [gr-qc]

work page arXiv 2025