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arxiv: 2511.00565 · v3 · submitted 2025-11-01 · 🌀 gr-qc

Reflectionless and echo modes in asymmetric Damour-Solodukhin wormholes

Wei-Liang Qian , Qiyuan Pan , Ramin G. Daghigh , Bean Wang , Rui-Hong Yue This is my paper

Pith reviewed 2026-05-18 01:31 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Damour-Solodukhin wormholesreflectionless modesecho modesquasinormal modesasymmetric wormholesgravitational wave echoescomplex frequency spectra
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The pith

In asymmetric Damour-Solodukhin wormholes, reflectionless modes and echo modes share asymptotically uniform spacing along the real frequency axis with coinciding real parts in the high-frequency limit.

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

This paper investigates the connection between reflectionless modes and echo modes in generalized asymmetric Damour-Solodukhin wormholes. It finds that their spectra in the complex frequency plane are asymptotically similar, both showing a roughly uniform distribution along lines parallel to the real axis with identical spacing between modes. The real parts of the echo modes align with those of the reflectionless modes in the limit where the real frequency greatly exceeds the imaginary part. This resemblance suggests that both sets of modes can effectively capture the echo waveforms produced by these objects, with reflectionless modes yielding stronger signals because they sit closer to the real frequency axis.

Core claim

By extending the definition of quasi-reflectionless modes to reflectionless ones and generalizing symmetric Damour-Solodukhin wormholes to asymmetric cases, the asymptotic properties of these two spectra exhibit a strong resemblance, featuring an approximately uniform distribution parallel to the real frequency axis with the same spacing between successive modes. Specifically, the real parts of echo modes coincide with those of reflectionless modes at the limit |Reω| ≫ |Imω|. While echo modes typically possess non-vanishing imaginary parts, the reflectionless modes of symmetric Damour-Solodukhin wormholes lie precisely on the real frequency axis, with any deviation serving as a measure of a

What carries the argument

The asymptotic analysis of high-overtone modes in the complex frequency plane for both reflectionless and echo spectra, enabled by extending the metric and mode definitions to asymmetric wormholes.

If this is right

  • Waveforms from a given source, computed via Green's functions, have larger amplitudes when constructed from reflectionless modes than from echo modes.
  • Reflectionless modes lie closer to the real frequency axis than echo modes, making them more prominent in the waveforms.
  • Any departure of reflectionless modes from the real axis directly quantifies the degree of asymmetry in the wormhole.
  • Both the reflectionless-mode perspective and the echo-mode perspective provide effective descriptions of the echo phenomenon.

Where Pith is reading between the lines

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

  • The observed spectral similarity may extend to other ultracompact objects that produce echoes, such as certain black-hole mimics.
  • Targeted numerical calculations for specific asymmetry values could directly test whether the real-part coincidence persists beyond the analytic asymptotic regime.
  • Stronger signals from reflectionless modes could offer a practical advantage when searching for echoes in gravitational-wave data from potential exotic objects.

Load-bearing premise

The generalization of symmetric Damour-Solodukhin wormholes to asymmetric cases preserves the necessary properties for defining and comparing reflectionless and echo modes.

What would settle it

Computing the complex frequencies of both sets of modes for a chosen nonzero asymmetry parameter and verifying whether the real parts match and the spacing between successive modes is identical when |Reω| is large compared to |Imω|.

Figures

Figures reproduced from arXiv: 2511.00565 by Bean Wang, Qiyuan Pan, Ramin G. Daghigh, Rui-Hong Yue, Wei-Liang Qian.

Figure 1
Figure 1. Figure 1: An illustration of the effective potential in an asymmetric Damour-Solodukhin wormhole in the tortoise coordinate x. The effective potential consists of two distinct black hole effective potentials separated by a distance 2xc, where the black hole effective potential on the l.h.s. is spatially reflected. As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The RSM and quasi-RSM modes and their asymptotic values for symmetric Damour-Solodukhin [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The RSM and quasi-RSM modes and their asymptotic values for asymmetric Damour-Solodukhin [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The echo modes and their asymptotic values for symmetric and asymmetric Damour-Solodukhin [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The RSM and quasi-RSM modes and their asymptotic values for asymmetric Damour-Solodukhin [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The echo modes and their asymptotic values for symmetric and asymmetric Damour-Solodukhin [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The frequency-domain Green’s functions, Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The time-domain waveforms for asymmetric Damour-Solodukhin wormholes composed of two [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
read the original abstract

It is understood that the echo waveforms in ultracompact objects can be regarded as composed mainly of the asymptotic high-overtone quasinormal modes, dubbed echo modes, which predominantly lie parallel to the real frequency axis. Alternatively, Rosato {\it et al.} recently suggested that high-frequency quasi-reflectionless scattering modes are primarily responsible for the echo phenomenon. In this work, by extending the definition of quasi-reflectionless modes to reflectionless ones and generalizing symmetric Damour-Solodukhin wormholes to asymmetric cases, we examine the underlying similarity between the reflectionless and echo mode spectra in the complex frequency plane. Through a primarily analytical treatment, we demonstrate that the asymptotic properties of these two spectra exhibit a strong resemblance, featuring an approximately uniform distribution parallel to the real frequency axis with the same spacing between successive modes. Specifically, the real parts of echo modes coincide with those of reflectionless modes at the limit $|\mathrm{Re}\omega| \gg |\mathrm{Im}\omega|$. While echo modes typically possess non-vanishing imaginary parts, the reflectionless modes of symmetric Damour-Solodukhin wormholes lie precisely on the real frequency axis, with any deviation serving as a measure of the degree of asymmetry of the wormhole. For a given identical source, the waveforms are calculated numerically using the Green's functions. The amplitudes of the waveforms associated with reflectionless modes are found to be more pronounced than those of the echo modes, because reflectionless modes typically lie closer to the real frequency axis than the latter. It is argued that both perspectives provide effective tools for describing the echo phenomenon.

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 extends Damour-Solodukhin wormholes to asymmetric cases and compares the spectra of reflectionless modes (extended from quasi-reflectionless) with echo modes. Through primarily analytical methods it claims that both families asymptotically distribute uniformly parallel to the real frequency axis with identical spacing; specifically, the real parts of the echo modes coincide with those of the reflectionless modes in the limit |Re ω| ≫ |Im ω|. Reflectionless modes of the symmetric case lie exactly on the real axis, with deviations quantifying asymmetry. Numerical Green's-function waveforms for a fixed source are presented, showing larger amplitudes for the reflectionless-mode contribution.

Significance. If the central spectral resemblance holds, the work supplies a concrete link between two descriptions of the echo phenomenon in ultracompact objects and demonstrates that both viewpoints remain effective after the generalization to asymmetry. The primarily analytical treatment of the high-frequency asymptotics together with the explicit numerical waveform comparison constitutes a clear strength; the observation that reflectionless-mode amplitudes are more pronounced because they lie closer to the real axis is a useful quantitative result.

major comments (1)
  1. [Analytical treatment of the spectra (high-overtone asymptotics)] The load-bearing step is the assertion that the WKB or monodromy quantization condition for both mode families yields identical spacing and real-part coincidence after the metric is made asymmetric. The abstract states that the resemblance is shown “through a primarily analytical treatment,” yet the text does not appear to derive the modified phase accumulation that arises from distinct redshift factors on the two sides of the throat. An explicit calculation of the round-trip phase for the asymmetric effective potential is required to confirm that the spacing formula remains unchanged in the |Re ω| ≫ |Im ω| limit.
minor comments (2)
  1. [Metric definition] The notation for the asymmetry parameter and the two distinct throat radii should be introduced once and used consistently; occasional switches between symbols obscure the comparison between symmetric and asymmetric cases.
  2. [Numerical results] Figure captions for the Green's-function waveforms should state the precise value of the asymmetry parameter and the source location used in each panel.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and indicate the revisions we will make to strengthen the presentation of the analytical results.

read point-by-point responses

  1. Referee: [Analytical treatment of the spectra (high-overtone asymptotics)] The load-bearing step is the assertion that the WKB or monodromy quantization condition for both mode families yields identical spacing and real-part coincidence after the metric is made asymmetric. The abstract states that the resemblance is shown “through a primarily analytical treatment,” yet the text does not appear to derive the modified phase accumulation that arises from distinct redshift factors on the two sides of the throat. An explicit calculation of the round-trip phase for the asymmetric effective potential is required to confirm that the spacing formula remains unchanged in the |Re ω| ≫ |Im ω| limit.

    Authors: We appreciate the referee drawing attention to the need for greater explicitness in the high-overtone analysis. The manuscript does derive the asymptotic spacing and real-part coincidence for the asymmetric case by extending the WKB/monodromy condition, taking into account the distinct redshift factors on each side of the throat (see the discussion following Eq. (3.12) and the subsequent limit |Re ω| ≫ |Im ω|). Nevertheless, we agree that a more self-contained calculation of the round-trip phase integral for the asymmetric effective potential would make the invariance of the spacing formula clearer. In the revised version we will insert a dedicated paragraph (or short subsection) that explicitly evaluates the phase accumulation across the two sides and demonstrates that the leading-order spacing remains unchanged. This addition will directly address the referee’s request without altering the central claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the analytical comparison of reflectionless and echo mode spectra.

full rationale

The paper extends the Damour-Solodukhin metric to asymmetric cases and derives the spectra of reflectionless modes and echo modes through separate analytical treatments of the wave equation, including WKB or monodromy quantization conditions. The claimed asymptotic resemblance—uniform spacing parallel to the real axis and coincidence of real parts when |Re ω| ≫ |Im ω|—follows from these independent derivations applied to the same generalized metric, without any step that defines one spectrum in terms of the other, fits a parameter from one to predict the other, or relies on a load-bearing self-citation whose content reduces to the present result. The extension to asymmetry is modeled explicitly in the metric and effective potential, preserving the ability to compare the two families without circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central comparison rests on standard assumptions of linear perturbation theory in general relativity and on the existence of a well-defined scattering problem for the chosen wormhole metric family.

free parameters (1)
  • asymmetry parameter
    Introduced to deform the symmetric Damour-Solodukhin metric; its value controls the imaginary-part offset of reflectionless modes from the real axis.
axioms (1)
  • domain assumption The wormhole spacetime admits a well-posed scattering problem whose reflection coefficient can be set to zero at discrete complex frequencies.
    Invoked when extending the definition of reflectionless modes to the asymmetric case.

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Forward citations

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