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Superradiant amplification by rotating viscous compact objects

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Rotating viscous stars amplify low-frequency gravitational waves

desk verdict Plausible qualitative claim, uncontrolled quantitative curves: the superradiance numbers come from a low-frequency regime the authors themselves flag as beyond their linear-in-spin approximation. read the letter →

arxiv 2506.13850 v1 pith:VYVZB3IE submitted 2025-06-16 gr-qc astro-ph.HEhep-th

Jaime Redondo-Yuste , Vitor Cardoso This is my paper
classification gr-qcastro-ph.HEhep-th MSC 83C3583C5585A15 PACS 04.30.-w04.40.Dg
keywords superradiancegravitationalwavesviscousstarsslowrotationcausalrelativistichydrodynamicsreflectivityultracompactobjectsrotatingneutron
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a rotating star made of viscous fluid will reflect incoming low-frequency gravitational waves with more energy than it receives, because viscosity lets the star absorb radiation while rotation supplies rotational energy to the outgoing wave. The authors derive, in a slow-rotation approximation from causal first-order relativistic viscous hydrodynamics, a coupled system of equations for axial gravitational-wave perturbations and internal viscous modes, and integrate it numerically to compute the reflectivity $R^2$. They find $R^2 > 1$ for frequencies $\omega \lesssim m\Omega$, where $m$ is the azimuthal number and $\Omega$ the star's angular velocity, with amplification growing with shear viscosity and rotation rate. They argue this is universal for dissipative rotating bodies, and that for uniformly rotating stars it does not seed a superradiant instability because the modes trapped near ultracompact stars oscillate above the amplification threshold.

What carries the argument

The argument runs on two coupled wave equations, one for an axial gravitational master variable $\psi$ and one for the viscous fluid mode $Z$, written in a tortoise coordinate. Their propagation speeds acquire imaginary corrections proportional to $m(\Omega-\varpi)[(\rho+p)\tau_Q - \eta]$, where $\varpi$ is the frame-dragging potential, and these corrections encode the dissipation that enables amplification. The equations are integrated from the center of a slowly rotating polytropic star to its surface, where junction conditions and a regularity boundary condition for $Z$ are imposed, then matched to the exterior axial gravitational-wave equation; the ratio of outgoing to incoming amplitude at infinity defines the reflectivity $R^2$. A simplified scalar equation with an absorption term $\alpha u^\mu \nabla_\mu \varphi$ plays the supporting role of probing trapped-mode frequencies in ultracompact stars.

What would settle it

Compute the same reflectivity at second order in $\Omega/\Omega_K$, or with a fully nonlinear evolution of the same viscous-fluid equations, and check whether $R^2 - 1$ stays positive for $\omega < m\Omega$. If the amplification disappears, changes sign, or moves entirely into the classically allowed superradiant band, the central claim as stated would be refuted.

Watch

Extended reading notes

Core claim

On the authors' terms, the central discovery is that dissipation plus rotation turns a compact star into a superradiant amplifier: for incident gravitational waves with frequency below about $m\Omega$, the reflected wave carries more energy than the incident one ($R^2 > 1$), an effect controlled by shear viscosity and absent for perfect fluids. The amplification mechanism is the same thermodynamic one as for rotating black holes—absorption of a wave with $\omega < m\Omega$ decreases the energy in the co-rotating frame and increases entropy—but it operates in a body with no event horizon and no ergoregion at the order treated. A companion scalar toy model with an absorption term confirms that sub-threshold frequencies are amplified and that the frequency of the fundamental trapped mode in ultracompact stars always satisfies $\mathrm{Re}\,\omega > m\Omega$, so amplified waves are not trapped and trapped waves are not amplified; the authors conclude that uniform rotation does not trigger a superradiant instability.

Load-bearing premise

The calculation keeps only terms linear in the rotation rate, while the amplifying frequencies are so low that the omitted quadratic corrections become comparable; the authors themselves call the low-frequency and high-rotation results merely informative.

Editorial extensions

If this is right

  • Low-frequency gravitational waves scattering off rotating neutron stars should return amplified, with the gain set by the star's shear viscosity, giving a direct wave probe of internal dissipation.
  • Horizonless black-hole mimickers cannot be assigned a naive low-frequency absorption coefficient; any reflectivity must be defined with the co-rotating frequency shift $\omega - m\Omega$, otherwise the second law of thermodynamics is violated.
  • Uniformly rotating ultracompact stars with light rings should remain linearly stable against this amplification, because their trapped-mode frequencies lie above $\omega = m\Omega$.
  • The amplification increases with both rotation rate and shear viscosity, so rapidly spinning, highly viscous objects are the most promising sources of observable superradiant reflection.
  • Objects with black-hole-scale entropy, and therefore large shear viscosity, are expected to show particularly strong amplification, tightening the link between mimicker microphysics and gravitational-wave echoes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A second-order-in-spin calculation would likely collapse the superradiant band to the classical window $0 < \omega < m\Omega$; the paper's finding of amplification above $m\Omega$ is probably a truncation artifact, by analogy with slowly rotating black-hole perturbation theory.
  • The same mechanism should be testable in tabletop analogues: a rotating absorbing cylinder or a viscous-fluid vortex should amplify low-frequency sound or electromagnetic waves, extending laboratory superradiance experiments.
  • Differentially rotating stars, and accretion disks, may be destabilized by the same dissipation-plus-rotation mechanism even though uniform rotation is stable, since differential rotation supplies more free energy and the authors' stability argument does not apply.
  • A practical observational extension is to search for viscosity-dependent reflectivity in gravitational-wave echoes from ultracompact objects, where the amplification predicted here changes echo amplitudes and phases.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies axial gravitational-wave perturbations of slowly rotating, viscous, relativistic stars modeled with BDNK first-order causal hydrodynamics. In a linear-in-spin approximation, the authors derive a coupled system of master equations for the gravitational-wave variable ψ and a viscous fluid mode Z, which they integrate numerically to compute the reflectivity R^2 of the star. They report superradiant amplification (R^2 > 1) for frequencies ω ≲ mΩ, controlled by the dimensionless shear viscosity η̂, and argue that this amplification is universal for dissipative rotating bodies. The paper also argues, using a scalar toy model with an absorption term, that uniformly rotating ultracompact stars do not suffer a superradiant instability because trapped modes have Re ω > mΩ. The authors explicitly note that their slow-rotation expansion is not controlled at low frequencies and label those results as 'merely informative.'

Significance. If the central claim survives scrutiny, the result is significant: it provides a concrete, microphysically motivated mechanism for superradiant amplification by horizonless rotating compact objects, directly relevant for black hole mimickers, neutron stars, and dark-matter candidates. The derivation is systematic within BDNK hydrodynamics, the numerical code is made available, the non-rotating limit is validated against prior work, and the thermodynamic superradiance condition is derived independently of the numerical scheme. The paper also offers a falsifiable scaling relation between viscosity and amplification. However, the quantitative predictions are weakened by the uncontrolled low-frequency regime, and the stability conclusion rests on a simplified toy model rather than the full viscous system.

major comments (3)
  1. [Discussion, and Results (Figs. 1-2)] The central quantitative claim — that R^2 > 1 for rotating viscous stars — is computed in a regime where the linear-in-spin truncation is formally inconsistent. The exterior potential (17) contains a term 24 m J_S (3r − 7M_S) / [ℓ(ℓ+1) ω r^6] that is linear in J_S but scales as 1/ω. Near the superradiant band ω ∼ mΩ, this term is of order (Ω/ω) relative to the centrifugal barrier, so the effective expansion parameter is Ω/ω, not Ω/Ω_K. The authors acknowledge this in the Discussion ('results at high angular velocities or low frequencies should be considered merely informative'), but the superradiant band is precisely the low-frequency regime. Thus the reported amplification curves are not controlled predictions; they should be regarded as suggestive of an effect whose magnitude and sign require a second-order-in-spin calculation.
  2. [Results, Fig. 2, and Eqs. (1)-(3)] The numerical reflectivity shows R^2 − 1 > 0 for frequencies ω > mΩ, in direct contradiction with the entropy-based absorption condition derived in Eqs. (1)-(3), which requires R^2 < 1 whenever ω > mΩ. The authors attribute this to the linear-in-spin truncation, citing the known black-hole case [64]. While this explanation is plausible, the presence of such an unphysical feature inside the frequency band being studied means the numerical reflectivity curves in that band cannot be taken as reliable quantitative predictions. The thermodynamic argument makes low-frequency amplification plausible, but it does not validate the specific curves or the claimed scaling with η̂ and Ω.
  3. [Appendix A and section 'Viscosity-driven instabilities'] The no-instability conclusion for uniformly rotating ultracompact stars is established only in the scalar toy model with an ad hoc absorption term α u^μ ∇_μ Φ (Eq. A1), not in the full viscous system (14). The toy model uses a radial absorption coefficient α(r) that is a free input, and it neglects the mode coupling and boundary conditions of the gravitational problem. The statement that 'amplified waves are not trapped, and trapped radiation is not amplified' is therefore a conjecture supported by a simplified model, not a derivation from the viscous equations. This weakens the paper's claim of linear stability of rotating viscous ultracompact objects, which is one of its advertised conclusions.
minor comments (6)
  1. [Eq. (3) and surrounding text] The notation Z_m is used both for the absorbed fraction and (in the Introduction) for the fluid perturbation variable Z in Eq. (12). This is potentially confusing; consider renaming the absorbed fraction (e.g., to A_m) or explicitly distinguishing the two.
  2. [Fig. 1 and Fig. 2 axes] The insets and axes would benefit from explicit labels: the inset of Fig. 1 shows R^2 − 1 but the axis label is not present in the text; Fig. 2's horizontal axis is ω/(mΩ), which should be stated consistently with the vertical axis being R^2 − 1.
  3. [Reference [59]] Reference [59] is a generic webpage URL rather than a stable repository identifier. Please provide the actual data/code repository DOI or a direct link so that the code availability claim is verifiable.
  4. [Transport-coefficient parametrization, section 'Setup'] The paper states that the transport coefficients and BDNK constraints are 'discussed in detail in Ref. [2].' Since Ref. [2] is a companion paper, this is acceptable, but a brief self-contained summary of the constraints (e.g., the allowed ranges of η̂ and τ̂) would make the present Letter more readable.
  5. [Appendix B, separation of equations] The projection formalism in Appendix B is standard, but the sentence 'the coefficients of the equations are lengthy and unilluminating, so we provide them as a Mathematica notebook' would benefit from a stable link or a deposited notebook file rather than 'available in other formats upon request.'
  6. [General presentation] The paper is generally well written, but there are occasional grammatical slips (e.g., 'the frequency of the fundamental mode is always larger than the superradiant threshold' in Appendix A should read 'is always larger than the superradiant threshold frequency'), and the use of 'superradiant' in the context of ω > mΩ is explicitly flagged as an artifact, which should be stated even more prominently in the results section.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical superradiance result is derived from BDNK hydrodynamics and independently computed; the only self-citation is a non-load-bearing transport-coefficient parametrization.

full rationale

The paper's derivation chain is self-contained. The master equations (14) are derived from the linearized Einstein equations with the BDNK stress-energy tensor (5)-(6), using the standard slow-rotation expansion; the exterior Regge-Wheeler equation (16)-(17) follows from the same metric. The reflectivity R^2 is extracted from asymptotic plane-wave amplitudes (19), not from any fitted parameter. The transport coefficients are chosen following the authors' prior work [2], but this is a physical modeling choice, not an input that by construction yields R^2>1; indeed the non-rotating limit reproduces absorption (R^2<1) as in [2], and amplification appears only when both rotation and viscosity are present in the numerically integrated system. The thermodynamic superradiance condition (1)-(3) is a standard independent argument predicting amplification for ω<mΩ when absorption is present; the numerical result is consistent with it but is not derived from it. The authors explicitly flag the low-frequency/linear-spin inconsistency as a limitation, which is a correctness risk, not circularity. There is no fitted parameter renamed as a prediction, no uniqueness theorem invoked from self-citations, and no ansatz smuggled in via citation.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the BDNK hydrodynamic model, the slow-rotation linearization, neglect of axial-polar coupling, and the specific polytropic background. The toy model for instabilities is an ad hoc absorption equation not derived from BDNK. No free parameters are fitted to data; all are model inputs scanned to explore the phenomenon.

free parameters (5)
  • dimensionless shear viscosity η̂ = 0.1, 0.2, 0.3, 0.5 (Fig. 1); 0.1, 0.3 (Fig. 2)
    Chosen by hand to explore the dependence of amplification on viscosity. The central claim requires η̂ > 0.
  • dimensionless relaxation time τ̂ = 500 (fixed)
    Chosen for the BDNK transport coefficients; the paper reports a very mild dependence on this parameter.
  • Polytropic EOS parameters = κ = 700 km^2.5, n = 0.8, ρ_c = 3 × 10^15 g/cm^3
    Define the background stellar model with M_S = 1.6 M_sun, R_S = 8.2 km, compactness 0.288. Not fitted to data.
  • rotation rate Ω/Ω_K = 0.21, 0.26, 0.32, 0.37 (Fig. 2); 0.26 (Fig. 1)
    Scanned to show rotation dependence; the amplification grows with rotation.
  • toy model absorption coefficient α_0 = 1, 10, 100, 1000 (Appendix A)
    Used in the scalar toy model to mimic dissipation; scanned to study the instability argument.
assumptions (6)
  • domain assumption BDNK first-order causal relativistic hydrodynamics with stress tensor (Eq. 5-6)
    The fluid model is assumed to be described by BDNK theory, cited to Refs. [39-43]. The central equations are derived from this stress tensor.
  • domain assumption Slow rotation approximation, linear order in Ω/Ω_K
    The background and perturbations keep only terms linear in rotation. This is explicitly stated in the Setup section and is the weakest assumption because the amplification regime enters where this expansion is inconsistent.
  • domain assumption Neglect of axial-polar mode coupling to linear order
    The ℓ-th axial multipole is treated separately; the paper states this coupling is negligible to linear order (following Ref. [55]).
  • domain assumption Barotropic equation of state p = p(ρ), constant temperature and chemical potential, β_E = c_s^2 τ_Q, β_N = 0
    Used to simplify the transport coefficients. Stated in the Setup section.
  • domain assumption Israel junction conditions with viscosity vanishing at the star surface
    The boundary condition (18) follows from requiring regularity of the Z equation at the surface, with the Israel conditions trivially satisfied because τ_Q, η → 0 at r = R_S. Stated in Appendix C.
  • ad hoc to paper Scalar toy model □Φ = α u^μ ∇_μ Φ as a proxy for viscous absorption
    Used in Appendix A to study the no-instability claim. This is not derived from BDNK, but is an effective model; the conclusions about stability rest on it.

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Cite this review

Pith. "Pith review of Superradiant amplification by rotating viscous compact objects." pith.science (2026). https://pith.science/paper/VYVZB3IE

@misc{pith2026250613850,
  author       = {Pith},
  title        = {Pith review of: Superradiant amplification by rotating viscous compact objects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYVZB3IE}},
  note         = {Machine review of arXiv:2506.13850}
}
read the original abstract

We study fluctuations of rotating viscous stars, using the causal relativistic hydrodynamics of Bemfica, Disconzi, Kovtun, and Noronha. We derive, in a slow-rotation approximation, a coupled system of equations describing the propagation of axial gravitational waves through the star, which couple to internal viscous modes. We show that rotating viscous stars amplify incoming low-frequency gravitational waves, a phenomenon which we argue to be universal. Superradiant amplification does not seem to trigger an instability for uniformly rotating stars, even if the object is compact enough to have light rings.

Figures

Figures reproduced from arXiv: 2506.13850 by the authors.

Figure 1
Figure 1. Reflectivity as a function of the dimensionless fre [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Amplification factor R2 − 1 as a function of the dimensionless frequency ω/mΩ, for different values of the an￾gular velocity of the star Ω/ΩK (see legend). Solid (dashed) lines correspond to ηˆ = 0.1(0.3), while τˆ = 500 is fixed. for frequencies ω ≲ mΩ. We also find that the maxi￾mum amplification increases with the dimensionless shear viscosity. This parameter controls the absorption rate in the high-frequency lim… view at source ↗
Figure 3
Figure 3. Reflectivity minus one R2 − 1 as a function of the dimensionless frequency ω/mΩ for the toy model (A1). Solid lines correspond to a constant profile of α, which has a sharp cutoff at the surface of the star, whereas dashed lines corre￾spond to a smooth profile of α, which goes to zero smoothly towards the surface. The amplification factor scales propor￾tionally with α, as expected. 10−4 10−3 10−2 100 101 102 103 Ω ℜ… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Real part of the frequency of the fundamental mode, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.