Quasi-normal Modes and Instabilities of Rotating Relativistic Stars

Masaru Shibata; Yong Gao

REVIEW

Quasi-normal Modes and Instabilities of Rotating Relativistic Stars

Yong Gao, Masaru Shibata

High Energy Astrophysical Phenomena · Submitted 2 Nov 2026 · 113 references

Pith's verdict: First direct complex QNM spectra for rapidly rotating GR stars, but the formalism sits in an unpublished companion and the one clean comparison with published work is an admitted contradiction.

T0 · AI review

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arXiv ID2611.03986 (version 1)

Pith IDPJDRIZX2, Pith's permanent record of this paper and its review

CitationMasaru Shibata, Yong Gao. "Quasi-normal Modes and Instabilities of Rotating Relativistic Stars." arXiv:2611.03986 (2026). https://pith.science/paper/PJDRIZX2

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Claim paper

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Editor's note

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Colleague,

Bottom line: this is the first direct computation of complex f-, r-, inertial-, and w-mode frequencies for rapidly rotating relativistic stars in full GR, and that is a genuine step. The prior literature really does seem to stop at neutral-mode calculations, time evolutions, and energy-based rate estimates, so the novelty claim holds up. It deserves a serious referee, not a desk reject. But the paper is not yet independently checkable, and the one clean confrontation with published work is an acknowledged contradiction.

What is good. The range is impressive: distinct CFS and Jacobi-like neutral points for the MIT-bag sequence, avoided crossings and weak interactions between f and inertial modes, r-mode growth at every sampled spin, quasiradial collapse thresholds, and strongly damped w-modes. The internal consistency checks are real evidence: the static K0 x = 0 calculation reproduces the f-mode CFS crossing, and the slow-rotation w-mode results agree with an independent Regge-Wheeler calculation to within about a percent. The literature review is honest, and the 'to our knowledge' claim is not inflated.

Soft spots. First, the load-bearing derivation is not in the paper. Equation (3) is stated, but the gauge-fixing term, displaced-surface matching, and exterior complex scaling are all deferred to ref. [58], 'In preparation'. No reader can audit whether the reduction injects spurious gauge or boundary modes. That matters a lot for a first claim of this kind. Second, the quasiradial ordering rho_c,N > rho_c,TP directly contradicts ref. [99] for a Gamma=2 polytrope, and the paper says so without analyzing why. The quoted offsets are tiny, roughly 10^-5 to 10^-3, and no convergence data accompany them, so 'always larger' is not yet established. Third, the claimed independent check via the F-mode zero is not genuinely independent: both the static neutral operator and the F-mode frequency come from the same action reduction, so a common-mode error would shift both markers together.

Who this is for: anyone working on compact-star asteroseismology, r-mode searches, or gravitational-wave instability predictions. The paper will be read and cited regardless. My recommendation: send it to peer review, but require the companion formalism to be posted or included, add convergence or error analysis for the small offsets, and demand a quantitative discussion of the discrepancy with ref. [99]. The authors deserve the referee time; the current manuscript just does not yet carry enough of the proof itself.

Referee report

Significance

If correct, the results would be a substantial advance: a full-GR, outgoing-wave eigenvalue calculation of the complex spectra of rapidly rotating relativistic stars, going beyond slow-rotation and Cowling approximations and providing direct complex frequencies and instability thresholds relevant to gravitational-wave asteroseismology. The authors are explicit that the central derivation is deferred to a companion paper listed as 'In preparation,' and the manuscript does not include convergence tests or numerical error estimates. The single externally checkable quasiradial result is stated to be in direct tension with Ref. [99] without analysis. The paper's strengths include an action-based approach that avoids inverting the target frequencies, use of high-accuracy KADATH backgrounds, and several clearly stated internal consistency checks; however, those checks share the same action reduction and therefore do not independently validate it.

Major objections (4)

[Formulation, Eq. (3)] The central derivation is not auditable in this paper. All complex frequencies in Figs. 1-4 are eigenvalues of the quadratic problem K0 + σK1 + σ²K2 = 0 assembled from the second-order Lorenz-gauge action with the added gauge-fixing term, displaced-surface matching, and exterior complex scaling. The paper describes these steps but explicitly defers the formalism to Ref. [58], which is listed as 'In preparation.' Because that source is unpublished, the reader cannot check that the reduction has no spurious gauge modes, that the surface matching is consistent with the displaced surface, or that the complex-scaling implementation selects exactly the outgoing quasi-normal spectrum. This is load-bearing: if any of these elements is incomplete or inconsistent, every reported frequency, damping/growth rate, and threshold would shift. The revision should include the derivation in an appendix or supplement, or provide stringent benchmarks (e.g., convergence with basis size and scaling angle, recovery of known spherical and slow-rotation limits), before the claim of first direct full-GR calculation can be evaluated.
[Quasiradial neutral point, Fig. 3] The paper's only direct confrontation with an independent published calculation is left unresolved. The text reports ρc,N > ρc,TP 'at every sampled J,' including a Γ=2 polytrope with K=100 M⊙² at J=1.75 M⊙², and concedes 'clear tension with Ref. [99],' which found the neutral line on the lower-density side of the turning-point line for a Γ=2 polytrope. Since this is the same type of model and the same physical quantity, the discrepancy must be analyzed rather than merely acknowledged: the authors should identify whether it arises from a different equilibrium-sequence construction, a different turning-point definition, a difference between the quasiradial mode and the static reduced operator, or a genuine disagreement. Without this, the abstract's claim that the dynamical threshold is 'always larger' than the secular one is not supported.
[Fig. 3, F-mode verification] The claimed independent verification of the static neutral operator by tracking the F-mode frequency to zero is not actually independent. Both diagnostics are obtained from the same Ritz-Galerkin reduction of the same Lorenz-gauge action, with the same gauge choice, surface treatment, and exterior boundary procedure; a common-mode error would move the circles and crosses in Fig. 3 together while preserving their agreement. A genuinely independent check is needed, for example the nonrotating limit against established radial-oscillation results, or a comparison of one rotating case with an independent time-evolution or alternative gauge calculation.
[Numerical accuracy, Figs. 1-3] No convergence tests or error estimates are reported, yet several load-bearing numbers are quoted to high precision and the quasiradial ordering involves very small differences. The relative offsets in Fig. 3 are as small as 10^-5-10^-3 in central density, the avoided-crossing gaps are quoted as approximately 31 Hz and 6.8 Hz, and the neutral chiralities are given to four decimal places in β. The paper should report spectral resolution tests (basis order), sensitivity to the complex-scaling angle and matching radius, and, ideally, provide the numerical data or convergence tables. Without this, the quoted offsets and the sign of (ρc,N - ρc,TP)/ρc,TP cannot be distinguished from numerical error.

Minor objections (4)

[Fig. 1 caption] The caption appears to contain missing glyphs in the vertical-axis label ('□1 0 1 2 □...'), making the plotted quantity hard to read; this must be corrected in the production version.
[Fig. 2] The caption should state explicitly which line styles denote the m=+2 and m=-2 outer branches and which markers denote axial-led versus polar-led inertial modes; the gray inertial branches are difficult to distinguish from the dashed partner curves.
[Spacetime modes, Fig. 4] The 'independent slow-rotation calculation in Regge-Wheeler gauge' used to check the w-mode splitting is neither cited nor specified; providing the equations, gauge conditions, and truncation order would make the reported 0.7% and 0.3% agreements meaningful.
[Radiation-time estimates, Sec. Neutron-star fluid modes] The fitted coefficients k_f and k_r for the approximate radiation-time curves in Fig. 2 should be accompanied by a statement of the fitting procedure, the data range used, and an estimate of their accuracy, since these curves are presented as quantitative comparisons to the full mode results.

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The paper

The paper claims

This paper reports the first direct full-GR calculation of the complex quasi-normal spectra of rapidly rotating stars: f-, r-, and inertial-mode frequencies with their radiative damping and growth rates, plus strongly damped w-modes and…

The reading

Rapidly rotating neutron stars and strange stars emit gravitational waves whose frequencies and damping or growth rates encode the star's interior and its stability. Those complex frequencies have mostly been estimated by approximations, because a direct full-general-relativity calculation at high spin is hard. This paper claims to close that gap with a spectral variational method applied to the second-order Einstein–fluid action, and reports what it calls the first direct calculation of the complex spectra of f-, r-, inertial, and strongly damped w-modes of rapidly rotating relativistic stars. If the calculation is right, the quoted thresholds—the spins at which modes first go unstable, the r-mode growth times, and the density at which collapse begins—are direct predictions of general relativity that gravitational-wave searches can use.

What carries the argument

The load-bearing object is the quadratic eigenvalue problem $(K_0+\sigma K_1+\sigma^2 K_2)x=0$, assembled by a spectral Ritz–Galerkin method: the ten Lorenz-gauge metric perturbations and three fluid displacement components are expanded in spectral bases, and stationarity of the second-order Einstein–fluid action fixes the coefficient matrices. The Lorenz gauge gives every metric equation the same wave operator as its principal part; the interior and exterior problems are matched at the displaced stellar surface with $\Delta p=0$; and outgoing radiation is selected by exterior complex scaling, deforming the exterior radial integration path so that the outgoing wave $e^{-i\sigma r}$ decays along it. The same action supplies the canonical energy $E_c$ and angular momentum $J_c$ that identify the CFS instability (negative energy in the inertial frame) and the Jacobi-like instability (negative energy in the corotating frame), and a Krein-collision reading of the two f-branches locates the conservative onset of dynamical bar-mode growth.

Extended reading notes

Core claim

The paper's central claim is that the entire complex quasi-normal spectrum of a rapidly rotating relativistic star follows from a single quadratic eigenvalue problem $(K_0+\sigma K_1+\sigma^2 K_2)x=0$, derived by varying the second-order Einstein–fluid action in Lorenz gauge with a displaced stellar surface and an exterior complex-scaling path that enforces outgoing radiation. On that basis the authors report quantitative stability statements: for the MIT-bag strange-star sequence, the CFS neutral point at $\beta=0.1023$, the Jacobi-like neutral point at $\beta=0.1665$, and the approach to dynamical bar-mode growth with growth time $\tau\simeq 1.0$ ms by $\beta=0.2715$; for the APR4 neutron-star sequence, an f-mode CFS threshold at $\chi=0.5638$ and r-mode growth at every sampled spin, with growth times shortening from 214 s at $\chi=0.3$ to 3.7 s at $\chi=0.65$. The paper also finds avoided crossings between f- and inertial modes, classifies them with a two-mode coupling model, and locates the quasiradial collapse threshold at a central density $\rho_{c,N}$ above the turning-point density $\rho_{c,TP}$ for every sampled angular momentum, verified by both a static neutral operator and the vanishing of the F-mode frequency.

Load-bearing premise

The load-bearing premise is that the Lorenz-gauge action reduction—including the gauge-fixing term, the displaced-surface matching, and the complex-scaled exterior integration path—yields exactly the physical quasi-normal spectrum with no spurious gauge or boundary artifacts; the paper defers the detailed derivation of that reduction to a reference listed as in preparation.

Editorial extensions

If this is right

For self-bound strange stars, the computed neutral points fix the spin sequence over which the f-mode branches are radiatively stable, secularly unstable via the CFS or Jacobi-like mechanisms, and finally dynamically unstable to bar formation, with $\tau\simeq1$ ms growth at $\beta=0.2715$.
r-mode growth at every sampled spin of the APR4 sequence, from a 214 s growth time at $\chi=0.3$ to 3.7 s at $\chi=0.65$, makes the r-mode a persistent gravitational-wave source in rapidly rotating neutron stars and gives continuous-wave searches concrete growth-rate targets.
The avoided-crossing analysis predicts where f- and inertial modes exchange their radiative character: a strong encounter with a 31 Hz real-frequency gap near $\beta=0.1362$, a weak encounter with a 6.8 Hz imaginary-part gap near $\beta=0.151$, and a weak f–5ip1 encounter near $\chi=0.529$ in the APR4 sequence.
The quasiradial collapse threshold $\rho_{c,N}>\rho_{c,TP}$ at every sampled angular momentum, verified two independent ways, means the dynamical collapse onset for uniform rotation sits above the secular turning-point density, which the paper notes is in tension with an earlier quasiradial stability result.
The w-mode spectra, with quality factors below 1.4 and agreement with slow-rotation results to within 0.7% in frequency and 0.3% in damping at low spin, show that the formalism also works in the regime where radiation damping is comparable to the oscillation frequency.

Reading between the lines

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

Inference: because the quadrupolar f-mode spectrum is the pole content of the tidal response, this action-based method should also be able to produce dynamical tidal Love numbers for rapidly rotating stars directly; comparing those with gravitational-wave measurements from coalescing binaries would test the computed frequencies independently.
Inference: if the ordering $\rho_{c,N}>\rho_{c,TP}$ survives in fully nonlinear evolutions, collapse of supramassive uniformly rotating remnants begins at slightly higher central density than turning-point arguments suggest, which would marginally lengthen estimated survival times of merger remnants.
Inference: at the weak-interaction crossings the f-mode radiative loss is partly redistributed to the inertial partner, so the f-mode gravitational-wave amplitude in that spin band should dip; a targeted search near $\chi\simeq0.53$ for the APR4 5ip1 encounter could test this prediction observationally.
Inference: extending the same machinery to differential rotation, which the authors describe as straightforward, would deliver the first full-GR complex spectra bearing on the low-$T/|W|$ corotation instability, a regime currently probed mainly by nonlinear simulations.

Original abstract

We derive the basic equations suitable for the coupled fluid--spacetime eigenvalue problem from the second-order action in full general relativity and accurately determine quasi-normal modes of a variety of rapidly rotating stars with outgoing-wave boundary conditions. For MIT-bag strange stars, the spectrum realizes Chandrasekhar's instability picture, with distinct Chandrasekhar--Friedman--Schutz and Jacobi-like neutral conditions and the transition to dynamical bar-mode growth. For neutron stars, we determine the $f$-, $r$-, and other inertial-mode spectra and their radiative damping or growth times. Interactions between $f$- and inertial modes reveal avoided crossings and exchanges for radiative damping or growth rates. We locate quasiradial collapse thresholds through a static stability analysis and independently verify them by tracking the fundamental $F$-mode frequency to zero, clarifying that the central density at the onset of the dynamical instability is always larger than that of the secular instability for uniform rotation. Our formalism and numerical implementation can also resolve highly damped $w$-modes.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-04: 1 request, 5,066 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.21 · compares with prior results 0.49 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We derive the basic equations suitable for the coupled fluid--spacetime eigenvalue problem from the second-order action in full general relativity and accurately determine quasi-normal modes of a variety of rapidly rotating stars with outgoing-wave boundary conditions. For MIT-bag strange stars, the spectrum realizes Chandrasekhar's instability picture, with distinct Chandrasekhar--Friedman--Schutz and Jacobi-like neutral conditions and the transition to dynamical bar-mode growth. For neutron stars, we determine the $f$-, $r$-, and other inertial-mode spectra and their radiative damping or growth times. Interactions between $f$- and inertial modes reveal avoided crossings and exchanges for radiative damping or growth rates. We locate quasiradial collapse thresholds through a static stability analysis and independently verify them by tracking the fundamental $F$-mode frequency to zero, clarifying that the central density at the onset of the dynamical instability is always larger than that of the secular instability for uniform rotation. Our formalism and numerical implementation can also resolve highly damped $w$-modes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We derive the basic equations suitable for the coupled fluid--spacetime eigenvalue problem from the second-order action in full general relativity and accurately determine quasi-normal modes of a variety of rapidly rotating stars with outgoing-wave boundary conditions.	contribution 0.70	0.97	0.15	asserted 1.00
s1	For MIT-bag strange stars, the spectrum realizes Chandrasekhar's instability picture, with distinct Chandrasekhar--Friedman--Schutz and Jacobi-like neutral conditions and the transition to dynamical bar-mode growth.	result 0.96	0.84	0.17	asserted 1.00
s2	For neutron stars, we determine the $f$-, $r$-, and other inertial-mode spectra and their radiative damping or growth times.	result 0.81	0.96	0.09	asserted 1.00
s3	Interactions between $f$- and inertial modes reveal avoided crossings and exchanges for radiative damping or growth rates.	result 1.00	0.91	0.07	asserted 1.00
s4	We locate quasiradial collapse thresholds through a static stability analysis and independently verify them by tracking the fundamental $F$-mode frequency to zero, clarifying that the central density at the onset of the dynamical instability is always larger than that of the secular instability for uniform rotation.	result 0.91	0.96	0.25	asserted 1.00
s5	Our formalism and numerical implementation can also resolve highly damped $w$-modes.	contribution 0.78	0.87	0.04	hedged possibility 0.58

What kind of paper the classifier reports this is: simulation study 0.34 · evidence from simulation or computation 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Quadrupolar mode frequencies of the MIT-bag se

**Figure 2.** FIG. 2. Fluid modes of the APR4 sequence with

Figures reproduced from arXiv:2611.03986 by the authors.

References

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Forward citations

Cited by 2 Pith papers

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Relativistic tidal response and mode excitation in neutron-star binaries
astro-ph.HE 2026-11 conditional novelty 7.0 of 10

A relativistic effective-field-theory matching yields the poles and residues of the tidal response for stratified neutron stars with crusts and phase transitions, enabling consistent mode excitation, gravitational-wav...
The $w$-mode instability of strangeon ergostars and its impact on energy extraction
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The w-mode instability does not grow along the maximum-energy-extraction paths of strangeon ergostars, supporting the availability of their rotational energy for powering short gamma-ray bursts.

Discussion

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