Axial instability of rotating relativistic stars

John L. Friedman; Sharon M. Morsink

REVIEW

Axial instability of rotating relativistic stars

John L. Friedman, Sharon M. Morsink

General Relativity and Quantum Cosmology · Submitted 23 Jun 1997 · 4 references

Pith's verdict: The Newtonian argument is complete and clean; the relativistic sign argument leans on a three-equation-of-state sample that the prose oversells as universal.

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arXiv IDgr-qc/9706073 (version 3)

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CitationJohn L. Friedman, Sharon M. Morsink. "Axial instability of rotating relativistic stars." arXiv:gr-qc/9706073 (1997). https://pith.science/paper/LSDWUYQI

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Editor's note

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Short version: the Newtonian half of this paper is a clean, complete argument, and the relativistic half is a good argument with one sampled number carrying the sign. Both halves are worth your time, and the paper is a founding document of the r-mode literature for a reason that holds up on reading the full text, not just the abstract's claim at [abstract:s2]. What is new here relative to Andersson is the canonical-energy demonstration: negative-canonical-energy initial data for every m >= 2 at arbitrarily slow rotation, in Newtonian theory and in slow-rotation general relativity. The Newtonian section is the part I would hand to a student. They impose the canonical condition (30), exhibit the displacement (34-36), recover the Papaloizou-Pringle frequency (40), and get Ec < 0 in (43) with no hand-waving. The relativistic section is more delicate and the authors are honest about it: sigma becomes a function of r (Eq. 81), so what they construct is canonical initial data, not a normal mode, and they say so and hedge the conclusion as instability or marginal stability. That is the right hedge. Soft spots, in order. The sign of Ec in the relativistic case rests on Eqs. (110)-(111): the ratio of metric to fluid contributions must stay below 1, and for m = 2 it reduces to 2 pi (epsilon + p) r^2. The text asserts the maximum over radius occurs at the maximum-mass star for any equation of state, then verifies < 0.4 for three equations of state from Arnett and Bowers. That is a three-point sample dressed as a universal statement. The margin is a factor of about 2.5 and the stiffest case is already excessively stiff, so I do not think it breaks anything, but the prose should have said for the equations of state we examined. Second, Sect. 4's timescale is an order-of-magnitude estimate, tau ~ M/(R Omega)^{2m+2}, explicitly handed off to Lindblom, Owen and Morsink; one supporting citation is to Friedman, Lockitch and Morsink 1998, listed as in preparation, which you cannot check. Minor. Who it is for: anyone working on r-modes, the CFS mechanism, or neutron-star spin evolution. Recommendation: send it out. The Newtonian result is a keeper, the relativistic caveat is fixable in revision, and the paper is candid about what it defers.

Referee report

Significance

This is a foundational result for the r-mode instability: it establishes that the CFS mechanism operates at any nonzero rotation for low m, in sharp contrast to polar modes, for which the minimum unstable m grows without bound as Ω → 0. The Newtonian derivation is self-contained and internally checked against the known inertial-mode frequency, and the relativistic extension is a genuine new step rather than a restatement of the cited variational framework. The result is falsifiable and has clear observational consequences for hot, rapidly rotating neutron stars, and it dovetails with the companion numerical study of Andersson. The paper is candid about the limits of its own estimate, noting that the growth time is long at small Ω and that viscosity will presumably restore stability except in hot, rapidly rotating stars.

Major objections (1)

[§3, Eqs. (110)–(111)] The sign of the relativistic canonical energy rests entirely on the bound 2π(ε+p)r² < 0.4 in Eq. (111), and the two steps that establish it are not proved. First, the sentence preceding Eq. (111) asserts that the maximum of 2π(ε+p)r² over radius 'for any equation of state' occurs at the maximum-mass star, but no argument for this monotonicity is given; the actual verification is a direct calculation for three equations of state (G, C, L from Arnett & Bowers 1977). Second, for m = 2 the prefactor 16/[(m−1)(m+2)]² in Eq. (110) is exactly unity, so the entire safety margin is the factor of about 2.5 between 0.4 and the threshold of 1; a stiffer or more compact configuration that pushed 2π(ε+p)r² above 1 would make the positive metric term Eg in Eq. (109) dominate the negative fluid term Ef in Eq. (108) and the relativistic instability claim would not follow from this construction. I would ask the authors either to supply an analytic argument (for example, from the TOV equations, that 2π(ε+p)r² is maximized at the maximum-mass model) or to extend the numerical survey to a wider set of equations of state, including very stiff and very compact configurations, and to state the resulting margin explicitly. Because the Newtonian result and Andersson's numerical frequency provide independent support for the qualitative conclusion, this is a soft joint rather than a refutation, but it is load-bearing for the relativistic half of the abstract's claim and should be shored up.

Minor objections (12)

[§2, Eq. (24)] The line reads 'ρ = ρ0 + O(Ω²), ρ = p0 + O(Ω²), ν = ν0 + O(Ω²)'; the second relation should evidently be p = p0 + O(Ω²).
[§2, Eq. (26)] The final term is written as ξ^aξ^b(∇_a∇_b p + ρ∇_a∇_aν), but the contraction with ξ^aξ^b requires the second index on the second derivative of ν to be b, i.e. ρ∇_a∇_bν; the same index slip should be checked in the surrounding text.
[§2, Eqs. (27)–(28)] The density-gradient term (1/ρ)∇_aρ∇_b p that appears in Eq. (27) is dropped in passing to Eq. (28), and the reason is not stated. It would help to note explicitly that this term vanishes for the axial displacements considered because ξ^r = 0 and ρ and p depend only on r, so that ξ^aξ^b∇_aρ∇_b p = 0; otherwise a reader may suspect an uncontrolled O(1) term has been discarded.
[§2, Eq. (31)] The operator £_z is written as if z were a vector, but z is a scalar coordinate; presumably £_{z^a} with z^a = ∇^a z is intended, as the preceding sentence defining z^a suggests.
[§3, Eqs. (94)–(95)] The solution Eq. (95) does not appear to follow from Eq. (94): with l = m the denominator (l−1)(l+2) equals (m−1)(m+2), whereas Eq. (94) contains (m+1)(m+2), and Eq. (95) carries r² in the numerator where Eq. (94) would give an inverse power. Eq. (109) uses the combination m/[(m+1)(m+2)], which is consistent with Eq. (94) rather than Eq. (95). Please reconcile the two equations and correct whichever is in error.
[§3, Eq. (110)] The inequality is written as strict, but the prefactor 16/[(m−1)(m+2)]² equals unity at m = 2, so the relation should be ≤ rather than <.
[§4, Eq. (117)] The estimate τ ∼ M/(RΩ)^{2m+2} is dimensionally inconsistent in gravitational units, where M/(RΩ)^{2m+2} has dimensions of length rather than time; presumably a factor of R (or M/R) has been dropped in the last step. Please restore the dimensional factors so the Ω-scaling can be read off unambiguously.
[§2, Eqs. (29) and (42); §3, Eq. (106)] The absolute-value bars around cosθ ξ^θξ^φ are ambiguous, since ξ^θ and ξ^φ are complex for e^{imφ} dependence and the sign of this cross term feeds directly into the sign of the bracket in Eq. (43) and of the corresponding term in Eq. (106). Please state whether the bars denote a modulus, a symmetrized real part, or something else, and give the convention used when performing the θ integration.
[§3, Eqs. (106)–(109)] The paper correctly notes that the split of Ec into fluid and metric pieces is gauge-dependent, but it does not state the reason the sign comparison is nevertheless meaningful, namely that Ec is gauge-invariant for canonical data satisfying the initial value equations. A single sentence making this explicit would close an apparent logical gap in the argument that Eg can be neglected relative to Ef.
[§3, Eq. (108)] The rewriting of Ef from Eq. (106) into the form Eq. (108) is not transparent, and the reader cannot easily verify the trigonometric identity used. An intermediate step, or a statement of the identity, would make the sign analysis checkable.
[§4, discussion of Andersson (1997)] The statement that the magnitude of the imaginary part Andersson computes 'is an artifact of an O(Ω²) approximation method' is stronger than the supporting argument, which is only a scaling estimate. Since the paper elsewhere is careful to hedge, it would be appropriate to soften this to a suggestion or to give the concrete reason the O(Ω²) truncation cannot capture the true growth rate.
[References] The citation labels do not match the reference list: the text cites Friedman and Schutz 1978a and 1978b, but the list contains two unlabeled 1978 entries (ApJ 221, 937 and ApJ 222, 281), and the text cites Stergioulas and Friedman (1997) while the list gives Stergioulas & Friedman 1998 (ApJ 492, 301). In addition, Friedman, Lockitch & Morsink 1998 is listed as 'paper in preparation' but is cited in §4 for the claim that the relativistic timescale has the same Ω dependence; that claim should either be softened or its unpublished status flagged in the text.

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The paper

The paper claims

For any rotating perfect-fluid star, however slowly it spins, the authors construct initial data for an axial r-mode perturbation whose canonical energy is negative, and read that as instability or marginal stability for every $m$ of 2 or更多

The reading

A nonrotating spherical star has axial perturbations — convective currents that swirl without changing density or pressure, and that neither oscillate nor radiate. The question this paper takes up is what happens to those currents once the star rotates at all. The authors show, first in Newtonian gravity and then in general relativity, that for angular dependence $e^{imφ}$ with $l=m$ and $m ≥ 2$ one can write down canonical initial data whose canonical energy is negative, and the standard criterion then says the configuration is unstable or marginally stable. What makes this worth attention is the contrast with the other parity class: unstable polar modes require a minimum $m$ that grows without bound as the rotation rate goes to zero, whereas here no amount of slow rotation removes the instability. The authors are careful that the growth is slow — the timescale behaves like a high power of $Ω$ — so viscosity should still save all but hot, rapidly rotating neutron stars.

What carries the argument

The load-bearing device is the canonical-energy instability criterion: a model is unstable to perturbations with angular dependence $e^{imφ}$ if there exists canonical initial data — data satisfying the perturbed circulation/vorticity condition — for which the canonical energy $E_c$ is negative. Everything else is in service of making $E_c$ negative. The relativistic computation splits $E_c$ into a fluid piece $E_f$, which is negative and has the same structure as the Newtonian expression, and a metric piece $E_g$, which is positive; the ratio of their angular integrals is $16/[(m-1)(m+2)]^2$ times $2π(ε+p)r^2$, so the sign is settled by a single dimensionless number built from the star's own equation of state.

Extended reading notes

Core claim

The claim is that rotating perfect-fluid stars are unstable (or at best marginally stable) to axial r-modes for every $m ≥ 2$, at arbitrarily small angular velocity $Ω$. In the Newtonian limit the authors take the known slow-rotation frequency $σ = -mΩ[1 - 2/(l(l+1))]$, which is negative for $l ≥ 2$ while the frequency relative to the fluid, $σ + mΩ$, is positive — the mode is dragged forward in an inertial frame while moving backward through the star. They then exhibit an explicit displacement with $l = m$ that satisfies the canonical condition (vanishing perturbed vorticity) and evaluate the canonical energy, finding it negative, Eq. (44). The relativistic treatment repeats the construction with a four-velocity perturbation built from the axial vector $ε_{αβ}∇^β Y_{lm}$, obtains a canonical displacement whose frequency varies with radius, satisfies the momentum constraint by choosing a suitable time derivative of the metric perturbation, and shows the fluid contribution to the canonical energy is negative and dominates the positive metric contribution, Eq. (112).

Load-bearing premise

The relativistic sign result depends on the numerical bound $2π(ε+p)r^2 < 0.4$ holding everywhere inside the star for every equation of state, but the paper verifies it by direct calculation only for maximum-mass models of three sample equations of state and asserts the general case in a single sentence.

Editorial extensions

If this is right

Every rotating perfect-fluid stellar model is unstable or marginally stable to axial r-modes with $m ≥ 2$, with no minimum spin threshold — the Chandrasekhar-Friedman-Schutz mechanism operates at any nonzero rotation rate for these modes.
The axial and polar instability thresholds behave oppositely as $Ω → 0$: the minimum unstable $m$ diverges for polar modes but stays at 2 for axial modes.
The growth time for a mode of given $m$ scales roughly as $M/(RΩ)^{2m+2}$, so slow rotation means very slow growth, and viscosity should suppress the instability except in hot, rapidly rotating neutron stars.
For isentropic stars the axial normal modes are the $l = |m|$ ones, so the gravitational radiation is dominated by current multipoles rather than mass multipoles.
Because the time-independent axial modes of a spherical star have nonvanishing shear, viscous damping necessarily acts on them, which is why the instability is not expected to survive at low spin.

Reading between the lines

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

The relativistic sign argument collapses to one dimensionless number, $2π(ε+p)r^2$, so the claim is really a statement about how compact realistic equations of state can be; any stellar model pushing that number above order unity would reverse the conclusion for $m=2$.
Because the constructed data have a frequency that varies with radius, they are not normal-mode data; the honest reading of 'instability or marginal stability' is that a full outgoing-mode calculation is still needed to decide whether the mode actually grows.
The extreme steepness of the growth time in $Ω$ makes the effect a sharp discriminator rather than a broad one: it predicts a detectable r-mode only in a narrow corner of the spin-temperature plane, which is a testable shape, not just a testable sign.
The same parity split and the same fluid-versus-field energy competition could be carried over to other parity-odd sectors of a rotating star, where the identical bookkeeping would decide stability.

Original abstract

Perturbations of rotating relativistic stars can be classified by their behavior under parity. For axial perturbations (r-modes), initial data with negative canonical energy is found with angular dependence $e^{im\phi}$ for all values of $m\geq 2$ and for arbitrarily slow rotation. This implies instability (or marginal stability) of such perturbations for rotating perfect fluids. This low $m$-instability is strikingly different from the instability to polar perturbations, which sets in first for large values of $m$. The timescale for the axial instability appears, for small angular velocity $\Omega$, to be proportional to a high power of $\Omega$. As in the case of polar modes, viscosity will again presumably enforce stability except for hot, rapidly rotating neutron stars. This work complements Andersson's numerical investigation of axial modes in slowly rotating stars.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,508 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.59 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.39 · compares with prior results 0.94 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Perturbations of rotating relativistic stars can be classified by their behavior under parity. For axial perturbations (r-modes), initial data with negative canonical energy is found with angular dependence $e^{im\phi}$ for all values of $m\geq 2$ and for arbitrarily slow rotation. This implies instability (or marginal stability) of such perturbations for rotating perfect fluids. This low $m$-instability is strikingly different from the instability to polar perturbations, which sets in first for large values of $m$. The timescale for the axial instability appears, for small angular velocity $\Omega$, to be proportional to a high power of $\Omega$. As in the case of polar modes, viscosity will again presumably enforce stability except for hot, rapidly rotating neutron stars. This work complements Andersson's numerical investigation of axial modes in slowly rotating stars.

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	Perturbations of rotating relativistic stars can be classified by their behavior under parity.	background 1.00	0.12	0.03	asserted 1.00
s1	For axial perturbations (r-modes), initial data with negative canonical energy is found with angular dependence $e^{im\phi}$ for all values of $m\geq 2$ and for arbitrarily slow rotation.	result 0.99	0.86	0.46	asserted 1.00
s2	This implies instability (or marginal stability) of such perturbations for rotating perfect fluids.	interpretation 1.00	0.66	0.68	asserted 0.68
s3	This low $m$-instability is strikingly different from the instability to polar perturbations, which sets in first for large values of $m$.	interpretation 0.50	0.49	0.08	asserted 1.00
s4	The timescale for the axial instability appears, for small angular velocity $\Omega$, to be proportional to a high power of $\Omega$.	result 1.00	0.72	0.07	hedged possibility 0.88
s5	As in the case of polar modes, viscosity will again presumably enforce stability except for hot, rapidly rotating neutron stars.	interpretation 0.76	0.09	0.05	hedged possibility 0.80
s6	This work complements Andersson's numerical investigation of axial modes in slowly rotating stars.	background 0.77	0.18	0.02	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.93.

What was examined, and what was not

sources read

arXiv listing abstract v3

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

References

Works cited (4)

[1]

Andersson, N. 1997 “A new class of unstable modes of rotating rela tivistic stars,” ApJ, to appear (gr-qc/9706075) Arnett, W.D., & Bowers, R. L. 1977, ApJ Sup., 33, 415 Chandrasekhar, S. 1970, Phys. Rev. Lett., 24, 611 Friedman, J. L., 1978, Commun. Math. Phys., 62, 247 Friedman, J. L., & Ipser, J. R. 1992, Phil. Trans. R. Soc. Lond., A34 0, 391 Friedman,...

View on Pith Pith review arXiv 1997
[23]

Provost, J., Berthomieu, G. and Rocca, A., 1981, Astron. Astrop hys. 94,

View on Pith 1981
[126]

Regge, T. & Wheeler, J. A. 1957, Phys. Rev. 108, 1063 Stergioulas, N., 1996, The Structure and Stability of Rotating Relat ivistic Stars, Ph. D. Thesis, University of Wisconsin-Milwaukee. Stergioulas, N. & Friedman, J. L., 1998, ApJ, 492, 301 Yoshida, S. & Eriguchi, Y. 1997, ApJ, 490, 779 This preprint was prepared with the AAS L ATEX macros v4.0

View on Pith 1957
[463]

Papaloizou, J. & Pringle, J. E., 1978, Mon. Not. R. astr. Soc., 182, 4

View on Pith 1978

Forward citations

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How later papers used it

56 later papers cite it

Evidence: thin. 56 of the citing papers share no author with it, and 1 of those carries a confident label from Pith's citation classifier.

Pith read 3 of the 62 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its data or values.

Uses its data or values 1 paper

Obtained its dimensionless parameters J = 0.01635 and I = 0.261 using a Γ = 2 polytropic model with a stellar radius of R = 10 km from this paper.

Superradiant and dynamical spin-down of neutron stars with gravitational wave implications (2025) · 0.94 · paraphrased: the licence does not allow quoting

How the literature uses it

32 arXiv papers cite this in 34 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 18 sentences without a confident label

Uses its data or values 1 paper

Here dimensionless parameters $J = 0.01635$ and $I = I/(MR^2) = 0.261$ are obtained using a $Γ= 2$ polytropic model with a stellar radius of $R = 10 km$ [ref] .

Superradiant and dynamical spin-down of neutron stars with gravitational wave implications · $r$-mode oscillation · 0.94

Cites it as background 15 papers

The remarkable property of $r$-modes is that, neglecting dissipation, they are subject to a gravitationally driven Chandrasekhar-Friedman-Schutz (CFS) instability at arbitrary spin frequency $ν$ of a NS [ref] .

Explaining observations of rapidly rotating neutron stars in LMXBs · Introduction · 0.97
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Neutrino cooling and spin-down of rapidly rotating compact stars · Introduction · 0.91
As it was shown in 1998 [ref] , in the absence of dissipation r-modes (predominantly toroidal oscillations of rotating stars restored by the Coriolis force [ref] ) are unstable with respect to radiation of gravitational waves at any rotation frequency of a star.

Resonance suppression of the r-mode instability in superfluid neutron stars: Accounting for muons and entrainment · Introduction · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Andersson, N. 1997 “A new class of unstable modes of rotating rela tivistic stars,” ApJ, to appear (gr-qc/9706075) Arnett, W.D., & Bowers, R. L. 1977, ApJ Sup., 33, 415 Chandrasekhar, S. 1970, Phys. Rev. Lett., 24, 611 Friedman, J. L., 1978, Commun. Math. Phys., 62, 247 Friedman, J. L., & Ipser, J. R. 1992, Phil. Trans. R. Soc. Lond., A34 0, 391 Friedman,...

View on Pith Pith review arXiv 1997

[2] [23]

Provost, J., Berthomieu, G. and Rocca, A., 1981, Astron. Astrop hys. 94,

View on Pith 1981

[3] [126]

Regge, T. & Wheeler, J. A. 1957, Phys. Rev. 108, 1063 Stergioulas, N., 1996, The Structure and Stability of Rotating Relat ivistic Stars, Ph. D. Thesis, University of Wisconsin-Milwaukee. Stergioulas, N. & Friedman, J. L., 1998, ApJ, 492, 301 Yoshida, S. & Eriguchi, Y. 1997, ApJ, 490, 779 This preprint was prepared with the AAS L ATEX macros v4.0

View on Pith 1957

[4] [463]

Papaloizou, J. & Pringle, J. E., 1978, Mon. Not. R. astr. Soc., 182, 4

View on Pith 1978