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Fully general relativistic description of rapidly-rotating axially-symmetric neutron stars for constraining nuclear matter equations of state

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Pith's one-line read Using fully general-relativistic rotating-star models, this paper claims that neutron stars can spin at 716 Hz in mass–radius regions that the standard empirical Kepler-limit formula marks as forbidden, so the 716 Hz constraint on…

desk verdict Careful KEH implementation with a genuinely interesting but overstated challenge to the 716 Hz constraint; the stability language goes beyond what is computed. read the letter →

arxiv 2505.20990 v3 pith:QQ4FC77F submitted 2025-05-27 nucl-th astro-ph.HEgr-qc

Hyukjin Kwon , Kazuyuki Sekizawa This is my paper
classification nucl-thastro-ph.HEgr-qc
keywords rotatingneutronstarsequationofstateSkyrmeenergydensityfunctionalgeneralrelativityKeplerlimitmass-radiusrelationmillisecondpulsarssymmetry
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

Using fully general-relativistic models of rotating neutron stars, this paper argues that fast spin changes how the nuclear equation of state should be constrained by observations. Its central case is the fastest known pulsar, PSR J1748–2446ad at 716 Hz: the paper finds that a substantial part of the mass–radius region that the usual empirical Kepler-limit formula marks as forbidden actually contains equilibrium solutions spinning below the true general-relativistic mass-shedding limit. With five Skyrme-type equations of state it shows that rotation raises both mass and radius along the M–R curve, that the effect is small below roughly 400 Hz but large above it, and that the maximum sustainable frequency depends on the stiffness of the equation of state and on the symmetry-energy slope $L$. The reason to care is that most nuclear-physics constraints still rely on the non-rotating TOV equation, and this paper gives concrete evidence that doing so can misstate which equations of state are observationally allowed.

What carries the argument

The load-bearing object is the KEH method, a fully general-relativistic self-consistent field scheme that constructs rigidly rotating, axially symmetric perfect-fluid stars. It solves elliptic integral equations for the metric potentials $\gamma$, $\rho$, $\alpha$ and the frame-dragging frequency $\omega$ in a quasi-isotropic coordinate system, together with the first integral of hydrostatic equilibrium that ties the fluid to the equation of state; a compactified radial coordinate brings spatial infinity onto the grid. The Kepler (mass-shedding) limit is evaluated from the geodesic condition for equatorial circular orbits, and the paper compares that general-relativistic limit against the Newtonian empirical formula. The other essential ingredient is a tabulated npeμ equation of state built from five Skyrme-type parameter sets plus standard crust equations of state, which lets the code output not only masses and radii but internal density and composition profiles.

What would settle it

Evolve one of the reported low-mass 716 Hz equilibria (for example the SLy4 configuration with central density $n_c \simeq 0.54\,n_0$ and mass about $0.75\,M_\odot$) with a fully general-relativistic hydrodynamics code: if it sheds mass, collapses, or develops an r-mode instability within a short dynamical time, the claim that this region is stable is falsified. A less expensive check is a linear radial-oscillation or r-mode analysis of the same equilibria looking for unstable modes.

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Extended reading notes

Core claim

On its own terms, the paper claims that a fully general-relativistic treatment of axisymmetric rotating stars changes how the 716 Hz spin constraint should be read. Using the KEH method, a fully general-relativistic self-consistent field scheme, with core matter described by five Skyrme-type equations of state in npeμ (neutron–proton–electron–muon) composition and standard crust equations of state, it computes mass–radius curves at fixed spin frequencies of 0, 205, 346, and 716 Hz. At 716 Hz, a significant portion of the gray 'forbidden' region defined by the empirical formula $\nu_K = 1045\,(M/1.4M_\odot)^{1/2}(10\,\mathrm{km}/R)^{3/2}\,\mathrm{Hz}$ is populated by convergent solutions whose angular frequency stays below the general-relativistic Kepler limit; the paper concludes that the parameter space accessible to stable rotating configurations is broader than previously estimated and that the 716 Hz constraint should be interpreted more conservatively. It also finds that the rotational increase in maximum mass at 346 Hz can push an EoS like SLy4 over the $2\,M_\odot$ boundary, and that radius enhancement at a given spin scales with the slope of the symmetry energy, $L$.

Load-bearing premise

The word 'stable' here only means the computed spin is below the mass-shedding limit; the paper does not check whether the 716 Hz configurations survive perturbations, and if any of them are dynamically unstable the claim of a broader allowed region is weakened.

Editorial extensions

If this is right

  • The empirical 716 Hz limit curve should be treated as approximate: fully general-relativistic equilibria occupy a substantial part of the mass–radius region the curve labels forbidden, so the constraint is less restrictive than usually stated.
  • For spin frequencies above about 400 Hz, rotation changes mass and radius enough that comparing TOV models with observations can bias equation-of-state constraints; rotating models become necessary.
  • At 346 Hz the rotational rise in maximum mass, though small, can matter for marginal cases: for SLy4 the maximum mass increases from $2.055$ to $2.062\,M_\odot$, moving it from barely consistent to more comfortably consistent with the observed mass of PSR J0740+6620.
  • The rotational increase in radius at fixed mass correlates with the symmetry-energy slope $L$, so equations of state with similar incompressibility but different $L$ respond differently to the same spin.
  • Low-mass stars deform more at a given frequency and develop thicker equatorial crusts, making low-mass rapidly rotating stars the most sensitive probes of rotation-induced structural changes.

Reading between the lines

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

  • Editorial inference: if the 716 Hz equilibria pass a full dynamical-stability test, pulsar searches should not dismiss low-mass candidates that fall inside the empirically 'forbidden' band, since such objects could be genuinely stable millisecond pulsars.
  • Editorial inference: a measured mass and radius for any pulsar near 716 Hz would turn the spin frequency from a veto into a direct EoS probe, because only equations of state stiff enough in the right density range can support that spin in full general relativity.
  • Editorial inference: extending the calculation to differential rotation, which the paper notes can support larger maximum masses, would likely shift the allowed region further outward, but it would also make the missing dynamical-stability analysis more urgent.
  • Editorial inference: a controlled scan that varies a single saturation parameter such as $L$ at a time, as the paper suggests for future work, could convert the observed correlations into a quantitative map from nuclear parameters to maximum spin frequency.
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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 / 4 minor

Summary. The paper presents a newly developed fully general relativistic code based on the KEH method for computing equilibrium structures of rapidly rotating, axially symmetric neutron stars. Using five Skyrme-type EoS parameter sets with npeμ core matter and BPS/BBP crusts, the authors compute mass-radius relations at fixed spin frequencies, study how rotation shifts these relations, correlate the shifts with nuclear matter parameters (especially the symmetry-energy slope L), and compare with the spin frequencies of PSR J0740+6620, PSR J0030+0451, and PSR J1748-2446ad. The central claim, stated in Sec. IV.C.3 and Fig. 7, is that fully general relativistic rotating configurations occupy a significant portion of the parameter space that the empirical 716 Hz constraint of Eq. (44) marks as forbidden, so that this empirical constraint should be interpreted more conservatively.

Significance. If the central claim holds, the paper provides a useful, fully relativistic reassessment of a widely used observational constraint on neutron star equations of state, and it offers concrete M-R curves for several modern Skyrme EoSs at realistic spin frequencies. The numerical work is self-contained: the KEH inputs are central density and axis ratio, the spin frequency is an output, the EoS parameters come from independent prior fits, and no parameter is fitted to the 716 Hz constraint. The convergence study in Table II shows mass and radius stable to about the third decimal across resolutions from 257×257 to 2049×2049, which supports the internal consistency of the code. However, the paper's main conclusion is weakened by the fact that 'stability' is identified only with being below the Kepler mass-shedding limit, and by the absence of any quantitative external validation of the code in the manuscript.

major comments (3)
  1. [Sec. IV.C.3, Fig. 7, Eq. (42)] The claim that a significant portion of the empirically forbidden region 'can actually correspond to stable rotating neutron star configurations' is not supported by the calculations shown. The only stability criterion applied is that the computed angular frequency lies below the general relativistic Kepler (mass-shedding) limit of Eq. (42). The KEH method solves for equilibrium configurations, not for dynamical, secular, or non-axisymmetric stability; radial/turning-point stability, the bar-mode instability, and the CFS r-mode instability are not examined. The manuscript itself states in Sec. V that 'stability analysis becomes crucial, particularly in connection with phenomena such as r-mode instabilities' and defers it to future work. Because the paper's central conclusion about the 716 Hz constraint depends on these equilibria being physically realizable and not merely formal solutions, the word 'stable' should either be replaced with 'equilibrium' or the statement should be accompanied by an explicit stability analysis or a clear caveat that stability has not been established.
  2. [Sec. III] The claimed validation of the new code against Stergioulas and Friedman (Ref. [18]) and Morsink and Stella (Ref. [41]) is only asserted verbally, without any quantitative comparison. Since the central conclusion rests on the accuracy of the KEH implementation and the code is not released, please include a table comparing masses, equatorial radii, and Kepler frequencies for identical input EoS, central density, and axis ratio against the published results, or at least provide a quantitative convergence statement for the Kepler frequency itself. The current convergence table (Table II) is for one configuration and does not demonstrate that the code reproduces established rotating-star results to the accuracy needed for the 716 Hz claim.
  3. [Sec. V] The discussion in Sec. V acknowledges that 'in order to systematically explore the 716 Hz constraint across a wider range of EoSs—especially to identify the lowest-mass neutron star that can stably rotate at this frequency—it will be necessary to develop a more numerically stable and automated framework.' This admission undercuts the strength of the Sec. IV.C.3 conclusion, because the paper does not actually identify the boundary of the allowed region or demonstrate that the configurations shown at 716 Hz are stable. Please either temper the central claim accordingly or provide the additional analysis needed to support it.
minor comments (4)
  1. [Abstract and Sec. II.D.2] The abstract describes the KEH method as providing 'stable solutions' for rotating equilibrium configurations; this wording conflates equilibrium with stability. Please change it to 'equilibrium configurations' or explicitly qualify that stability is meant only with respect to mass shedding.
  2. [Sec. V] There are several typographical and grammatical errors, e.g., 'relays on' should be 'relies on' and 'metioned' should be 'mentioned'. A careful proofread is recommended.
  3. [Table I and Ref. [25]] The SkT5a parameter set is cited only as 'M. Duta, private communication'. Please provide a citable published source for this parameter set, since it is used in several quantitative conclusions.
  4. [Sec. IV.C.1] The statement that rotation at 346 Hz makes the SLy4 EoS 'more satisfactory' relative to the PSR J0740+6620 mass constraint is based on a single EoS and a small mass increase (ΔM ≈ 0.007 M⊙). It would be helpful to state explicitly that the shift is only marginally larger than the numerical uncertainty implied by Table II, and that this is not a decisive EoS test.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central results are self-contained and validated against independent external benchmarks.

full rationale

The paper's derivation chain is self-contained. Rotating neutron star structures are computed with a newly developed code implementing the KEH method, where the inputs are the central density and the axis ratio, and the angular frequency, mass, and radius are self-consistently obtained outputs. The five Skyrme EoS parameter sets (BSk24, SkI4, SkM*, SkT5a, SLy4) are taken from independent prior fits to nuclear data, and the paper verifies that their saturation properties match the original references. The code is validated against published external results: stellar structures agree with Stergioulas and Friedman and Kepler frequencies are consistent with Morsink and Stella. The central claim about the 716 Hz constraint compares the computed Kepler-limited configurations against the empirical Lattimer-Prakash formula, which is an external benchmark and not a quantity fitted in this work. No parameter is fitted to the target result, and the maximum angular frequencies are not defined in terms of the empirical constraint. The paper's use of the word 'stable' relies on the general relativistic Kepler (mass-shedding) limit rather than a full dynamical stability analysis; this is a scientific caveat, not a circularity, because the equilibrium solutions are obtained independently and the stability criterion is not an input to the calculation. Self-citations are limited to methodological references (KEH, CST, RNS) that are independently established and externally validated by the community. Therefore, no load-bearing step reduces to its own input or to a self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No entities are invented. The only hand-chosen control is the rigid-rotation limit (A→∞). All nuclear parameters come from published Skyrme fits; the crust EoS and KEH equations are standard. The central claim depends on the rigid-rotation and npeμ-composition assumptions.

free parameters (1)
  • A (differential rotation parameter) = ∞ (rigid rotation)
    Set to the rigid-body rotation limit in Eq. (38); the central 716 Hz results assume rigid rotation and would need to be revisited for differential rotation.
assumptions (5)
  • domain assumption Rigid-body rotation with A tending to infinity
    The first integral of motion uses j(Ω)=A^2(Ω_c−Ω) and the large-A limit is taken (Sec. II.D.2); real pulsars may have differential rotation.
  • domain assumption npeμ composition throughout the core, no exotic degrees of freedom
    Sec. II.C assumes only neutrons, protons, electrons and muons, excluding hyperons or quarks; the inner core is not modeled.
  • domain assumption BPS and BBP crust EoS adopted
    Sec. II.C uses the Baym-Pethick-Sutherland and Baym-Bethe-Pethick equations of state for the outer and inner crusts.
  • standard math Validity of the KEH formulation
    The Einstein equations in Eq. (32) and the first integral Eq. (36) are adopted from Komatsu et al. 1989 without re-derivation.
  • domain assumption Skyrme EoS parameters from published fits
    Five parameter sets (BSk24, SkI4, SkM*, SkT5a, SLy4) are taken from the literature; their saturation properties are checked against published values.

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Cite this review

Pith. "Pith review of Fully general relativistic description of rapidly-rotating axially-symmetric neutron stars for constraining nuclear matter equations of state." pith.science (2026). https://pith.science/paper/QQ4FC77F

@misc{pith2026250520990,
  author       = {Pith},
  title        = {Pith review of: Fully general relativistic description of rapidly-rotating axially-symmetric neutron stars for constraining nuclear matter equations of state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQ4FC77F}},
  note         = {Machine review of arXiv:2505.20990}
}
abstract

Background: Constraining the nuclear matter equation of state (EoS) from neutron star observations is one of the main subjects in nuclear physics today. In general, neutron stars rotate rapidly and structure of neutron stars can be affected, especially in millisecond pulsars. To better constrain the nuclear EoS, it is important to describe neutron star structure taking into account the effects of rotation in a fully relativistic manner. Purpose: In this study, we investigate the internal structure of neutron stars under the influence of rotation. We explore correlations between rotational effects and EoS parameters, based on realistic calculations of rapidly rotating neutron stars based on the KEH method, which provides stable solutions for axially symmetric rotating equilibrium configurations. Results: Using 5 different Skyrme EoS parameter sets, we find that the maximum angular frequency achievable by rotating neutron stars, as calculated via the KEH method, varies depending on the stiffness of the equation of state. We confirm that an increase in the rotating frequency leads to an overall increase in both the mass and radius along the M-R curve. By performing calculations at two frequently referenced neutron stars, we further examine how the changes in mass and radius correlate with the nuclear matter properties at saturation density. Our results suggest that the 716Hz rotational constraint may require a more conservative interpretation when accounting for realistic stellar deformation effects. Conclusions: To place stringent constraints on the nuclear EoS based on observational data, it is sometimes essential to account for the effects of rotation in neutron star models. In particular, the influence of rotation becomes increasingly significant at higher spin frequencies and cannot be neglected in rapidly rotating systems with $>$400Hz.

Figures

Figures reproduced from arXiv: 2505.20990 by the authors.

Figure 1
Figure 1. FIG. 1. A schematic illustration of the cross section of a neutron star, [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Equations of state calculated under the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Angular frequency [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Mass-radius relations for rotating neutron stars with angular [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Mass–radius relation of rotating neutron stars at various [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Mass-radius relations for rotating neutron stars with angular [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Mass-radius relations for rotating neutron stars with an [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Pith tools

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