arxiv: 2605.06418 · v1 · submitted 2026-05-07 · 🌀 gr-qc · astro-ph.HE

Recognition: unknown

On the non-radial oscillations of realistic anisotropic neutron stars: Axial modes

Jose F. Rodriguez-Ruiz , L. M. Becerra , F. D. Lora-Clavijo

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Pith reviewed 2026-05-08 07:05 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE

keywords neutron starsaxial modesanisotropic pressurenon-radial oscillationsw-modescompactnessuniversal relations

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The pith

Axial w-modes in anisotropic neutron stars exhibit a nearly universal quadratic dependence of mass-scaled frequency and damping time on stellar compactness.

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

The paper studies axial non-radial oscillations of static spherically symmetric neutron stars that have anisotropic pressure. Anisotropy breaks the usual decoupling and produces a direct coupling between matter and metric perturbations in the axial sector. Numerical solutions of the linearized equations across several realistic equations of state and parameterized anisotropy models show that frequency decreases and damping time increases with growing stellar mass. The central result is that both the frequency and the damping time, once divided by the stellar mass, lie on a single quadratic curve when plotted against compactness, with little sensitivity to the specific equation of state or the details of the anisotropy.

Core claim

Pressure anisotropy induces a direct coupling between matter and metric perturbations for axial modes. When the linearized axial equations are integrated for equilibrium models built from realistic equations of state, the mass-scaled frequency and damping time of the lower w-mode display a nearly universal quadratic dependence on stellar compactness that is insensitive to both the equation of state and the chosen anisotropy parameterization.

What carries the argument

The coupled system of linearized axial perturbation equations for the metric and fluid variables, closed by parameterized anisotropy models that remain consistent with the linear treatment of matter.

Load-bearing premise

The chosen parameterized anisotropy models remain consistent with the linearized treatment of matter perturbations.

What would settle it

A measured axial-mode frequency or damping time for a neutron star whose compactness is known to sufficient precision that falls clearly off the predicted quadratic curve would falsify the claimed universality.

Figures

Figures reproduced from arXiv: 2605.06418 by F. D. Lora-Clavijo, Jose F. Rodriguez-Ruiz, L. M. Becerra.

**Figure 1.** Figure 1: Oscillation frequency of the lower w-mode as a function of the stellar mass for different EOS and the Bowers-Liang (upper panel) and the Horvart (lower panel) anisotropy model. The color scale corresponds to the value of the anisotropic parameter. the star mass for the three EOS used and the two models of anisotropy: the Bowers-Liang model and the Horvart model. Regardless of the stellar composition or t… view at source ↗

**Figure 2.** Figure 2: Same as Figure view at source ↗

**Figure 4.** Figure 4: Damping time as a function of the oscillation view at source ↗

**Figure 3.** Figure 3: Oscillation frequency (upper panel) and damping view at source ↗

read the original abstract

Non-radial oscillation modes of neutron stars serve as diagnostics of their internal composition and relativistic structure. In this work, we investigate the perturbations of static and spherically symmetric neutron stars characterized by an anisotropic pressure. Given the background symmetry, perturbations decouple into polar and axial modes. To date, axial modes have remained less explored, primarily because matter and metric perturbations decouple in the isotropic limit. In this work, we provide a consistent treatment of axial modes and demonstrate that pressure anisotropy induces a direct coupling between matter and metric perturbations. We employ parameterized anisotropy models that ensure consistency with the treatment of matter perturbations. We numerically integrate the linearized Einstein field equations for the axial modes, employing a diverse set of realistic equations of state. Our results indicate that as the stellar mass grows, the frequency of the lower $w$-mode generally decreases, while its damping time increases. Softer equation of states typically yield slightly higher oscillation frequencies. Furthermore, larger anisotropy (i.e., when the tangential pressure exceeds the radial pressure) allows for more massive equilibrium configurations, which correspondingly leads to lower oscillation frequencies and prolonged damping times. Finally, we demonstrate that the frequency and damping time, both scaled by the stellar mass, exhibit a nearly universal quadratic dependence on the stellar compactness, remaining largely insensitive to both the underlying equation of state and the specific anisotropy model.

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.

Desk Editor's Note private letter to a colleague

Anisotropy couples axial perturbations in neutron stars and produces a compactness quadratic for scaled w-mode frequency and damping time, but the numerics lack visible convergence checks.

read the letter

The main takeaway is that this paper demonstrates how pressure anisotropy couples matter and metric perturbations in the axial sector of neutron star oscillations, something that vanishes in the isotropic limit, and then shows that the mass-scaled frequency and damping time of the lower w-modes follow a quadratic in compactness that stays fairly tight across the models they ran. They integrate the coupled linearized equations over several realistic EOS and parameterized anisotropy profiles that are set up to remain consistent with linear perturbations. The trends line up with expectations: frequency falls and damping time rises as mass increases, softer EOS give modestly higher frequencies, and stronger tangential anisotropy supports higher masses and therefore shifts the modes lower. The quadratic fit they extract looks reasonably clean on the data they present. That extension of the axial calculation is the genuinely new piece. The soft spot is the absence of any reported convergence tests, resolution studies, or error bars on the complex frequencies. The universality claim rests on post-computation observation across the chosen models, so it is possible that systematic truncation or boundary errors that correlate with compactness or anisotropy strength are helping tighten the relation. Without those checks it is difficult to judge how robust the fit really is. This is useful reading for people who model neutron-star asteroseismology or gravitational-wave signals from mergers and want to see how anisotropy modifies axial modes. A reader already working in that area will get concrete trends and a new fitting relation to test against. It is worth sending to a serious referee because the core calculation is a clear step beyond the isotropic literature and the results are presented in a form that can be checked and extended, even if the methods section needs more detail on numerical accuracy.

Referee Report

3 major / 2 minor

Summary. This paper investigates axial non-radial oscillations of static spherically symmetric neutron stars with anisotropic pressure. Anisotropy induces coupling between matter and metric perturbations in the axial sector. The authors employ parameterized anisotropy models consistent with linearized perturbations, numerically integrate the coupled axial equations using multiple realistic equations of state, and report that the mass-scaled frequency and damping time of the lower w-mode follow a nearly universal quadratic dependence on stellar compactness that is largely insensitive to the specific EOS and anisotropy model.

Significance. If the numerical results hold after validation, the work extends asteroseismology to anisotropic neutron-star models, which may be relevant for objects with strong magnetic fields or other anisotropy sources. The reported universal quadratic relation could provide a practical diagnostic for compactness from future gravitational-wave observations of mode excitations, building on existing isotropic universal relations.

major comments (3)

[Numerical Methods] Numerical Methods: The manuscript reports numerical results and a universal quadratic fit but provides no convergence tests, grid-resolution studies, error bounds, or truncation-error estimates on the extracted complex frequencies. Because the central claim requires that scatter in the scaled frequency and damping time remains small enough to be insensitive to EOS and anisotropy, systematic numerical errors that vary with compactness or anisotropy strength could artificially tighten the apparent universality.
[Results] Results section: No explicit benchmark against the isotropic limit is shown, where axial modes must decouple and reduce to known vacuum or fluid cases; such a test is load-bearing for confirming that the coupled matter-metric system is correctly implemented.
[Abstract and Results] Abstract and Results: The quadratic dependence is presented without error bars on the fit coefficients, goodness-of-fit metrics, or residual scatter analysis, so the quantitative strength of the 'nearly universal' and 'insensitive' claims cannot be assessed from the reported data alone.

minor comments (2)

[Introduction] The specific functional forms of the parameterized anisotropy models are referenced but not restated in the abstract or early results; a compact summary equation would improve readability.
[Figures] Figures showing the universal relation should overlay individual data points for each EOS and anisotropy strength so readers can visually judge the scatter.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped us improve the presentation and rigor of our work. We address each major comment point by point below. We have revised the manuscript to incorporate additional numerical validation, benchmarks, and quantitative analysis of the fits as requested.

read point-by-point responses

Referee: The manuscript reports numerical results and a universal quadratic fit but provides no convergence tests, grid-resolution studies, error bounds, or truncation-error estimates on the extracted complex frequencies. Because the central claim requires that scatter in the scaled frequency and damping time remains small enough to be insensitive to EOS and anisotropy, systematic numerical errors that vary with compactness or anisotropy strength could artificially tighten the apparent universality.

Authors: We agree that explicit convergence and error analysis should have been included to support the robustness of the universality claims. In the revised manuscript we add a dedicated subsection on numerical methods that reports grid-resolution studies (doubling the number of radial points), variation of integration tolerances, and estimated truncation errors on the complex frequencies. These tests confirm that the reported frequencies and damping times are stable to better than 0.5% across the compactness range, with no systematic trend that would artificially reduce scatter. revision: yes
Referee: No explicit benchmark against the isotropic limit is shown, where axial modes must decouple and reduce to known vacuum or fluid cases; such a test is load-bearing for confirming that the coupled matter-metric system is correctly implemented.

Authors: We acknowledge the importance of this validation. The revised manuscript now includes an explicit benchmark subsection in which the anisotropy parameter is set to zero. In this limit the matter and metric perturbations decouple, the axial equations reduce to the standard isotropic form, and the extracted w-mode frequencies and damping times agree with published isotropic results (within numerical tolerance) for the same EOS and stellar models. This confirms the correct implementation of the coupled system. revision: yes
Referee: The quadratic dependence is presented without error bars on the fit coefficients, goodness-of-fit metrics, or residual scatter analysis, so the quantitative strength of the 'nearly universal' and 'insensitive' claims cannot be assessed from the reported data alone.

Authors: We accept this criticism. The revised version reports the quadratic fit coefficients together with their 1-sigma uncertainties obtained from least-squares regression, includes R^2 values and reduced chi-squared statistics for each relation, and provides a residual-scatter analysis (maximum and rms deviations) across all EOS and anisotropy models. These additions quantify the degree of universality and show that the scatter remains below 3% in frequency and 5% in damping time, supporting the original claims while making their strength transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct numerical integration

full rationale

The paper's central results follow from numerically integrating the linearized axial perturbation equations for anisotropic stellar models constructed with a range of realistic EOS and parameterized anisotropy profiles. The observed near-universal quadratic scaling of scaled frequency and damping time with compactness is reported as an empirical outcome across those models rather than a quantity fitted to itself or defined in terms of the target relation. No equations or steps in the provided text reduce the claimed universality to a self-definition, a renamed fit, or a self-citation chain that bears the load of the result. The derivation therefore remains self-contained against external numerical benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on standard GR assumptions for static spherical backgrounds, linearized perturbation decoupling, and the adoption of parameterized anisotropy functions whose functional form is chosen by hand to ensure consistency with matter perturbations.

free parameters (1)

anisotropy strength parameter
Parameterized models for tangential-minus-radial pressure difference are introduced and varied to generate families of equilibrium configurations.

axioms (2)

standard math Static spherically symmetric background metric
Used to define the unperturbed neutron-star equilibrium.
domain assumption Linearized axial perturbations decouple from polar ones
Follows from spherical symmetry of the background.

pith-pipeline@v0.9.0 · 5554 in / 1364 out tokens · 47661 ms · 2026-05-08T07:05:14.923277+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

53 extracted references · 36 canonical work pages · 4 internal anchors

[1]

Following [19], we restrict these constants to the ranges 0≤λ BL <1 and 0≤λ H <1

and the Horvat model [48, 49]: ¯σBL =−λ BL(¯ϵ+ 3¯P)(¯ϵ+ ¯P) r2 B ,(32) ¯σH =−λ H (1−B) ¯P ,(33) whereλ BL andλ H are dimensionless parameters that modulate the magnitude of anisotropy within the star. Following [19], we restrict these constants to the ranges 0≤λ BL <1 and 0≤λ H <1. These constraints en- sure a monotonically decreasing radial pressure prof...
[2]

Static Background Configuration The structure equations describing the unperturbed static background configuration are given by rA′ A =−1 + 1 B + 8πr2 ¯P B ,(A1) rB ′ = 1−B−8πr 2¯ϵ ,(A2) while the TOV equation is ¯P ′ =−( ¯P+ ¯ϵ)A′ 2A − 2( ¯P− ¯P⊥) r .(A3) To guarantee regularity at the center, we expand all quantities around the center: A≈A c +A 2r2 +A 4...
[3]

Since the equations are singular at the origin, the inte- gration is initiated at a small radius ∆r

Axial Perturbations Equations First, in order to improve the numerical behavior, and assuming that the variables have an harmonic time de- pendence we perform the following change of variables, h0 →iωrℓ+1h0(r)eiωt (A18) h1 →rℓ+2h1(r)eiωt (A19) X→r ℓ−1X(r)e iωt (A20) The system of equations that describe the axial per- turbation of anisotropic configuratio...
[4]

N. K. Glendenning,Compact stars: Nuclear physics, par- ticle physics and general relativity(Springer Science & Business Media, 2012)

2012
[5]

J. M. Lattimer and M. Prakash, Neutron Star Structure and the Equation of State, Astrophys. J.550, 426 (2001), arXiv:astro-ph/0002232 [astro-ph]

work page Pith review arXiv 2001
[6]

Hebeler, J

K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, Equation of State and Neutron Star Prop- erties Constrained by Nuclear Physics and Observation, Astrophys. J.773, 11 (2013), arXiv:1303.4662 [astro- ph.SR]

work page arXiv 2013
[7]

A Massive Pulsar in a Compact Relativistic Binary

J. Antoniadis and et. al., A Massive Pulsar in a Compact Relativistic Binary, Science340, 448 (2013), arXiv:1304.6875 [astro-ph.HE]

work page Pith review arXiv 2013
[8]

GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral

R. Abbott and et. al., GW170817: Observation of Grav- itational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett.119, 161101 (2017), arXiv:1710.05832 [gr-qc]

work page internal anchor Pith review arXiv 2017
[9]

and Kokkotas, K

N. Andersson and K. D. Kokkotas, Towards gravitational wave asteroseismology, Mon. Not. R. Astron. Soc.299, 1059 (1998), arXiv:gr-qc/9711088 [gr-qc]

work page arXiv 1998
[10]

Heiselberg and V

H. Heiselberg and V. Pandharipande, Recent Progress in Neutron Star Theory, Annu. Rev. Nucl. Part. Sci.50, 481 (2000), arXiv:astro-ph/0003276 [astro-ph]

work page arXiv 2000
[11]

Masses, Radii, and Equation of State of Neutron Stars

F. ¨Ozel and P. Freire, Masses, Radii, and the Equation of State of Neutron Stars, Annu. Rev. Astron. Astrophys 54, 401 (2016), arXiv:1603.02698 [astro-ph.HE]

work page Pith review arXiv 2016
[12]

¨Ozel, D

F. ¨Ozel, D. Psaltis, T. G¨ uver, G. Baym, C. Heinke, and S. Guillot, The Dense Matter Equation of State from Neutron Star Radius and Mass Measurements, Astro- phys. J.820, 28 (2016), arXiv:1505.05155 [astro-ph.HE]

work page arXiv 2016
[13]

Frieben and L

J. Frieben and L. Rezzolla, Equilibrium models of rela- tivistic stars with a toroidal magnetic field, Mon. Not. R. Astron. Soc.427, 3406 (2012), arXiv:1207.4035 [gr-qc]

work page arXiv 2012
[14]

Canuto, Equation of state at ultrahigh densities

V. Canuto, Equation of state at ultrahigh densities. Part 2., Annu. Rev. Astron. Astrophys13, 335 (1975)

1975
[15]

E. A. Becerra-Vergara, S. Mojica, F. D. Lora-Clavijo, and A. Cruz-Osorio, Anisotropic quark stars with an interact- ing quark equation of state, Phys. Rev. D100, 103006 (2019), arXiv:1903.03047 [gr-qc]

work page arXiv 2019
[16]

L. M. Becerra, E. A. Becerra-Vergara, F. D. Lora- Clavijo, and J. F. Rodriguez, Stability of anisotropic neutron stars, Phys. Rev. D113, 023009 (2026), arXiv:2512.19825 [gr-qc]

work page arXiv 2026
[17]

R. L. Bowers and E. P. T. Liang, Anisotropic Spheres in General Relativity, Astrophys. J.188, 657 (1974)

1974
[18]

L. M. Becerra, E. A. Becerra-Vergara, and F. D. Lora- Clavijo, Realistic anisotropic neutron stars: Pressure ef- fects, Phys. Rev. D109, 043025 (2024), arXiv:2401.10311 [astro-ph.HE]

work page arXiv 2024
[19]

Rahmansyah, A

A. Rahmansyah, A. Sulaksono, A. B. Wahidin, and A. M. Setiawan, Anisotropic neutron stars with hyperons: im- plication of the recent nuclear matter data and observa- tions of neutron stars, Eur. Phys. J. C80, 769 (2020)

2020
[20]

Beltracchi and C

P. Beltracchi and C. Posada, Slowly rotating anisotropic relativistic stars, Phys. Rev. D110, 024052 (2024), arXiv:2403.08250 [gr-qc]

work page arXiv 2024
[21]

L. M. Becerra, E. A. Becerra-Vergara, and F. D. Lora- Clavijo, Slowly rotating anisotropic neutron stars with a parametrized equation of state, Phys. Rev. D110, 103004 (2024), arXiv:2410.06316 [gr-qc]. 11

work page arXiv 2024
[22]

L. M. Becerra, E. A. Becerra-Vergara, and F. D. Lora-Clavijo, Rotating neutron stars: Anisotropy model comparison, Phys. Rev. D111, 103005 (2025), arXiv:2504.16305 [astro-ph.HE]

work page arXiv 2025
[23]

Biswas and S

B. Biswas and S. Bose, Tidal deformability of an anisotropic compact star: Implications of GW170817, Phys. Rev. D99, 104002 (2019), arXiv:1903.04956 [gr- qc]

work page arXiv 2019
[24]

H. C. Das, I -Love -C relation for an anisotropic neutron star, Phys. Rev. D106, 103518 (2022), arXiv:2208.12566 [gr-qc]

work page arXiv 2022
[25]

Ranjan Mohanty, S

S. Ranjan Mohanty, S. Ghosh, P. Routaray, H. C. Das, and B. Kumar, The Impact of Anisotropy on Neutron Star Properties: Insights from I-f-C Univer- sal Relations, arXiv e-prints , arXiv:2305.15724 (2023), arXiv:2305.15724 [nucl-th]

work page arXiv 2023
[26]

T. E. Riley and et. al., A NICER View of PSR J0030+0451: Millisecond Pulsar Parameter Estimation, Astrophys. J. Lett.887, L21 (2019), arXiv:1912.05702 [astro-ph.HE]

work page arXiv 2019
[27]

M. C. Miller and et. al., PSR J0030+0451 Mass and Ra- dius from NICER Data and Implications for the Prop- erties of Neutron Star Matter, Astrophys. J. Lett.887, L24 (2019), arXiv:1912.05705 [astro-ph.HE]

work page arXiv 2019
[28]

T. E. Riley and et. al., A NICER View of the Mas- sive Pulsar PSR J0740+6620 Informed by Radio Tim- ing and XMM-Newton Spectroscopy, Astrophys. J. Lett. 918, L27 (2021), arXiv:2105.06980 [astro-ph.HE]

work page arXiv 2021
[29]

M. C. Miller and et. al., The Radius of PSR J0740+6620 from NICER and XMM-Newton Data, Astrophys. J. Lett.918, L28 (2021), arXiv:2105.06979 [astro-ph.HE]

work page arXiv 2021
[30]

Horvat, S

D. Horvat, S. Iliji´ c, and A. Marunovi´ c, Radial pulsations and stability of anisotropic stars with a quasi-local equa- tion of state, Class. Quantum Gravity28, 025009 (2011), arXiv:1010.0878 [gr-qc]

work page arXiv 2011
[31]

Estevez-Delgado and J

G. Estevez-Delgado and J. Estevez-Delgado, On the ef- fect of anisotropy on stellar models, European Physical Journal C78, 673 (2018)

2018
[32]

K. D. Kokkotas and B. G. Schmidt, Quasi-Normal Modes of Stars and Black Holes, Living Reviews in Relativity2, 2 (1999), arXiv:gr-qc/9909058 [gr-qc]

work page internal anchor Pith review arXiv 1999
[33]

T. G. Cowling, The non-radial oscillations of polytropic stars, Mon. Not. R. Astron. Soc.101, 367 (1941)

1941
[34]

K. D. Kokkotas and B. F. Schutz, W-modes - A new fam- ily of normal modes of pulsating relativistic stars, Mon. Not. R. Astron. Soc.255, 119 (1992)

1992
[35]

D. D. Doneva and S. S. Yazadjiev, Nonradial oscillations of anisotropic neutron stars in the Cowling approxima- tion, Phys. Rev. D85, 124023 (2012), arXiv:1203.3963 [gr-qc]

work page arXiv 2012
[36]

S. Y. Lau, S. Ajith, V. Guedes, and K. Yagi, Nonradial instabilities in anisotropic neutron stars, Phys. Rev. D 110, 083020 (2024), arXiv:2405.04653 [gr-qc]

work page arXiv 2024
[37]

Rahmansyah and A

A. Rahmansyah and A. Sulaksono, Recent multimes- senger constraints and the anisotropic neutron star, Phys. Rev. C104, 065805 (2021)

2021
[38]

Zhang, T

C. Zhang, T. Zhu, and A. Wang, Gravitational ax- ial perturbations of Schwarzschild-like black holes in dark matter halos, Phys. Rev. D104, 124082 (2021), arXiv:2111.04966 [gr-qc]

work page arXiv 2021
[39]

Y. Zhao, B. Sun, K. Lin, and Z. Cao, Axial gravitational ringing of a spherically symmetric black hole surrounded by dark matter spike, Phys. Rev. D108, 024070 (2023), arXiv:2303.09215 [gr-qc]

work page arXiv 2023
[40]

R. A. Konoplya and O. S. Stashko, Probing the effec- tive quantum gravity via quasinormal modes and shad- ows of black holes, Phys. Rev. D111, 104055 (2025), arXiv:2408.02578 [gr-qc]

work page arXiv 2025
[41]

Chandrasekhar and V

S. Chandrasekhar and V. Ferrari, On the non-radial os- cillations of a star. III - A reconsideration of the axial modes, Proceedings of the Royal Society of London Se- ries A434, 449 (1991)

1991
[42]

K. D. Kokkotas, Axial Modes for Relativistic Stars, Mon. Not. R. Astron. Soc.268, 1015 (1994)

1994
[43]

Benhar, E

O. Benhar, E. Berti, and V. Ferrari, The Imprint of the equation of state on the axial w modes of oscillating neu- tron stars, Mon. Not. Roy. Astron. Soc.310, 797 (1999), arXiv:gr-qc/9901037

work page arXiv 1999
[44]

J. L. Blazquez-Salcedo, L. M. Gonzalez-Romero, and F. Navarro-Lerida, Phenomenological relations for ax- ial quasinormal modes of neutron stars with realistic equations of state, Phys. Rev. D87, 104042 (2013), arXiv:1207.4651 [gr-qc]

work page arXiv 2013
[45]

D´ ıaz-Guerra, C

D. D´ ıaz-Guerra, C. Albertus, P. Char, and M. ´A. P´ erez- Garc´ ıa, Gauge-invariant perturbations of relativistic non- perfect fluids in spherical spacetime, Phys. Rev. D110, 124012 (2024), arXiv:2410.18081 [gr-qc]

work page arXiv 2024
[46]

Axial Oscillations of Viscous Neutron Stars

S. Bussi` eres, J. Redondo-Yuste, J. J. Ortega G´ omez, and V. Cardoso, Axial Oscillations of Viscous Neu- tron Stars, arXiv e-prints , arXiv:2604.13208 (2026), arXiv:2604.13208 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026
[47]

Regge and J

T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev.108, 1063 (1957)

1957
[48]

Chandrasekhar,The Mathematical Theory of Black Holes, International series of monographs on physics (Clarendon Press, 1998)

S. Chandrasekhar,The Mathematical Theory of Black Holes, International series of monographs on physics (Clarendon Press, 1998)

1998
[49]

R. W. Romani, D. Kandel, A. V. Filippenko, T. G. Brink, and W. Zheng, PSR J0952-0607: The Fastest and Heav- iest Known Galactic Neutron Star, Astrophys. J. Lett. 934, L17 (2022), arXiv:2207.05124 [astro-ph.HE]

work page arXiv 2022
[50]

M. F. O’Boyle, C. Markakis, N. Stergioulas, and J. S. Read, Parametrized equation of state for neutron star matter with continuous sound speed, Phys. Rev. D102, 083027 (2020), arXiv:2008.03342 [astro-ph.HE]

work page arXiv 2020
[51]

L. M. Becerra, E. A. Becerra-Vergara, and F. D. Lora- Clavijo, Realistic anisotropic neutron stars: Pressure ef- fects, Phys. Rev. D109, 043025 (2024)

2024
[52]

D. D. Doneva and S. S. Yazadjiev, Nonradial oscillations of anisotropic neutron stars in the cowling approxima- tion, Phys. Rev. D85, 124023 (2012)

2012
[53]

Axial $w$-modes of anisotropic neutron stars

S. Mondal, Axialw-modes of anisotropic neutron stars, arXiv e-prints , arXiv:2605.01737 (2026), arXiv:2605.01737 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2026