Radial Oscillations of Viscous Stars

arXiv:2603.23622 · gr-qc, astro-ph.HE · Submitted 2026-03-24 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Today's paper: "Radial Oscillations of Viscous Stars".

Jocelyn: The gist: Viscosity damps radial modes on millisecond timescales and induces fractional shifts in the oscillation frequency which increase both with the compactness and viscosity of the star,

Vera: First, who's behind it and why it matters.

Title and authors: Vera: So we’re diving into "Radial Oscillations of Viscous Stars" today. We’re looking at how adding viscosity changes the way neutron stars wobble when they oscillate radially. It sounds like a topic that gets a lot of attention in gravitational wave studies, and I'm curious what kind of new constraints this paper puts on those predictions.

Jocelyn: Yeah, it seems like the core idea is that viscosity doesn't just add some noise; it actually dampens these radial modes on timescales of milliseconds and causes shifts in the oscillation frequency. That sounds like a pretty big effect for how we model what we see in gravitational waves during mergers.

Subrahmanyan: Exactly. We’re seeing these effects increase with both the compactness and the viscosity of the star, which is interesting because it links internal physics right to external observations, like those from third-generation detectors.

Vera: Right, so if we look at what this paper actually found in terms of numbers, it says that for compact stars with bulk viscosities around ten to the power of thirty g/cm/s, viscosity shifts the fundamental mode downwards by up to a percent level. That’s a significant shift.

Jocelyn: A percent shift is substantial when you're trying to measure tiny frequency changes in gravitational wave signals. I wonder what kind of follow-up observations would be needed to actually pin down that level of precision in the data.

Subrahmanyan: The paper also points out that for more viscous stars, the oscillation frequency can become arbitrarily small, vanishing entirely when the bulk viscosities are around ten to the power of thirty-one g/cm/s across two different equations of state and several levels of compactness. That suggests a very specific kind of star we might be looking at.

Vera: It’s like they’ve mapped out where viscosity starts to completely change the dynamics, pushing the frequencies down until they essentially disappear for those higher viscosity values. That gives us a clear boundary for what we might expect in certain stellar models.

Jocelyn: And it makes sense that this is tied to how dissipation works in these systems, which is why the paper sets up two different hydrodynamic frameworks to look at this, Eckart and the causal one from Bemfica, Disconzi, Noronha, and Kovtun.

Subrahmanyan: That choice between those two frameworks highlights how important it is to have a well-posed theory when you're dealing with dissipation in gravity problems. The text mentions that the Eckart frame simplifies things but the underlying theory is ill-posed and acausal, while the BDNK formulation gives us a stable, causal, and well-posed first-order theory of dissipative hydrodynamics.

Title and authors: Vera: So if we look at the way they set up these equations, it shows that when you include viscosity in Eckart hydrodynamics, you get a non-hermitian problem for finding the radial modes because viscosity provides that dissipation channel.

Jocelyn: That non-hermitian nature is what introduces an imaginary part to those eigenvalues, which means we get damping on the modes rather than just real frequencies. The paper shows that this damping time tau is related to the oscillation frequency omega by omega equals two pi f minus i over tau.

Subrahmanyan: And it’s important because it means we are dealing with complex frequencies, not just simple oscillations, which is a direct consequence of having that dissipation channel in the equations of motion.

Vera: What I found interesting about the structure of these modes is that an initial perturbation on an Eckart star that excites a single overtone n greater than zero will eventually be dominated by the fundamental mode. That’s a key structural result they found there.

Jocelyn: That suggests that even if you start with something specific, the system tends to settle into the most fundamental mode in this viscous scenario. It simplifies things a bit when you think about long-term evolution of these systems.

Subrahmanyan: From a cosmological perspective, this dominance of the fundamental mode when approaching stability thresholds hints at how structure might form or evolve under dissipative conditions in dense matter. It's about which modes survive the damping process over time.

Vera: Now let’s talk about the improvements they suggest for their work itself. They point out that for causal frames, direct numerical integration of system nineteen can be numerically unstable, so they transform it into a system where the change in lambda is evolved as a constrained degree of freedom alongside the change in u and epsilon by wave equations.

Jocelyn: That sounds like a smart way to handle the numerical challenges inherent in those causal treatments, which is good because you want robust results regardless of how you solve the equations on your computer.

Subrahmanyan: It’s about making sure that when we use these theoretical frameworks, the numerical implementation doesn't introduce artifacts that mess with the physics they are trying to study. The authors are trying to ensure stability across those different causal frames.

Title and authors: Vera: So, in terms of model selection guidance, they suggest that close agreement between the results from these viscous fluid models is shown in figures two and three, which implies that Eckart hydrodynamics might be sufficient for describing the effects of small transport coefficients.

Jocelyn: That's a practical thing to know for anyone trying to use these tools. If you’re dealing with small viscosity, you can likely rely on the results from the simpler Eckart model without needing the more complex causal ones immediately.

Subrahmanyan: And that ties back into how we connect theory to observation; it suggests that if we are looking at stars where viscosity is very small, those simpler models might give us a good approximation of reality for stability analysis.

Vera: So, to wrap things up on the implications of "Radial Oscillations of Viscous Stars," the main takeaway is that viscosity doesn't prevent gravitational collapse at this linear level, but it significantly slows down the rate at which it happens.

Jocelyn: They quantify that this shift in the instability timescale scales proportionally with the viscosity, dominating any inviscid contribution when those bulk viscosities are around ten to the power of thirty g/cm/s or higher.

Subrahmanyan: That proportionality is crucial because it means we can predict how much slower a star collapses based on its internal viscosity, which connects directly to the matter physics we’re trying to understand inside neutron stars.

Vera: It also provided evidence that the threshold for instability might actually depend on viscosity when using those BDNK theories, which is a subtle point they brought up.

Jocelyn: It’s a lot of information packed into this paper, showing how viscosity acts as both a damping agent and a modifier of fundamental stability thresholds in these stellar oscillations.

Subrahmanyan: Overall, the paper on "Radial Oscillations of Viscous Stars" shows that viscosity plays a decisive role in the late inspiral and post-merger phases by dampening modes and affecting the collapse timescale.

Vera: It’s a solid piece of work for understanding how internal transport properties translate into observable gravitational wave signatures.

Jocelyn: It definitely opens up some avenues for how we interpret those signals, especially when we look at stars with potentially high core viscosities.

Subrahmanyan: We need to keep looking at these kinds of viscous effects because they connect the microscopic physics of matter directly to the macroscopic gravitational waves that detectors like Einstein Telescope will eventually measure.

The paper's summary: Vera: So, this paper is looking at how adding viscosity messes with the way neutron stars wobble when they vibrate radially.

Jocelyn: Yeah, it’s about these radial modes and how that extra friction causes them to dampen on millisecond timescales and shift their frequency.

Subrahmanyan: It connects the internal physics—like whether there are strange quarks or hyperons—to what we actually see in gravitational waves from mergers.

Vera: Right, they found that for compact stars with bulk viscosities around one ten to the power of thirty g per centimeter per second, this viscosity shifts the fundamental mode down by up to a percent.

Jocelyn: A percent shift is something you have to be really careful about when trying to measure tiny frequency changes in those gravitational wave signals.

Subrahmanyan: And they also found that if the viscosity gets even higher, around one ten to the power of thirty-one g per centimeter per second, the oscillation frequency can just become purely imaginary, meaning it stops oscillating altogether.

Vera: That tells us there's a real physical limit to how much viscosity can affect these modes before they essentially vanish.

Jocelyn: It’s like finding where the system transitions from being a simple wave to something that is completely overdamped and just slowly dying out.

Subrahmanyan: From a cosmological standpoint, this helps us figure out how much internal friction might be required in stellar cores for certain collapse scenarios to happen at all.

Vera: The paper also looked at two different ways to set up the math, Eckart and one from Bemfica, Disconzi, Noronha and Kovtun.

Jocelyn: That choice of framework shows how important it is to use a theory that actually makes sense for describing dissipation in these systems.

Subrahmanyan: The causal framework they used, the BDNK one, proved to be really stable and well-posed compared to the Eckart one, which they noted was acausal and ill-posed.

Vera: When you look at the results, even with a purely viscous star using Eckart’s setup, there’s still this signature of non-hermitian dynamics appearing when viscosity hits that one ten to the power of thirty g per centimeter per second level.

Jocelyn: That non-hermitian nature is what gives us that damping time tau, which links the real oscillation frequency f to the decay rate i over tau.

Subrahmanyan: And this damping is directly tied to how viscous a star is, so if you know its bulk viscosity, you can predict exactly how fast it’s going to lose energy through these modes.

Vera: So what does this mean for us who are actually looking at the data? It suggests that when we analyze gravitational wave signals from mergers, we need to account for this damping effect when trying to isolate the star's structure.

Jocelyn: It means those frequency shifts aren't just noise; they could be real fingerprints of the viscosity inside that neutron star.

Subrahmanyan: We need observational data that can actually resolve these small frequency changes, maybe with the Einstein Telescope or Cosmic Explorer, to see if we can find these viscous signatures.

Vera: It opens up a whole new way to probe the core matter composition through these oscillation modes.

Jocelyn: And it sets up a clear goal for future observations: looking for those precise frequency shifts in the post-merger phase.

The paper's improvements: Vera: So, we've talked about how viscosity dampens radial modes and shifts frequencies in neutron stars. Now let's talk about how they’re trying to make this work better for the next round of research.

Jocelyn: Right, they are suggesting some way to make these oscillation predictions more useful by incorporating the viscosity effects into prediction tools.

Subrahmanyan: So, it’s not just a static calculation anymore; it should be a system that can predict how viscosity parametrically shifts the radial modes of a star and also affect the rate of gravitational collapse.

Vera: That’s interesting because we’re moving beyond just looking at what happens in a perfectly stable star to modeling how viscosity modifies stability thresholds.

Jocelyn: And they are also focusing on making sure their results are robust against different causal frames when using the BDNK hydrodynamics.

Subrahmanyan: So, they want to check that the results hold up even if you change the way you frame the equations, ensuring that viscosity in that BDNK theory only slightly modifies the threshold of collapse.

Vera: They’re essentially doing a sanity check across different mathematical approaches to make sure we aren't just getting artifacts from a specific model setup.

Jocelyn: And they’ve also focused on the very high viscosity regime, where stars become so overdamped that their fundamental mode is purely imaginary.

Subrahmanyan: That tells us we need to develop methods for detecting that non-oscillatory evolution when stars are extremely viscous, which is a key piece of information for structure formation.

Vera: They also provided guidance on how to pick the right model, showing that close agreement between the results from Eckart and the BDNK fluid models is shown in figures two and three.

Jocelyn: That means if you’re dealing with small transport coefficients, you can probably get away with using the simpler Eckart model for your stability analysis without needing to jump straight into those more complex causal ones.

Subrahmanyan: So, this gives us a practical roadmap for theoretical astrophysicists choosing which hydrodynamic framework is most appropriate based on how much viscosity you expect to have in the star.

Vera: It’s about connecting the microscopic physics of transport coefficients directly to which computational tools are most efficient and accurate for the job.

Jocelyn: And this whole process helps us narrow down what kind of observational data would actually be needed to verify these effects.

Subrahmanyan: We need high precision in those gravitational wave measurements to confirm if we see those predicted frequency shifts in the post-merger phase.

Conclusion: Vera: So we’re wrapping up our look at "Radial Oscillations of Viscous Stars." Basically, this paper confirms that viscosity isn't just a minor correction; it actively shapes how these neutron stars vibrate and collapse.

Jocelyn: Right, they show that this friction causes damping on millisecond timescales and shifts the oscillation frequency based on how compact or viscous the star is.

Subrahmanyan: From a cosmological view, this means we can start to link internal matter physics—like strange quarks—to gravitational wave signals during mergers.

Vera: The main thing is that viscosity shifts the instability timescale by an amount proportional to its value when it’s around one ten to the power of thirty g per centimeter per second or higher.

Jocelyn: That means we can predict how much slower a star will collapse based on its internal friction, which is really important for modeling that post-merger phase.

Subrahmanyan: It also provided evidence that the threshold for instability might actually depend on viscosity when you use those BDNK theories, which is a subtle point.

Vera: The paper did a lot of groundwork here to make sure their mathematical tools are robust across different causal frames, which is good because it means we can trust the results more.

Jocelyn: It’s also helpful that they gave us guidance on model selection, suggesting Eckart might be enough for small transport coefficients.

Subrahmanyan: The limitation they flagged is that they haven't fully explored the non-linear regime yet, so we need to watch those gradients in more detailed simulations.

Vera: It’s a solid piece of work on "Radial Oscillations of Viscous Stars," showing how internal transport properties translate into observable gravitational wave signatures.

Jocelyn: This opens up avenues for future observations to look specifically for those frequency shifts during the late inspiral and post-merger.

Subrahmanyan: We really need those precise measurements to confirm if we are seeing these viscous effects in reality, or if the stars are behaving more like perfect fluids.

Department of Applied Mathematics and Theoretical Physics, University of Cambridge · Center of Gravity, Niels Bohr Institute, University of Copenhagen · William H. Miller III Department of Physics & Astronomy, Johns Hopkins University

gr-qc, astro-ph.HE

Submitted: 2026-03-24

Updated: 2026-10-15

Comments: 15 + 3 pages, 9 figures; V3: fixed typo in Eq. 9

Journal ref: Phys. Rev. D 113, 124058, 2026

DOI: 10.1103/41kq-fj3h

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 78/100

The gist: The gist: Viscosity damps radial modes on millisecond timescales and induces fractional shifts in the oscillation frequency which increase both with the compactness and viscosity of the star,

Key concepts

Radial Oscillations
These are specific types of vibrations where the star oscillates purely along its radial direction, like a spring stretching and relaxing. These modes are crucial because they carry information about the internal structure and composition of neutron stars, which is what scientists hope to learn from gravitational waves.
Bulk Viscosity ($ζ$)
Bulk viscosity represents the internal resistance of the neutron star matter to changes in its volume or density. It acts as a dissipative force, meaning it converts kinetic energy from oscillations into heat. High bulk viscosity is expected if the core contains exotic particles like strange quarks.
Hydrodynamic Frameworks (Eckart vs. BDNK)
These are mathematical models used to describe how the fluid inside the star moves and dissipates energy due to viscosity. Eckart's framework is simpler but less physically complete, while BDNK provides a more causal and stable description of dissipative fluids, making it better for accurate calculations.

Terminology

Summary

The gist: Viscosity damps radial modes on millisecond timescales and induces fractional shifts in the oscillation frequency which increase both with the compactness and viscosity of the star, reaching up to the percent level for the fundamental mode with bulk viscosities ζ ∼ 1030 g/cm/s

Introduction

Neutron stars are a fascinating laboratory for new physics, since the four fundamental interactions of nature play a major role in their dynamics. One key puzzle is related to the composition of matter in the interior of neutron stars, where the density is most extreme [1, 2]. Gravitational wave observations of neutron stars merging seemingly offer a promising avenue to answer this question. Third-generation detectors such as the Einstein Telescope and Cosmic Explorer will likely observe most of the neutron star binaries merging in the Universe, with greatly improved accuracy compared to current observations [7–9]. A particularly promising avenue for inferring properties of matter in the interior of neutron stars is gravitational wave asteroseismology. This program aims to deduce the composition of stars by characterizing their oscillation modes. Stellar oscillation modes leave an imprint in the gravitational wave signal during the late inspiral (as tidal resonances), and in the post-merger phase. The impact of viscosity on neutron star oscillations has received limited attention. Recent research has demonstrated that if neutron stars feature strange quarks in their core – either isolated, or forming hyperons – the star’s bulk viscosity cannot be safely neglected. In fact, the bulk viscosity can be as large as ζ ∼ 1028 − 1030 g/cm/s during the late inspiral and the post-merger. An observational detection of viscosity becomes a smoking gun of strangeness in the core of neutron stars.

Hydrodynamic Frameworks and Theory

The study focuses on purely radial oscillations and works perturbatively to linear order within two hydrodynamic frameworks: the acausal covariant generalization of the Navier-Stokes equations proposed by Eckart, and the causal generalization formulated by Bemfica, Disconzi, Noronha, and Kovtun (BDNK). The problem of finding the radial modes of a star is rendered non-hermitian by viscosity because it provides a dissipation channel. The treatment in Eckart theory is simplified but the underlying hydrodynamic theory is ill-posed and acausal. The causal generalization formulated by BDNK [42–46] provides a well-posed, causal, and stable first-order theory of dissipative hydrodynamics. In the BDNK frame, the stress-energy tensor is given by A BDNK = τe [u a∇aϵ + (ϵ + p) ∇au a], Π BDNK = −ζ∇au a + τp [u a∇aϵ + (ϵ + p) ∇au a], Q BDNK = τq h(ϵ + p) u b∇bu a + c 2s∆ab∇bϵi, where ζ is the bulk viscosity. The causality constraints for linear perturbations require η, ζ ≥ 0, τe, τp, τq > 0 and specific inequalities involving the dimensionless transport coefficients.

Numerical Methods

The numerical methods employed to solve the equations of motion are described in Sec. IV. In the frequency domain for Eckart stars, the method combines a matrix method and a shooting method. The matrix method constructs a system of linear equations A⃗x = λ⃗x and numerically computes its eigenvalues and eigenvectors using package LinearAlgebra.jl. The shooting method refines results by directly solving Eq. (18) as an ordinary differential equation (ODE) in ξ, using initial conditions such as ξ(0, r) = 0 and Ξ(0, r) = 0. In the time domain for both Eckart and BDNK stars, the equations of motion are discretized on a uniform computational grid with radial step size h and time step ∆t = 0.05h. For causal frames, direct numerical integration of system (19) is prone to numerical instabilities, so the authors transform to a system in which δλ is evolved as a constrained degree of freedom and (δu, δϵ) by wave equations.

Results on Viscosity Effects

The radial modes of a viscous star are complex, unlike for a perfect fluid, due to the presence of dissipation. These frequencies are split into a real oscillation frequency f and a damping time τ, where ω = 2πf − i/τ. For compact stars with bulk viscosities ζc ∼ 1030 g/cm/s, viscosity shifts the fundamental mode downwards by up to the percent-level. For more viscous stars, the oscillation frequency can become arbitrarily small, vanishing for bulk viscosities ζ ≥ 1031 g/cm/s across two equations of state and several compactnesses. Consistent with stability analysis in the zero heat conductivity limit, viscosity in Eckart hydrodynamics cannot stabilize an unstable inviscid star. However, it can slow down the rate of collapse by several orders of magnitude in sufficiently viscous stars, slowing the timescale from milliseconds to seconds for bulk viscosities ζ ≥ 1033 g/cm/s.

Impact on Gravitational Collapse

Viscosity in Eckart hydrodynamics does not modify the perfect fluid threshold of gravitational collapse, as shown through analytic arguments. However, it can significantly affect how rapidly this instability ensues; viscosity shifts the timescale of the instability by an amount which scales proportionally with ζ, dominating the inviscid contribution for ζ ≥ 1030 g/cm/s. In contrast, in BDNK hydrodynamics, viscosity modifies the threshold of collapse slightly. The results suggest that stars which are unstable within a perfect fluid description are also unstable in the viscous theories considered in this work.

Non-Hermitian Dynamics

The study finds that an initial perturbation of an Eckart star which dominantly excites a single overtone n > 0 will eventually be dominated by the fundamental mode. The structure of the eigenvectors of the fluid depends strongly on the compactness of the star as the threshold of stability is approached. Viscous stars exhibit strong signatures of non-hermitian dynamics already at bulk viscosities ζ ∼ 1030 g/cm/s. For BDNK stars, the non-trivial coupling between gravity and the viscous fluid strongly affects the qualitative structure of the eigenvectors near the stability limit of the star.

Conclusion

Overall, we find that viscosity is unable to prevent gravitational collapse at the linear level. Viscosity does, however, slow down collapse: the viscous shift in the instability timescale is directly proportional to the viscosity of the star, dominating the inviscid contribution for ζ ≥ 1030 g/cm/s. The results also provide evidence that the threshold for instability may depend on viscosity in BDNK stars. Further study is required to verify that this shift is physical and to explore the fully nonlinear regime to check that gradients do not grow too large near the instability threshold.

Acknowledgments

J. R.-Y. is indebted to Laura Michelutti and Mathilde Menu for discussions on the oscillations of a viscous bubble. The authors thank Yago Bea, Alejandro CárdenasAvendaño, Vitor Cardoso, and Hengrui Zhu for valuable discussions, and Daniel Caballero, Marcelo Disconzi, Kostas Kokkotas, and Nicolás Yunes for comments on a draft of this manuscript. The Tycho supercomputer hosted at the SCIENCE HPC center at the University of Copenhagen was used for supporting this work.

References

[1] Gordon Baym, Tetsuo Hatsuda, Toru Kojo, Philip D. Powell, Yifan Song, and Tatsuyuki Takatsuka, “From hadrons to quarks in neutron stars: a review,” Rept. Prog. Phys. 81, 056902 (2018), arXiv:1707.04966 [astroph.HE].

[2] James M. Lattimer and Madappa Prakash, “The Equation of State of Hot, Dense Matter and Neutron Stars,” Phys. Rept. 621, 127–164 (2016), arXiv:1512.07820 [astro-ph.SR].

[3] Wit Busza, Krishna Rajagopal, and Wilke van der Schee, “Heavy Ion Collisions: The Big Picture, and the Big Questions,” Ann. Rev. Nucl. Part. Sci. 68, 339–376 (2018), arXiv:1802.04801 [hep-ph].

Improvements for AI systems

  1. textbfWeighting Viscosity Effects in Oscillation Modes: Incorporate Viscous Damping into Mode Characterization and Prediction Tools. This improved system can now predict how viscosity parametrically shifts the radial modes by quantifying how viscosity (i) parametrically shifts the radial modes of a stable inviscid star, and (ii) affects the threshold and rate of gravitational collapse.

  2. textbfFrame-Independent Results Validation: Implement Robust Frame Transformation Analysis for Hydrodynamic Codes. The system can now perform checks to ensure that the remainder of this section is structured as follows by verifying that results are robust against numerical dissipation, resolution, and changes of causal frame, confirming that viscosity in BDNK theory does slightly modify the threshold of collapse without being overly dependent on a specific choice among the three causal frames.

  3. textbfLarge Viscosity Regime Modeling: Develop Aperiodic Mode Detection Capabilities for Overdamped States. The improved system can now identify the non-oscillatory overdamped evolution characteristic of very viscous stars, specifically finding that Eckart stars with ζc ∼ 1031 g/cm3 are so viscous that their fundamental mode is purely imaginary.

  4. textbfCollapse Instability Rate Prediction: Quantify Viscosity's Role in Slowing Collapse Timescales. The system can now calculate the modification to the collapse timescale, finding that viscosity shifts the timescale of the instability by an amount which scales proportionally with ζ, dominating the inviscid contribution for ζ ≳ 1030 g/cm/s.

  5. textbfModel Selection Guidance: Provide Viscosity-Dependent Model Suitability Assessments. The system can now use viscosity as a metric to guide model choice, recognizing that the remainder of this section is structured as follows by highlighting that close agreement between the results of these viscous fluid models is shown in Figs. 2–3, suggesting Eckart may be sufficiently accurate to describe the effects of small transport coefficients.

Abstract

Oscillation modes of neutron stars, a key target for third-generation gravitational wave detectors, encode key information about their constituent nuclear matter. In this work, we study the effect of viscosity on oscillations of cold, polytropic, spherically symmetric neutron stars. We focus on purely radial oscillations and work perturbatively to linear order within two hydrodynamic frameworks: the acausal covariant generalization of the Navier-Stokes equations proposed by Eckart, and the causal generalization formulated by Bemfica, Disconzi, Noronha, and Kovtun (BDNK). We find that viscosity damps the radial modes on millisecond timescales and induces fractional shifts in the oscillation frequency which increase both with the compactness and viscosity of the star, reaching up to the percent level for the fundamental mode with bulk viscosities ζ about10 30 g/cm/s. For more viscous stars, the oscillation frequency decreases, becoming zero (i.e., an overdamped mode) for ζ 10 31 g/cm/s. We also study the linear threshold of gravitational collapse. Consistent with recent analytic results in the zero heat conductivity limit, we find that viscosity in Eckart theory cannot stabilize an unstable inviscid star. We provide numerical evidence that viscosity in BDNK theory is similarly unable to prevent gravitational collapse, but it slightly modifies the threshold of collapse. Overall, our results advance our understanding of the impact of viscosity on the oscillation modes of neutron stars, a key component of viscous asteroseismology with next-generation gravitational wave detectors.

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