Limiting cases of second-order moments of relativistic stars and their universality

Abstract: Extending the work presented in a workshop ``From Quarks to Neutron Stars: Insights from kHz gravitational waves'', we discuss some limiting cases of the moment of inertia, tidal deformability, and spin-induced quadrupole moment for relativistic stars. First, conjecturing that a hierarchy of the length scale is the key to proposed universality among these second-order moments, we revisit the relation for incompressible relativistic stars (known as Schwarzschild's interior solution) as a candidate of the possible stiff limit. Second, we present the limiting form for the weak-field limit. In particular, we demonstrate how relativistic computations of tidal deformation are related to the traditional Newtonian counterpart, which might not have been presented explicitly in the literature.

• The paper shows that incompressible-star fits for moment of inertia and spin-induced quadrupole reproduce reference data within 0.1%, while realistic equations of state deviate systematically by roughly 0.8–1.5%.
• The paper demonstrates that relativistic tidal equations reduce in the weak-field limit to classical Clairaut-Radau theory, establishing an explicit connection between relativistic Love numbers and Newtonian stellar perturbations.
• The paper argues that exact universality may depend on identifying a causal stiff-limit equation of state, since incompressible matter violates causality and realistic neutron-star models do not lie exactly on its universal curves.

Overview

This paper by Kyutoku examines limiting cases of second-order moments of relativistic stars — the moment of inertia, tidal deformability $\Lambda$, and spin-induced quadrupole moment — and their bearing on the so-called universal relations among these quantities (2608.16376). The work builds on a simplified reformulation for computing the spin-induced quadrupole moment presented in Ref. (Kyutoku, 28 Feb 2025), and pursues two specific goals: first, to test whether incompressible matter (Schwarzschild's interior solution) can serve as the "stiff limit" that defines stellar universality; second, to derive explicitly the weak-field limit of relativistic tidal computations and connect it to the classical Newtonian treatment via Clairaut-Radau theory. The motivation is partly observational: tidal deformability was measured for GW170817 (Collaboration et al., 2018), but the spin-induced quadrupole moment is currently inferred from $\Lambda$ through universal relations accurate at the $\sim 1\%$ level (Yagi et al., 2013, Yagi et al., 2013), an approximation acceptable only while measurement errors dominate; future kilohertz-sensitive or space-borne detectors such as DECIGO (Isoyama et al., 2018) would demand independent determination.

Reformulation of second-order moments

The paper adopts a streamlined computational framework in which all three moments follow from solving single first-order ordinary differential equations on a Tolman-Oppenheimer-Volkoff background. The background is characterized by the compactness $C = M/R$ and the logarithmic enthalpy $H = \int_0^P dP'/(\varepsilon + P')$, with $e^{2\nu} = (1-2C)e^{-2H}$.

The normalized moment of inertia $\bar{I}$ follows from Hartle's rotational perturbation variable $w$ [1967ApJ...150.1005H, 1987A&A...172...95Z] through $\bar{I} = w_s/[C^3(2w_s+6)]$. The spin-induced quadrupole moment $\bar{Q}$ is obtained from a two-variable system $\Lambda$0 introduced in Ref. (Kyutoku, 28 Feb 2025), which avoids solving the full fourth-order structure of the original Hartle-Thorne problem [1968ApJ...153..807H]. The tidal deformability follows from Hinderer's equation for $\Lambda$1 (0711.2420) with the standard surface matching formula. The author notes one technical subtlety relevant later: for incompressible stars, the density discontinuity at the surface requires the shifted value $\Lambda$2, as established in Ref. (0906.0096).

This reformulation is what makes the limiting-case analyses below tractable as clean numerical exercises.

The stiff limit and Schwarzschild's interior solution

The central conjecture of the paper is that universality arises from a hierarchy of length scales: when the variation length scale of physical quantities inside the star greatly exceeds the stellar radius, equation-of-state dependence washes out — analogous to universality in critical phenomena and scattering problems. Under this view, the universal relation should be defined by the stiffest possible matter, and the natural candidate is the incompressible fluid, whose configuration in general relativity is Schwarzschild's interior solution.

The author computes $\Lambda$3 and $\Lambda$4 numerically for this solution over $\Lambda$5 and fits them with eighth-degree polynomials in $\Lambda$6, fitting $\Lambda$7 (subtracting the black-hole limit $\Lambda$8) rather than $\Lambda$9 itself because this improves fit quality; subtracting $\sim 1\%$0 does not help. The fits reproduce the incompressible data to better than $\sim 1\%$1 for $\sim 1\%$2. A notable practical advantage is that these reference curves involve no arbitrariness in equation-of-state selection and are immune to updates in observational constraints.

Comparing against nuclear-theory-based equations of state (APR4, SLy, SFHo, DD2), the maximum relative deviations of $\sim 1\%$3 from the incompressible fit are $\sim 1\%$4, $\sim 1\%$5, $\sim 1\%$6, and $\sim 1\%$7 respectively, and those of $\sim 1\%$8 are $\sim 1\%$9, $C = M/R$0, $C = M/R$1, and $C = M/R$2 over $C = M/R$3. These deviations are consistent with the scatter of empirically fitted universal relations (Yagi et al., 2013, Yagi et al., 2013). More significantly, the deviations are systematic: nuclear-matter models yield smaller $C = M/R$4 and larger $C = M/R$5 than the incompressible case. The author draws a strong interpretive conclusion: if incompressibility defines universality, then realistic neutron stars do not merely exhibit equation-of-state-dependent scatter around a universal curve — they fail to achieve exact universality altogether.

The paper concedes an important caveat here. Incompressibility implies infinite sound speed and violates causality, so Schwarzschild's interior solution may be ruled out as a viable candidate even in this context. A causally consistent alternative is luminal-sound matter, $C = M/R$6, which lies between the incompressible and nuclear-theory cases and should therefore deviate by less than $C = M/R$7 from both, barring an extreme choice of the additive constant. The author deliberately refrains from surveying candidates, suggesting instead that if the stiffness-based picture holds, analytic tools such as renormalization group analysis of the governing differential equations could identify the matter at the critical point. This identification of the true fixed-point equation of state remains open.

Weak-field limit and the connection to Clairaut-Radau theory

The second half of the paper recovers $C = M/R$8 and $C = M/R$9 and takes the Newtonian limit of each perturbation equation, clarifying how relativistic computations reduce to classical results. In this limit $H = \int_0^P dP'/(\varepsilon + P')$0 coincides with the gravitational potential $H = \int_0^P dP'/(\varepsilon + P')$1, which satisfies Newtonian Bernoulli's theorem $H = \int_0^P dP'/(\varepsilon + P')$2.

For the moment of inertia, the weak-field equation for $H = \int_0^P dP'/(\varepsilon + P')$3 retains an apparent $H = \int_0^P dP'/(\varepsilon + P')$4 dependence because $H = \int_0^P dP'/(\varepsilon + P')$5 is defined from the spacetime angular momentum rather than a fluid integral, though an equivalent fluid-based expression exists in Hartle's original work. For the spin-induced quadrupole, the author observes that substituting $H = \int_0^P dP'/(\varepsilon + P')$6 transforms the weak-field equation for $H = \int_0^P dP'/(\varepsilon + P')$7 into exactly Clairaut-Radau's equation of degree $H = \int_0^P dP'/(\varepsilon + P')$8. This suggests the relativistic equation could be rewritten more concisely and possibly used to derive tidal deformability as well, though the author states this has not yet been attempted.

The most explicit result concerns tidal deformation. The Newtonian limit of Hinderer's equation for $H = \int_0^P dP'/(\varepsilon + P')$9 [(0711.2420), 2009ApJ...697..964H] is shown to be equivalent to the traditional Newtonian computation via the variable transformation $e^{2\nu} = (1-2C)e^{-2H}$0, under which the resulting Love number formula $e^{2\nu} = (1-2C)e^{-2H}$1 matches the classical definition [2014grav.book.....P]. The derivation proceeds from the metric perturbation $e^{2\nu} = (1-2C)e^{-2H}$2 (with $e^{2\nu} = (1-2C)e^{-2H}$3): in the Newtonian limit $e^{2\nu} = (1-2C)e^{-2H}$4 is the potential perturbation $e^{2\nu} = (1-2C)e^{-2H}$5, and perturbed Poisson's equation combined with Bernoulli's theorem (which holds even under static tidal fields) reproduces the relativistic equation exactly. Eliminating the density perturbation in favor of the fractional displacement $e^{2\nu} = (1-2C)e^{-2H}$6 then yields Clairaut's equation, equivalent to Clairaut-Radau form with $e^{2\nu} = (1-2C)e^{-2H}$7. While the equivalence itself may be folklore, the author notes it "might not have been presented explicitly in the literature," and the derivation given here closes that gap for general $e^{2\nu} = (1-2C)e^{-2H}$8.

Finally, for incompressible Newtonian stars, all four perturbation variables ($e^{2\nu} = (1-2C)e^{-2H}$9, $\bar{I}$0, $\bar{I}$1, $\bar{I}$2) are satisfied by their central regularity conditions alone, giving closed-form results $\bar{I}$3, $\bar{I}$4, $\bar{I}$5 (after the surface correction), and hence $\bar{I}$6, $\bar{I}$7, $\bar{I}$8. Interpreting these as the Newtonian limit of universality yields power laws $\bar{I}$9 and $w$0, consistent with earlier findings (Yagi et al., 2013).

Limitations and open questions

Several limitations qualify the results. The stiffness conjecture rests on numerical evidence from polytropic sequences and remains speculative; the paper offers no proof that length-scale hierarchy is the operative mechanism behind universality. The candidate defining the stiff limit is unresolved: incompressible matter violates causality, and the luminal equation of state is proposed only qualitatively without computation. The systematic $w$1 deviations of nuclear-theory models from the incompressible fits are interpreted conditionally — they signify failure of exact universality only if the incompressible (or some other stiff) limit genuinely defines the universal relation. The observation that the weak-field $w$2-equation maps onto Clairaut-Radau form hints at a deeper unification of the rotational and tidal perturbation schemes, but this restructuring is left undone. Finally, the suggestion that renormalization group methods could locate the critical equation of state is stated as a possibility, not carried out.

Conclusion

This paper provides two concrete contributions anchored in a simplified perturbative framework: polynomial fits to $w$3 and $w$4 for Schwarzschild's interior solution, accurate to $w$5, serving as an equation-of-state-independent reference for the universality debate; and an explicit demonstration that the relativistic tidal equations of Hinderer reduce, via a simple variable transformation, to the classical Clairaut-Radau formalism. The systematic deviations of nuclear-matter models from the incompressible curves sharpen the question of whether universality has a well-defined limiting origin, and identifying the correct causal stiff limit — or proving the length-scale conjecture analytically — remains the central open problem this work poses.