Static Tidal Perturbations of Relativistic Stars: Corrected Center Expansion and Love Numbers-I

Abstract: We revisit static tidal perturbations of relativistic stars with emphasis on two technical issues in the standard quadrupolar formulation. First, we derive the regular-center Frobenius expansion of the interior even-parity master function and obtain a corrected subleading coefficient, which differs from the expression commonly used in the literature. Second, we derive the static even-parity master equation on a Schwarzschild-de Sitter background, extending the usual asymptotically flat problem to a two-horizon geometry. To place these results on a common footing, we also show how the general interior even-parity system in Regge-Wheeler gauge reduces to the standard quadrupolar equation used in Love-number calculations. Numerical integrations for polytropic equations of state show that the corrected center coefficient affects only subleading initial data and leaves the extracted Love number $k_2$ unchanged within numerical accuracy. Taken together, these results fix the regular-center input to the standard quadrupolar problem and extend the static even-parity formalism to Schwarzschild-de Sitter backgrounds.

• The paper corrects the subleading coefficient in the regular-center Frobenius expansion, establishing a more accurate framework for tidal Love number computations.
• Direct numerical integration for polytropic EOSs demonstrates that the corrected coefficient minimally influences the computed tidal Love number k2 due to its subdominant role.
• The work extends the master equation for tidal perturbations to Schwarzschild–de Sitter spacetimes, broadening the analysis to include cosmological effects.

Static Tidal Perturbations of Relativistic Stars: Corrected Center Expansion and Love Numbers

Introduction and Motivation

The computation of tidal Love numbers in compact astrophysical objects such as neutron stars plays a critical role in modeling gravitational waveforms in binary mergers and constraining the nuclear equation of state (EOS). Love numbers characterize the linear response of a self-gravitating object to an external tidal field through perturbative analysis of the spacetime geometry, providing a bridge between observable tidal deformabilities and the underlying microphysics. This paper revisits foundational aspects of static, quadrupolar metric perturbations in general relativity, explicitly corrects technical aspects of the regular-center (Frobenius) expansion for the interior master function, and generalizes the formalism to Schwarzschild--de Sitter (SdS) backgrounds.

The work systematically addresses three issues:

1. Correction of the Regular-Center Expansion: The widely used subleading term in the Frobenius expansion of the quadrupolar master function near the center is shown analytically to be erroneous; the correct coefficient is derived and justified.
2. Impact on Love Number Extraction: Direct numerical integration demonstrates that, for polytropic EOSs, the corrected coefficient has a negligible effect on the computed tidal Love number $k_2$, as it influences only subdominant initial data.
3. Even-Parity Master Equation on Schwarzschild--de Sitter: The quadrupolar equation is extended to include a vacuum exterior with a positive cosmological constant, formulating the master equation in the presence of both a black-hole and a cosmological horizon.

These results provide analytic and computational clarification for relativistic tidal perturbation theory, with further implications for more general backgrounds.

Formalism: Perturbative Structure and Master Equations

Stellar Models and Perturbation Setup

A non-rotating relativistic star is modeled as a static, spherically symmetric perfect fluid (metric in Schwarzschild coordinates), subject to the TOV equations for background quantities. Static perturbations due to an external tidal field are imposed, and the system is linearized to first order in the metric perturbation:

$g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$

The metric perturbation is decomposed into spherical harmonics, and, working in Regge--Wheeler gauge, the even- and odd-parity sectors decouple. The analysis focuses on the $l=2$ even-parity ("electric-type") response relevant for the standard Love number $k_2$.

Exterior Problem: Schwarzschild and Schwarzschild--de Sitter

For $r > R$, the background is vacuum, and the exterior solution of the master equation reduces to a second-order ODE for the relevant mode $h_{tt}^{lm}(r)$ (or equivalently the conventional master function $H(r)$ in the single-variable formalism [Hinderer 2008, (0711.2420)]). In Schwarzschild, the equation is analytically solvable in terms of associated Legendre functions; matching to the interior solution determines the Love number via a surface value for the logarithmic derivative $y_s = r H'(r)/H(r)|_{r=R}$.

This work extends the formulation to Schwarzschild--de Sitter geometry, identifying a new master equation where the standard asymptotic analysis at $r\to\infty$ is supplanted by a two-horizon (black hole and cosmological) problem.

Interior Problem and Center Expansion

Inside the star ($r < R$), the even-parity perturbation couples to the fluid variables. The conventional approach reduces the system, after a series of algebraic and gauge constraints, to a master ODE for $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$0 that depends on the background EOS, mass profile, and pressure. Near the stellar center, regularity determines the admissible solution via a Frobenius expansion:

$g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$1

The key analytic result is the correction of the subleading coefficient, affecting the formal structure of the initial data for numerical integration.

Analytic Correction: Regular-Center Expansion

The paper provides a careful Frobenius analysis:

• The unperturbed TOV system and EOS are expanded about $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$2.
• The $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$3 master equation is linearized, and the series expansion is matched order-by-order.
• The corrected subleading coefficient $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$4 is derived:

$g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$5

This explicit form replaces the previous, incorrect expressions found in the literature and should be used in future integrations.

Numerical Results

A systematic study is presented for relativistic polytropic stellar models, with compactnesses ranging from the Newtonian to strongly relativistic regime and several polytropic indices. The tidal master equation and TOV system are integrated with both the old ($g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$6, from Hinderer) and the corrected ($g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$7) center expansions.

The strong claim derived: The computed Love numbers $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$8 agree to all reported digits for both prescriptions. The numerical insensitivity arises because the difference in initial data is suppressed as $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$9 when starting the integration at a small radius $l=2$0, and the observable $l=2$1 depends only on the ratio $l=2$2. The correction is analytically essential but has no practical effect for double-precision integrations initiated sufficiently close to the origin.

Generalization to Schwarzschild--de Sitter

The authors derive and present a static, even-parity master equation for metric perturbations on Schwarzschild--de Sitter spacetime. This provides a new starting point for analyzing Love numbers or tidal interactions in more general, cosmological contexts where the notion of an "asymptotic" tidal field must be reconsidered due to the presence of both black hole and cosmological horizons.

Implications and Future Directions

Theoretical

• The explicit correction of the Frobenius coefficient eliminates a subtle, previously unknown analytic error in tidal Love number computations, enhancing the mathematical rigor of neutron star perturbation theory.
• The construction illuminates the precise way in which the relativistic perturbation system admits a reduction to the common single-function ($l=2$3) master equation, clarifying the relation between general (multipolar, gauge-fixed) formulations and practical numerical prescriptions.
• The generalization to SdS backgrounds opens up the study of Love numbers beyond asymptotically flat spacetimes, with implications for settings where de Sitter or more exotic asymptotics play a role.

Practical

• For standard equations of state and typical numerical schemes, the revised expansion can be implemented without impacting computed values of $l=2$4, though future extremely high-accuracy calculations or very stiff/soft EOSs may eventually encounter regimes where subleading terms become numerically relevant.
• Maintaining formal accuracy in code implementations is vital, given the increasing precision of gravitational wave measurements and EOS constraints.

Outlook

• The extension to SdS motivates the study of tidal response in cosmological or early-universe scenarios and may interface with quantum-gravitational considerations or black hole tidal deformations.
• The analytic clarity of the center behavior may be essential for systematic uncertainty analysis, rigorous perturbation theory proofs, and the extension to more complex objects (e.g., anisotropic or phase-transition stars).

Conclusion

This paper fully resolves a long-standing technical ambiguity concerning regular interior solutions for relativistic tidal perturbations, without altering the practical extraction of $l=2$5 for the models and resolutions in use. It provides a mathematically consistent basis for future developments in relativistic tidal theory, including applications to gravitational wave astrophysics and perturbations on broader classes of backgrounds. The explicit master equation for Schwarzschild--de Sitter backgrounds lays the groundwork for a next generation of analyses in gravitational perturbation theory.

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Reference:

"Static Tidal Perturbations of Relativistic Stars: Corrected Center Expansion and Love Numbers-I" (2604.15195)