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

TL;DR

This paper corrects the mathematical formulation of static tidal perturbations in relativistic stars, ensuring accurate initial conditions and extending the analysis to Schwarzschild-de Sitter backgrounds without altering the Love numbers.

Contribution

It derives a corrected center expansion coefficient and extends the formalism to Schwarzschild-de Sitter backgrounds, improving the accuracy of tidal perturbation calculations.

Findings

Corrected the subleading coefficient in the center expansion.

Extended the master equation to Schwarzschild-de Sitter backgrounds.

Found that the correction does not change the Love number $k_2$ within numerical accuracy.

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.