Time-dependent tidal interactions during the late inspiral of binary neutron stars encode valuable information about neutron-star structure, but systematically extending the familiar Newtonian mode-sum picture into full general relativity is nontrivial. In this paper, we develop a practical relativistic implementation of mode-sum tidal response for nonrotating neutron stars in Regge-Wheeler gauge. Using near-zone boundary conditions, we systematically define the interior tidal field, the relativistic overlap integrals, and the corresponding mode amplitudes. The good is that the dominant $f$-mode contribution is remarkably robust, reproducing the direct matching calculation to within $\sim 3\%$ across the equations of state we consider. The bad is that the operator governing mode inner product is not positive definite on the full Regge-Wheeler-gauge function space, so the relativistic mode sum truncated at $\mathcal{O}({\omega }^2)$ is not expected to strictly converge to the direct matching solution. The subtle is that the tidal field inside the star is not unique, although this ambiguity has only a limited impact on the dominant $f$-mode response for the classes of extensions studied here. Our results establish the practical utility of relativistic mode-sum approximations while making clear that their predictive power comes from a controlled low-mode description rather than from a formally convergent strong-field expansion.

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• General relativity formalism
• Gravitational wave sources