Neutron Stars as Perfect Fluids: Extracting the Linearized Response Function

We develop an effective field theory framework for the conservative linear tidal response of a relativistic neutron star modeled as a perfect fluid with conserved particle number. Starting from the covariant fluid action in a curved background, we linearize about a static equilibrium configuration and obtain the quadratic action for fluid displacements coupled to metric perturbations. We then split the metric perturbation into induced and externally sourced parts, and integrate out the induced metric on a conservative matched domain using a symmetric Green kernel. The resulting effective fluid theory contains metric-mediated fluid interactions through a self-adjoint operator acting on the displacement, while the external perturbation acts as a tidal source. This self-adjoint structure permits a modal expansion of the pole sector: after projection onto conservative eigenmodes, the dynamics reduce to tidally driven oscillators with couplings fixed by relativistic inner products and overlap integrals. Matching these oscillator variables to the quadrupolar worldline theory gives analytic expressions for the modal dynamical tidal deformabilities in terms of mode frequencies, normalizations, and overlap integrals. We also identify a source-only non-pole sector generated by terms quadratic in the external perturbation. This sector does not affect the mode amplitudes, but its electric-quadrupolar projection can contribute to the full tidal response. This formulation clarifies which part of the relativistic tidal response is captured by explicit fluid modes and where a possible source-only non-pole contribution enters the worldline description.