Neutron Stars as Perfect Fluids: Extracting the Linearized Response Function

We derive the general relativistic linear tidal response of a neutron star modeled as a barotropic perfect fluid. From the covariant fluid effective action, we linearize about equilibrium and obtain the action for fluid displacements coupled to metric perturbations. Splitting the latter into external and induced parts and integrating out the induced field yields a Hermitian operator and a discrete gapped spectrum of driven modes. Projecting the displacement onto this eigenbasis and integrating out the spatial dependence over the stellar radius reduces the dynamics to tidal-driven oscillators, with couplings set by relativistic inner products and overlap integrals. Matching to the quadrupolar worldline effective action gives a mode-sum response function and analytic dynamical tidal deformabilities from mode frequencies, normalizations, and overlaps.

💡 Research Summary

In this paper the authors develop a covariant effective‑field‑theory framework to compute the linear tidal response of a neutron star modeled as a barotropic perfect fluid within general relativity. Starting from the covariant fluid action S = ∫d⁴x √−g F(b), where b = √det B_{IJ} and B_{IJ}=g^{μν}∇_μϕ^I∇νϕ^J encodes the internal comoving coordinates, they expand around a static, spherically symmetric equilibrium configuration. Small fluid displacements π^I(x) are introduced, and the action is expanded to quadratic order, yielding kinetic terms for π^I, gradient terms proportional to the sound speed c_s, and linear couplings to metric perturbations h{μν}.

The background spacetime is the Tolman‑Oppenheimer‑Volkoff (TOV) solution with metric ds² = −e^{2Φ}dt² + e^{2Λ}dr² + r²dΩ². In Regge‑Wheeler gauge the even‑parity metric perturbations reduce to a single scalar potential φ, which obeys a Sturm–Liouville radial operator O_r that is self‑adjoint with respect to the weight W_φ = e^{−Φ+Λ}r². The eigenfunctions φ_{nℓ}(r) and eigenvalues Ω_{nℓ}² form a complete basis for the external tidal field.

The fluid sector yields a Hermitian operator O_{IJ} acting on the displacement field. Solving the eigenvalue problem O_{IJ}π^J_n = ω_n²δ_{IJ}π^J_n provides a discrete set of normal modes with frequencies ω_n and eigenfunctions π_n^I(x). The modes are normalized by the relativistic inner product ∫√−g w₀ π_n·π_m d³x = M_*R_*² N_n δ_{nm}, where w₀ = ρ₀ + p₀ is the background enthalpy density.

The external tidal field is introduced as a weak‑field perturbation h^{ext}_{00}=−2φ^{ext}, with φ^{ext} generated by a distant point mass. By integrating out φ^{ext} and projecting the source term onto the fluid eigenbasis, the authors obtain a set of driven harmonic oscillators:

L = ∑_{nℓm}