Nonradial oscillations of realistic anisotropic neutron stars: Polar modes
In this work, we study the polar perturbations of static, spherically symmetric neutron stars with anisotropic pressure in full general relativity, including linear-order perturbations of both the metric and the fluid. We calculate the $f$-mode frequencies and the corresponding damping times using a consistent treatment of the perturbation of the radial vector $k^α$. In particular, its Lagrangian perturbation $Δk^α$ is determined by the metric perturbations and the fluid Lagrangian displacement and is constrained to the $(\tilde{u},\tilde{k})$ plane, where $\tilde{u}^α$ is the normalized fluid four-velocity. This constraint introduces an additional dynamical degree of freedom into the perturbation equations. Considering three equations of state and the Horvat and Bowers-Liang prescriptions for pressure anisotropy, we find that the $f$-mode frequency increases with stellar mass, ranging from $1$ to $3$~kHz, while the damping time decreases, ranging from $0.5$ to $1.25$~s. Increasing anisotropy, in the sense of tangential pressure exceeding radial pressure, generally lowers the oscillation frequency, while its effect on the damping time depends on the anisotropy prescription: the damping time decreases with increasing anisotropy for the Horvat model but increases with increasing anisotropy for the Bowers-Liang model. We further find quasi-universal relations between the real and imaginary parts of the $f$-mode frequency, $Mω_R$ and $Mω_I$, and the stellar compactness $\mathcal{C}=M/R$, which are largely insensitive to the equation of state. Polynomial fits to these relations achieve an accuracy better than $10\%$, providing a simple phenomenological framework for constraining neutron-star pressure anisotropy through future asteroseismology observations.