Nonradial instabilities in anisotropic neutron stars

Nonradial oscillation modes of a neutron star possess valuable information about its internal structure and nuclear physics. Starting from the quadrupolar order, such modes under general relativity are known as quasi-normal modes since they dissipate energy through gravitational radiation and their frequencies are complex. The stability of these modes is governed by the sign of the imaginary part of the frequency, which determines whether the mode would decay or grow over time. In this paper, we develop a fully consistent framework in general relativity to study quasi-normal modes of neutron stars with anisotropic pressure, whose motivation includes strong internal magnetic fields and nonvanishing shear or viscosity. We employ parametrized models for the anisotropy, with a specific form under perturbation that is consistent with previous studies under the Cowling approximation, and solve the perturbed Einstein field equations numerically. The assumed form of the perturbed anisotropy can be easily extended to more general cases. We find that, unlike the case for isotropic neutron stars, the imaginary parts of some of the pressure ($p$-)modes flip signs as the degree of anisotropy deviates from zero, depicting a transition from stable modes to unstable modes. This finding indicates that some anisotropic neutron star models are unstable, potentially restricting the form of sustained anisotropy.