Nonradial oscillation modes of neutron stars serve as diagnostics of their internal composition and relativistic structure. In this work, we investigate the perturbations of static and spherically symmetric neutron stars characterized by an anisotropic pressure. Given the background symmetry, perturbations decouple into polar and axial modes. To date, axial modes have remained less explored, primarily because matter and metric perturbations decouple in the isotropic limit. In this work, we provide a consistent treatment of axial modes and demonstrate that pressure anisotropy induces a direct coupling between matter and metric perturbations. We employ parametrized anisotropy models that ensure consistency with the treatment of matter perturbations. We numerically integrate the linearized Einstein field equations for the axial modes, employing a diverse set of realistic equations of state. Our results indicate that as the stellar mass grows, the frequency of the base $w$ mode generally decreases, while its damping time increases. Softer equations of state typically yield slightly higher oscillation frequencies. Furthermore, larger anisotropy (i.e., when the tangential pressure exceeds the radial pressure) allows for more massive equilibrium configurations, which correspondingly leads to lower oscillation frequencies and prolonged damping times. Finally, we demonstrate that the frequency and damping time, both scaled by the stellar mass, exhibit a nearly universal quadratic dependence on the stellar compactness, remaining largely insensitive to the underlying equation of state, but slightly dependent to the specific anisotropy model.

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• Fluids & classical fields in curved spacetime
• General relativity equations & solutions
• Gravitational wave sources
• Numerical simulations in gravitation & astrophysics