Rotational modes of relativistic stars: Analytic results - art. no. 024019

We study the r modes and rotational "hybrid" modes (inertial modes) of rela tivistic stars. As in Newtonian gravity, the spectrum of low-frequency rota tional modes is highly sensitive to the stellar equation of state. If the s tar and its perturbations obey the same one-parameter equation of state (as with barotropic stars), there exist no pure r modes at all-no modes whose limit, for a star with zero angular velocity, is an axial-parity perturbati on. Rotating stars of this kind similarly have no pure g modes, no modes wh ose spherical limit is a perturbation with polar parity and vanishing pertu rbed pressure and density. In spherical stars of this kind, the r modes and g modes form a degenerate zero-frequency subspace. We find that rotation s plits the degeneracy to zeroth order in the star's angular velocity Omega, and the resulting modes are generically hybrids, whose limit as Omega -->0 is a stationary current with both axial and polar parts. Because each mode has definite parity, its axial and polar parts have alternating values of l . We show that each mode belongs to one of two classes, axial-led or polar- led, depending on whether the spherical harmonic with the lowest value of I that contributes to its velocity field is axial or polar. Newtonian barotr opic stars retain a vestigial set of purely axial modes (those with l=m); h owever, for relativistic barotropic stars, we show that these modes must al so be replaced by axial-led hybrids. We compute the post-Newtonian correcti ons to the l=m modes for uniform density stars. On the other hand, if the s tar is nonbarotropic (that is, if the perturbed star obeys an equation of s tate that differs from that of the unperturbed star), the r modes alone spa n the degenerate zero-frequency subspace of the spherical star. In Newtonia n stars, this degeneracy is split only by the order-Omega (2) rotational co rrections. However, when relativistic effects are included, the degeneracy is again broken at zeroth order. We compute the I modes of a nonbarotropic, uniform density model to first post-Newtonian order.