### 3.2 Effect of rotation on quasi-normal modes

In a continuous sequence of rotating stars that includes a nonrotating member, a quasi-normal mode of
index l is defined as the mode which, in the nonrotating limit, reduces to the quasi-normal mode of
the same index l. Rotation has several effects on the modes of a corresponding nonrotating
star:
- The degeneracy in the index is removed and a nonrotating mode of index l is split into
different modes.
- Prograde () modes are now different from retrograde () modes.
- A rotating “polar” l-mode consists of a sum of purely polar and purely axial terms [292], e.g., for
,
that is, rotation couples a polar l-term to an axial term (the coupling to the term is,
however, strongly favoured over the coupling to the term [61]). Similarly, for a rotating
“axial” mode with ,
- Frequencies and damping times are shifted. In general, frequencies (in the inertial frame) of
prograde modes increase, while those of retrograde modes decrease with increasing rate of
rotation.
- In rapidly rotating stars, apparent intersections between higher order modes of different l can occur.
In such cases, the shape of the eigenfunction is used in the mode classification.

In rotating stars, quasi-normal modes of oscillation have been studied only in the slow rotation
limit, in the post-Newtonian, and in the Cowling approximations. The solution of the fully
relativistic perturbation equations for a rapidly rotating star is still a very challenging task and only
recently have they been solved for zero-frequency (neutral) modes [292, 295]. First frequencies of
quasi-radial modes have now been obtained through 3D numerical time evolutions of the nonlinear
equations [105].

Going further The equations that describe oscillations of the solid crust of a rapidly rotating relativistic
star are derived by Priou in [246]. The effects of superfluid hydrodynamics on the oscillations of
neutron stars have been investigated by several authors, see, e.g., [202, 67, 8, 10] and references
therein.