### 4.2 Pulsations of rotating stars

Pulsations of rotating relativistic stars are traditionally studied (when possible) as a time independent,
linear eigenvalue problem, but recent advances in numerical relativity also allow the study
of such pulsations via numerical time evolutions. The first quasi-radial mode frequencies of
rapidly rotating stars in full general relativity have been recently obtained in [105], something
that has not been achieved yet with linear perturbation theory. The fundamental quasi-radial
mode in full general relativity has a similar rotational dependence as in the relativistic Cowling
approximation, and an empirical relation between the full GR computation and the Cowling
approximation can be constructed (Figure 18). For higher order modes, apparent intersections of
mode sequences near the mass-shedding limit do not allow for such empirical relations to be
constructed.
In the relativistic Cowling approximation, 2D time evolutions have yielded frequencies for the to
axisymmetric modes of rapidly rotating relativistic polytropes with [104]. The higher
order overtones of these modes show characteristic apparent crossings near mass-shedding (as was observed
for the quasi-radial modes in [330]).

Numerical relativity has also enabled the first study of nonlinear r-modes in rapidly rotating relativistic
stars (in the Cowling approximation) by Stergioulas and Font [293]. For several dozen dynamical
timescales, the study shows that nonlinear r-modes with amplitudes of order unity can exist in a star
rotating near mass-shedding. However, on longer timescales, nonlinear effects may limit the r-mode
amplitude to smaller values (see Section 3.5.3).