  We consider both mode calculations and time evolutions of axial r-modes for
relativistic uniformly rotating non-barotropic neutron stars, using the
slow-rotation formalism, in which rotational corrections are considered up to
linear order in the angular velocity \Omega. We study various stellar models,
such as uniform density models, polytropic models with different polytropic
indices n, and some models based on realistic equations of state. For weakly
relativistic uniform density models, and polytropes with small values of n, we
can recover the growth times predicted from Newtonian theory when standard
multipole formulae for the gravitational radiation are used. However, for more
compact models, we find that relativistic linear perturbation theory predicts a
weakening of the instability compared to the Newtonian results. When turning to
polytropic equations of state, we find that for certain ranges of the
polytropic index n, the r-mode disappears, and instead of a growth, the time
evolutions show a rapid decay of the amplitude. This is clearly at variance
with the Newtonian predictions. It is, however, fully consistent with our
previous results obtained in the low-frequency approximation.
