  We study the magnetosphere of a slowly rotating magnetized neutron star
subject to toroidal oscillations in the relativistic regime. Under the
assumption of a zero inclination angle between the magnetic moment and the
angular momentum of the star, we analyze the Goldreich-Julian charge density
and derive a second-order differential equation for the electrostatic
potential. The analytical solution of this equation in the polar cap region of
the magnetosphere shows the modification induced by stellar toroidal
oscillations on the accelerating electric field and on the charge density. We
also find that, after decomposing the oscillation velocity in terms of
spherical harmonics, the first few modes with $m=0,1$ are responsible for
energy losses that are almost linearly dependent on the amplitude of the
oscillation and that, for the mode $(l,m)=(2,1)$, can be a factor $\sim8$
larger than the rotational energy losses, even for a velocity oscillation
amplitude at the star surface as small as $\eta=0.05 \ \Omega \ R$. The results
obtained in this paper clarify the extent to which stellar oscillations are
reflected in the time variation of the physical properties at the surface of
the rotating neutron star, mainly by showing the existence of a relation
between $P\dot{P}$ and the oscillation amplitude. Finally, we propose a
qualitative model for the explanation of the phenomenology of intermittent
pulsars in terms of stellar oscillations that are periodically excited by star
glitches.
