  In this paper, we study in detail the role of general relativity on the
global dynamics of giant pulsar glitches as exemplified by Vela. For this
purpose, we carry out numerical simulations of the spin up triggered by the
sudden unpinning of superfluid vortices. In particular, we compute the exchange
of angular momentum between the core neutron superfluid and the rest of the
star within a two-fluid model including both (non-dissipative) entrainment
effects and (dissipative) mutual friction forces. Our simulations are based on
a quasi-stationary approach using realistic equations of state (EoSs). We show
that the evolution of the angular velocities of both fluids can be accurately
described by an exponential law. The associated characteristic rise time
$\tau_{\text{r}}$, which can be precisely computed from stationary
configurations only, has a form similar to that obtained in the Newtonian
limit. However, general relativity changes the structure of the star and leads
to additional couplings between the fluids due to frame-dragging effects. As a
consequence, general relativity can have a large impact on the actual value of
$\tau_{\text{r}}$: the errors incurred by using Newtonian gravity are thus
found to be as large as $\sim 40 \%$ for the models considered. Values of the
rise time are calculated for Vela and compared with current observational
limits. Finally, we study the amount of gravitational waves emitted during a
glitch. Simple expressions are obtained for the corresponding characteristic
amplitudes and frequencies. The detectability of glitches through gravitational
wave observatories is briefly discussed.
