  Vortex-mediated mutual friction governs the coupling between the superfluid
and normal components in neutron star interiors. By, for example, comparing
precise timing observations of pulsar glitches with theoretical predictions it
is possible to constrain the physics in the interior of the star, but to do so
an accurate model of the mutual friction coupling in general relativity is
needed. We derive such a model directly from Carter's multifluid formalism, and
study the vortex structure and coupling time-scale between the components in a
relativistic star. We calculate how general relativity modifies the shape and
the density of the quantized vortices and show that, in the quasi-Schwarzschild
coordinates, they can be approximated as straight lines for realistic neutron
star configurations. Finally, we present a simple universal formula (given as a
function of the stellar compactness alone) for the relativistic correction to
the glitch rise-time, which is valid under the assumption that the superfluid
reservoir is in a thin shell in the crust or in the outer core. This universal
relation can be easily employed to correct, a posteriori, any Newtonian
estimate for the coupling time-scale, without any additional computational
expense.
