  We analyse the possibility that topological defects can act as a source of
magnetic fields through the Harrison mechanism in the radiation era. We give a
detailed relativistic derivation of the Harrison mechanism at first order in
cosmological perturbations, and show that it is only efficient for temperatures
above T ~ 0.2 keV. Our main result is that the vector metric perturbations
generated by the defects cannot induce vorticity in the matter fluids at linear
order, thereby excluding the production of currents and magnetic fields. We
show that anisotropic stress in the matter fluids is required to source
vorticity and magnetic fields. Our analysis is relevant for any mechanism
whereby vorticity is meant to be transferred purely by gravitational
interactions, and thus would also apply to dark matter or neutrinos.
