  Considering a moving vortex line in a dilute atomic Bose-Einstein condensate
within time-dependent Hartree-Fock-Bogoliubov-Popov theory, we derive a
criterion for the quasiparticle excitations to follow the vortex core rigidly.
The assumption of adiabaticity, which is crucial for the validity of the
stationary self-consistent theories in describing such time-dependent
phenomena, is shown to imply a stringent criterion for the velocity of the
vortex line. Furthermore, this condition is shown to be violated in the recent
vortex precession experiments.
