  In this paper we calculate the linear perturbations of the cosmological
redshift drift. We show explicitly that our expressions are gauge-invariant and
compute the power spectrum of the redshift drift perturbations and its
correlations with galaxy number counts within linear perturbation theory. Our
findings show that the perturbations are small, and that the peculiar velocity
and acceleration terms are dominating and cannot be neglected when modeling the
full perturbative expression for the redshift drift. We also find that the
cross-correlations with galaxy number count fluctuations might increase the
detectability of the effect and can help to separate the perturbative effects
from the background cosmological redshift drift signal.
