  It is widely believed that the cosmological redshift is not a Doppler shift.
However, Bunn & Hogg have recently pointed out that to settle properly this
problem, one has to transport parallelly the velocity four-vector of a distant
galaxy to the observer's position. Performing such a transport along the null
geodesic of photons arriving from the galaxy, they found that the cosmological
redshift is purely kinematic. Here we argue that one should rather transport
the velocity four-vector along the geodesic connecting the points of
intersection of the world-lines of the galaxy and the observer with the
hypersurface of constant cosmic time. We find that the resulting relation
between the transported velocity and the redshift of arriving photons is not
given by a relativistic Doppler formula. Instead, for small redshifts it
coincides with the well known non-relativistic decomposition of the redshift
into a Doppler (kinematic) component and a gravitational one. We perform such a
decomposition for arbitrary large redshifts and derive a formula for the
kinematic component of the cosmological redshift, valid for any FLRW cosmology.
In particular, in a universe with Omega_m = 0.24 and Omega_Lambda = 0.76, a
quasar at a redshift 6, at the time of emission of photons reaching us today
had the recession velocity v = 0.997c. This can be contrasted with v = 0.96c,
had the redshift been entirely kinematic. Thus, for recession velocities of
such high-redshift sources, the effect of deceleration of the early Universe
clearly prevails over the effect of its relatively recent acceleration. Last
but not least, we show that the so-called proper recession velocities of
galaxies, commonly used in cosmology, are in fact radial components of the
galaxies' four-velocity vectors. As such, they can indeed attain superluminal
values, but should not be regarded as real velocities.
