  At low redshift, it is possible to combine spectroscopic information of
galaxies with their luminosity or angular diameter distance to directly measure
the projection of peculiar velocities (PV) along the line-of-sight. A PV survey
probing a large fraction of the sky is subject to so-called wide-angle effects,
arising from the variation of the line-of-sight across the sky, and other
sub-leading projection effects due to the propagation of the photons in a
perturbed cosmological background. In this work, for the first time, we provide
a complete description, within linear theory and General Relativity, of the
power spectrum of luminosity distance fluctuations, clarifying its relation to
the observables in a PV survey. We find that wide-angle effects will be
detected at high significance by future observations and will have to be
included in the cosmological analysis. Other relativistic projections effects
could also be detected provided accurate, per object, distances are available.
