Metallicity has followed local gravitational potential of galaxies since
  z=3
  The MZ relation between stellar mass (M*) and metallicity (Z) of nearby
galaxies has been described as both a global and local property, i.e. valid
also on sub-galaxy scales. Here we show that Z has remained a local property,
following the gravitational potential, since z=3. In absorption the MZ relation
has been well studied, and was in place already at z=5.1. A recent absorption
study of GRB galaxies revealed a close match to Damped Ly{\alpha} (DLA)
galaxies, surprising due to their vastly different impact parameters and
leading the authors to suggest that local metallicity follows the local
gravitational potential. In this paper we formulate an observational test of
this hypothesis. The test, in essence, forms a prediction that the velocity
dispersion of the absorbing gas in galaxy halos, normalized by the central
velocity dispersion, must follow a steep log scale slope of -0.015 dex/kpc as a
function of impact parameter out to at least 20-30 kpc. We then compile an
archival data and literature based sample of galaxies seen in both emission and
absorption suitable for the test, and find that current data confirm the
hypothesis out to 40-60 kpc. In addition we show that the distribution of the
velocity offsets between z em and z abs favours a model where DLA systems are
composed of individual sub-clouds distributed along the entire path through the
halo, and disfavours a model where they are one single cloud with a bulk motion
and internal sub-structure.
