  Previous studies have found that the Galactic rotation velocity-metallicity
(V-[Fe/H]) relations for the thin and thick disk populations show negative and
positive slopes, respectively. The first Gaia Data Release includes the
Tycho-Gaia Astrometric Solution (TGAS) information, which we use to analyze the
V-[Fe/H] relation for a strictly selected sample with high enough astrometric
accuracy. We aim to arrive at an explanation for the slopes of the V-[Fe/H]
relationship. We measure the V-[Fe/H] relation for thin and thick disk stars
classified on the basis of their [$\alpha$/Fe] and [Fe/H] abundances. We find
dV/d[Fe/H]= -18 +/- 2 km/s/dex for stars in the thin disk and dV/d[Fe/H]= +23
+/- 10 km/s/dex for thick disk stars, so we confirm the different signs for the
slopes. The negative value of dV/d[Fe/H] for thick disk stars is consistent
with previous studies, but the combination of TGAS and APOGEE data provide
higher precision, even though systematic errors could exceed +/-5 km/s/dex .
Our average measurement of dV/d[Fe/H] for local thick disk stars shows a
somewhat flatter slope than the previous studies, but we confirm a significant
spread and a dependence of the slope on the [alpha/Fe] ratio of the stars.
Using a simple N-body model, we demonstrate that the observed trend for the
thick and thin disk can be explained by the observed radial metallicity
gradients and the correlation between orbital eccentricity and metallicity in
the thick disk.
