We present the first stellar velocity dispersion measurement of a massive quenching galaxy at z = 4.01 .
The galaxy is first identified as a massive z \geq 4 galaxy with suppressed star formation from photometric redshifts based on deep multi-band data in the UKIDSS Ultra Deep Survey field .
A follow-up spectroscopic observation with MOSFIRE on Keck revealed strong multiple absorption features , which are identified as Balmer absorption lines , giving a secure redshift of z = 4.01 .
Thanks to the high S/N of the spectrum , we are able to estimate the stellar velocity dispersion , \sigma = 268 \pm 59 km s ^ { -1 } .
This velocity dispersion is consistent with that of massive galaxies today , implying no significant evolution in stellar velocity dispersion over the last 12 Gyr .
Based on a stringent upper limit on its physical size from deep optical images ( r _ { eff } < 1.3 kpc ) , we find that its dynamical mass is consistent with the stellar mass inferred from photometry .
Furthermore , the galaxy is located on the mass fundamental plane extrapolated from lower redshift galaxies .
Combining all these results , we find that the velocity dispersion does not significantly evolve with redshift , although the size and mass of massive quenched galaxies do .
This suggests that the mass in the core of massive galaxies does not evolve significantly , while most of the mass growth occurs in the outskirts of the galaxies , which also increases the size .
This picture is consistent with a two-phase formation scenario in which mass and size growth is due to accretion in the outskirts of galaxies via mergers .