To date , PMN J2134–0419 ( at a redshift z = 4.33 ) is the second most distant quasar known with a milliarcsecond-scale morphology permitting direct estimates of the jet proper motion .
Based on two-epoch observations , we constrained its radio jet proper motion using the very long baseline interferometry ( VLBI ) technique .
The observations were conducted with the European VLBI Network ( EVN ) at 5 GHz on 1999 November 26 and 2015 October 6 .
We imaged the central 10-pc scale radio jet emission and modeled its brightness distribution .
By identifying a jet component at both epochs separated by 15.86 yr , a proper motion of \mu = 0.035 \pm 0.023 mas yr ^ { -1 } is found .
It corresponds to an apparent superluminal speed of \beta _ { \mathrm { a } } = 4.1 \pm 2.7 c .
Relativistic beaming at both epochs suggests that the jet viewing angle with respect to the line of sight is smaller than 20 \degr , with a minimum bulk Lorentz factor \Gamma = 4.3 .
The small value of the proper motion is in good agreement with the expectations from the cosmological interpretation of the redshift and the current cosmological model .
Additionally we analyzed archival Very Large Array observations of J2143 - 0419 and found indication of a bent jet extending to \sim 30 kpc .