@RaphaelJ.F.Berger The four-momentum divided by the rest mass is the four-velocity. The spatial momentum is not a covariant quantity and therefore has no special meaning in SR.
@RaphaelJ.F.Berger note that "relativist mass" is an outdated concept, and most physicists only think of rest mass when you say "mass" physics.stackexchange.com/questions/133376/…
incidentally, your question is an example of why invariant mass is a more useful concept: as John said, it relates four-momentum and four-velocity, which are two useful four-vectors.
I have obtained some expressen in rel qm which is like beta c alpha, where c alpha is the velocity operator, its anti hermitian and it seems to me it corresponds to the frame dependent mess. Possibly the name will stick to it.
note that if the $\gamma$ is the Lorentz factor for the object moving with that 4-velocity (and not that of another observer), the "mess" is just $(c,v^i)$, where $v^i$ is the non-relativistic 3-velocity.
OK! Thats in interesting. So its about an electron moving around a atomic nucleus. I suppose the non-relativistic velocity has no particular physical meaning except that its equal to the 4-velocity for slow motions, right?