4-velocity

4 velocity is a four-vector whose elements are given by the contravariant expression

$U^{\mu }={\frac {dx^{\mu }}{d\tau }}$

where $\tau$ is the proper time.

For special relativity an inertial frame observer finds the proper time from his own coordinate time $t$ and the coordinate speed $u$ of the thing being observed by

$dt={\frac {d\tau }{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}=\gamma d\tau$

So we can write

$U^{\mu }=\gamma {\frac {dx^{\mu }}{dt}}$

Giving us the relation between 4-velocity and coordinate velocity as

$U^{\mu }=\gamma u^{\mu }$