4-velocity

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

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

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

$U^{\mu }=\gamma \frac{d{x}^{\mu }}{dt}$

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

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