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

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

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