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

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

where {\displaystyle \tau }{\displaystyle \tau } is the proper time.

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

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

So we can write

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

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

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