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

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

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

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

{\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}}} |

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

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