# 4-velocity

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

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