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 […]