4-Velocity

In some frame where an object is measured to have 3-velocity $\tv{v}$, it has 4-velocity $\fv v$ such that $\fv v^0 = \gamma(v) c$ and $\fv v^a = \gamma(v) \tv v^a$. This is the rate of movement through spacetime with resepct to proper time, i.e. the clock of the object in motion.

4-velocity must be time-like. We can’t have light-like 4-velocity because $\gamma$ diverges, and it can’t be space-like because then the object would move faster than light. 4-velocity doesn’t work for light.

We derive 4-velocity from 4-momentum my requiring that $\fv p = m \fv u$.