Four-velocity ($U^{\mu }=d{X}^{\mu }/d\tau$)

A massive observer's four-velocity is the tangent $U^{\mu }=d{X}^{\mu }/d\tau$ to its worldline, parametrized by proper time. Its invariant squared Minkowski norm is $c^2$ in the $+---$ metric signature, and $-c^2$ in the opposite convention. Unlike the coordinate values $X^{\mu }$, the tangent transforms as a vector under a general coordinate change.

The average of future-directed normalized four-velocities is generally timelike but is not normalized. For an isotropic ensemble with common Lorentz factor $\gamma$, the mean is $(\gamma c,0,0,0)$ and has squared norm $-{\gamma }^2{c}^2$. A normalized bulk velocity instead equals $c⟨v⟩/\sqrt{-g(⟨v⟩,⟨v⟩)}$. In a perfect fluid in general relativity its rest-frame energy eigenvector is the natural bulk choice; the rest-frame energy density includes random kinetic energy.

With signature $(-,+,+,+)$ and units $c=1$, parametrizing a timelike worldline by proper time makes its four-velocity satisfy $U^a{U}_a=-1$.