Normalization of an averaged four-velocity

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.