= 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 $-\gamma^2c^2$. A normalized bulk velocity instead equals $c\langle v\rangle/\sqrt{-g(\langle v\rangle,\langle v\rangle)}$. 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.
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