= Vector-fluid momentum dilution
{title2=$[a^4(\bar\rho+\bar P)\mathbf v_\perp]'=0$}
In <linear cosmological perturbation theory>, an unsourced <vector cosmological perturbation> of a <perfect fluid> in <synchronous gauge> obeys <stress-energy conservation> in the form $\mathbf q'+4\mathcal H\mathbf q=0$, where $\mathbf q=(\bar\rho+\bar P)\mathbf v_\perp$ is the <momentum density>, $\mathcal H$ the <conformal Hubble parameter>, and the prime a <conformal time> <derivative>. Its conserved quantity is $a^4\mathbf q$. For constant <equation-of-state parameter> $w\ne-1$, the background <cosmological continuity equation> gives $\mathbf v_\perp\propto a^{3w-1}$. Thus a <pressureless matter> vector velocity falls as $a^{-1}$, while a radiation vector velocity is constant. This statement assumes vanishing vector <anisotropic stress> and no momentum exchange with another component.
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