= Redshifting of peculiar momentum
{title2=$a\,p=\mathrm{constant},\quad a\gamma v=\mathrm{constant}$}
A freely moving massive test particle in an <FLRW metric> has physical <momentum> $p=m\gamma v$ relative to a <comoving observer>, where $v$ is its <peculiar velocity>. For a radial <geodesic>, the <geodesic Lagrangian> gives $C=a^2d\chi/d\tau$ constant, and the local radial <momentum> is $p=ma\,d\chi/d\tau=mC/a$. The <proper-time normalization> gives $dt/d\tau=\sqrt{1+C^2/(a^2c^2)}$, so
$$
v=\frac{C/a}{\sqrt{1+C^2/(a^2c^2)}}.
$$
Consequently nonrelativistic <peculiar velocity> scales as $a^{-1}$, while the exact relativistic law keeps $|v|<c$. The <momentum> magnitude of a free <photon> also scales as $a^{-1}$, giving <cosmological redshift> through $E=cp$. These statements concern force-free propagation; a bound system need not follow the cosmological expansion.
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