 Solution

The mass-shell condition and $P_{\mu }^c=-m{g}_{\mu \nu }{\dot{x}}^{\nu }$ imply $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-1$, so the affine parameter is proper time up to orientation. Choose its future orientation. With conformal time $t$,In a local comoving orthonormal frame both time and spatial coordinate intervals acquire the same factor $a$, so the measured peculiar velocity is precisely $\mathbf{v}=d\mathbf{x}/dt$. Thus the physical momentum obeysThe fixed vector direction also gives $\mathbf{P}(t)=a(t_{\ast })\mathbf{P}(t_{\ast })/a(t)$. With the conventional opposite overall sign for the affine particle action, the familiar comoving momentum is $\mathbf{q}=a\mathbf{P}={\mathbf{P}}^c$ instead. The redshift law is identical under either sign convention.