= Solution
In proper-time gauge the particle <Lagrangian> is $L=m(-\dot t^2+a^2\dot{\mathbf x}^{\,2}-1)/2$, now with dots referring to <proper time> $s$. The canonical spatial <momentum> is
$$
\boxed{p_i^C=\frac{\partial L}{\partial\dot x^i}=ma^2\delta_{ij}\frac{dx^j}{ds}.}
$$
The <FRW metric> is independent of all spatial coordinates. Hence $\partial L/\partial x^i=0$ and the <Euler-Lagrange equation> gives $dp_i^C/ds=0$. This <conserved quantity> is the <comoving momentum>.
The mass-shell <mass shell> constraint gives $(dt/ds)^2-a^2|d\mathbf x/ds|^2=1$. For future-directed motion and proper <peculiar velocity> $\mathbf v=a\,d\mathbf x/dt$, it follows that $dt/ds=(1-v^2)^{-1/2}=\gamma$. Therefore
$$
\mathbf p^C=ma^2\gamma\frac{d\mathbf x}{dt}=a\gamma m\mathbf v=a\mathbf p^K,
\qquad \boxed{\mathbf p^K=\frac{\mathbf p^C}{a}.}
$$
The kinetic <momentum> here is the <physical momentum in an FRW universe>, measured in the <orthonormal frame> of a <comoving observer>. It <redshifts> as $a^{-1}$ irrespective of whether the particle is relativistic or nonrelativistic.
Back to article page
