Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 62 / 1 / c / Solution

The mass shell constraint and the conserved comoving momentum giveDividing yieldsThe comoving direction is constant because ${\mathbf{p}}^C$ is constant. With $a(t_0)=1$ and physical momentum magnitude $p_0>0$ at $t_0$, we have $p^C=p_0$. Integrating the speed along that fixed direction givesThis is a distance travelled, rather than a light-ray distance: the extra factor is the particle's proper peculiar speed $v=p^K/\sqrt{m^2+(p^K{)}^2}$. For a particle at rest, $p_0=0$, the distance is zero, understood separately or as the limit. The formula assumes collisionless motion over the interval. Its existence at the lower endpoint depends on the early scale factor; in radiation domination the integral is finite.