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## Homework Statement

Consider a flat FRW model whose metric is (in polar coordinates) [tex]ds^{2} = -dt^{2} + a^{2}(t)[dr^{2} + r^{2}d\theta ^{2} + r^{2}sin^{2}\theta d\phi ^{2}] [/tex] where a(t) is the scale factor. Show that, if a particle is shot from the origin at a time and with velocity [tex]t_{0},V_{0}[/tex] respectively, with respect to a co - moving observer then asymptotically it comes to rest with respect to the co - moving frame. Express the co - moving coordinate radius at which it comes to rest as an integral over a(t).

## The Attempt at a Solution

First off, since the particle is shot from the origin, [tex]d\theta = d\phi = 0[/tex]. Doing this then dividing both sides of the metric by dt, I got [tex](\frac{ds}{dt})^{2} = -1 + a^{2}(t)(\frac{dr}{dt})^{2} [/tex]. Since the particle is shot with respect to a co - moving frame, the distance with respect to the two never changes so ds / dt = 0. Doing this then getting dr / dt on its own I got, [tex]\frac{dr}{dt} = \frac{1}{a(t)}[/tex]. Since in an expanding universe, a(t) always increases, as a(t) (or as t) increases dr / dt will asymptotically go to zero. I hope this is right so far? The problem came with the second part (as long as the first part is correct): [tex]dr = \frac{dt}{a(t)}[/tex] and, denoting by R the distance at which the particle comes to rest with respect to the co - moving frame, [tex]R = \int_{t_{0}}^{?} \frac{dt}{a(t)}[/tex] do I use t = infinity when integrating the right - hand side to denote the time at which dr / dt = 0 or do I just denote any t = final time to represent it since it is asymptotic? Sorry if this is an absurdly stupid question.