# How does proper distance relate to the Robertson-Walker metric?

I'm sort of confused by what the Robertson-Walker metric describes. For example in the book I am reading, the author says that when our universe is assumed to be flat, then $S_k(r)=r$, and $d_p(t_0)=r$, where $d_p(t_0)$ is the proper distance at the time of our observation, and $S_k(r)$ is a part of the RW metric, that changes depending on whether we assume space to be flat or pos/neg curved: $S_k(r)= R_0\sin(r/R_0)$ if $k= +1$, $S_k(r)=r$ if $k=0$, and $S_k(r)=R_0\sinh(r/R_0)$ if $k=-1$. So my question has two parts: what does the Robertson-Walker metric describe conceptually, and how does the proper distance relate to it?

The RW metric describes a space time whose spatial part is expanding, homogeneous and isotropic. Since we think of our universe to be expanding, homogeneous and isotropic, this metric is used in cosmology.

There are two distance measures in cosmology (actually there are a few more, but these are the most important ones), the proper and the comoving distance. Imagine a balloon with dots on it. If you inflate the balloon, the distance between the dots will increase. This is the proper distance. Then you can think of a coordinate system which is expanding such that the coordinates of the dots in this coordinate system exactly stay fixed all the time. If you compute the distance in these coordinates, you will get a fixed distance by construction. This is the comoving distance.

Now, we are free to choose a scale for the comoving distance once. We choose this scale such that at $t=t_0$, that is, now, both distance measures coincide: $$d_p(t_0)=d_\text{comoving}$$ For all other times the distances do not coincide, since the proper distance grows with time and the comoving distance doesn't.

• This metric does not necessarily require space to be expanding, e.g., a flat spacetime with radiation and $\Lambda<0$ will collapse, and so will the closed Tolman spacetime. – William J. Cunningham May 3 '17 at 13:49

There are various distance measures and comoving and proper distance are two very important ones.

And those can be pretty confusing, it's easy to loose the mental image and the clarity on how to compute them.

See the wiki article on comoving distance (which also describes the proper distance). It is all from the FLRW solution. See it at https://en.m.wikipedia.org/wiki/Comoving_distance

Comoving distance is the easiest to define and forms the basis for any others. It is easily related to redshift z. It is the distance that you measure to another galaxy, say from earth or the center of the Milky Way, NOW (meaning at this cosmic or comoving time, for all of a whole bunch of observers between here and there all measuring their distances with their rulers, and adding all that up). The comoving time is critically important, it is the time defined by the FLRW solution. The spatial hypersurfaces at each comoving time are isotropic and homogeneous.

It does factor out the expansion, so you'd have to lay down these comoving coordinates, and have those expand with the universe, but never change their label - so if the merged black holes are now 1.3 billion ly away, we'd have a coordinate system marking them at comoving coordinate system with comoving label distance of 1.3 billion years.

As the universe expands the proper distance increases by the scale factor.

Right now, comoving and proper distances are the same. But they differ in the past and the future.

The equation is d(proper) = d(comoving) $a(t)$/$a(now)$, where the scale factor now is 1.

See also the other distance measures, and a graph of how they diverge for large redshifts (really z greater than about 0.5, and very noticeably for z greater than 1). See the figures at https://en.m.wikipedia.org/wiki/Distance_measures_(cosmology)