I found this part in the book of Bernard Schutz on GR. (Sec 12.2: General Relati

Question

I found this part in the book of Bernard Schutz on GR. (Sec 12.2: General Relativistic Cosmological Models).

Robertson-Walker metrics: We shall adopt the following assumptions about the universe: (i) spacetime can be sliced into hypersurfaces of constant time which are perfectly homogeneous and isotropic; and (ii) the mean rest frame of the galaxies agrees with the definition of simultaneity.

Can someone kindly explain the second assumption, in particular what is a "mean rest frame" for a galaxy.

Explanation / Answer

First, we pick a global definitions of "time" in (i). Then in (ii) we ask, "In my coordinates, do the galaxies have a statistically preferred direction they are moving? Or is the distribution of velocities centered on 0?" We demand that in fact the distribution be centered on 0 (isotropy tells us the distribution is symmetric).

In this way we pick out the most natural frame to work in. Now some people will take the coordinate-invariance of GR a bit too far and say things like "we shouldn't pick reference frames artificially, since all reference frames are equivalent." But in fact the real universe we live in does have a most natural rest frame, and there's nothing wrong with choosing to work in those coordinates, since it often makes life easier.

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