PeterDonis
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You can do that for the flat and open cases (i.e., zero and negative curvature), yes. For the closed case (positive curvature), the handling of the spatial part of the chart has to be somewhat different (strictly speaking, there can't be a single chart covering all of a spacelike 3-surface since an n-sphere can't be covered by a single chart).cianfa72 said:strictly speaking, there is a complete family of such charts since we can just continuously remap the values of spatial coordinates assigned to each of the worldlines orthogonal to the spacelike hypersurfaces of constant by adding the same constant to the old values assigned to them.
More generally, there are of course a variety of transformations you can apply to the spatial part of the chart in any FRW spacetime, without changing the property that comoving worldlines have constant spatial coordinates in the chart. So yes, in that sense no chart is truly unique.