I guess it's an either/or scenario, that is, either the metric expansion is purely along the spatial coordinates, or can be accounted only by the time component. Since $ds^2=a(t)(dx^2+dy^2+dz^2)-dt^2$, why are we implicitly assuming only the spatial components are affected by the metric expansion?
A friend suggested that the curvature cannot be absorbed by the time coordinate since it is impossible to come up with such a transform.
Since some people did not get my question; I might rephrase it: Is there an a priori reason why we choose co-moving coordinates in FLRW metric? Is it possible to construct a similar coordinate system where the metric expansion could be accounted by an expanding time?
Question was inspired from the answer of the third question in: (https://physics.stackexchange.com/q/24327)
Feel free to include as much math as you want. Thank you.