What is the Ricci curvature tensor in a region of sparse matter? In empty space, where the energy-momentum tensor $T_{\mu\nu} = 0$, any solution to the Einstein field equation has Ricci tensor $R_{\mu\nu} = 0$. So what does a solution look like in a region of space with sparse matter, say one molecule per cubic meter? At any point in the empty space away from the molecules of matter, any metric should have $R_{\mu\nu} = 0$. However, the standard solution in this situation is the RWF metric with $R_{\mu\nu} \neq 0$.
I understand the FLRW metric treats the matter as a continuous fluid.   But it’s not.   The exact solution has $R=0$ everywhere except at the isolated points of mass.   So in what sense does the FLRW metric approximate the actual solution?    Does $R$ for FLRW equal the average R for the exact solution?   If so, then, for the exact solution, $R$ must take on large values at the isolated points of mass.
 A: I suggest as a way in you first consider the simpler case of electromagnetism, and then take the case you asked about.
For electromagnetism you could, for example, note that $\nabla \cdot {\bf E}=0$ in the vacuum between charges, and $\nabla \cdot {\bf E} = \rho/\epsilon_0$ in a region where there is charge. So then the question arises, how can we describe the electric field of a cloud of charged particles? And in particular, what happens to $\nabla \cdot {\bf E}$?
One thing to note is that you must not try to treat point-like entities with finite charge. They are a fiction (if there were such a thing then it would have diverging field at locations near to it and the energy density of this field would diverge so strongly that the total field energy would be infinite too). So now the model is a sparse collection little charged lumps of matter with finite $\rho$, and everything is finite. You can now define an average field: it is the volume-average of the electric field. Its divergence will be non-zero. It will probably only be a useful quantity when you are considering effects on distance scales large compared to the distance between the charged lumps.
We can now make similar points about gravity. We treat lumps of matter of finite density. We have a non-zero curvature tensor everywhere. The Ricci tensor meanwhile is non-zero inside the lumps and zero in the empty space between the lumps, as you correctly point out. When modelling this situation as if it were a continuous fluid, the standard approach is the begin with the average density of the matter, after averaging over volume. From that starting-point one obtains a curvature tensor and a Ricci tensor. I expect (but I do not know) that in this scenario the curvature tensor will prove to be a volume-average of the curvature tensor of the actual situation (i.e. lumps of matter separated by empty space). I'm not sure if that answers your question; if it does not then perhaps it would be useful to say what aspect of the behaviour you are interested to explore. I think it is an interesting and valid question to ask whether taking the average matter-density as a starting point is a safe way of arriving at an approximate treatment. I don't know if anyone has ever figured that out more fully, but I expect that if there were a problem with it then it would have shown up by now, in numerical studies for example.
