Concerning then the Einstein Field Equations, today we have a plethora of solutions $$, $$, $$. But, when can we call a solution "cosmological"? Because, suppose the Kerr spacetime; region I (the "Minkowski diamond") describes a compactfied universe far from Kerr black hole, i.e., Minkowski spacetime region from past time-like infinity to future time-like infinity, for instance. On the other hand astrophysicists use it to describe black holes, not the whole universe.
Conversely Godel's metric, FRW metric and so on... describes a whole universe.
So, which are the mathematical facts that says "this metric describes a cosmological solution rather than a single body inside a universe"?
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$$ STEPHANI.H; et al; Exact Solutions of Einstein’s Field Equations
$$ MULLER.T; GRAVE.F; Catalogue of Spacetimes
$$ PODOLSKI.J; GRIFFITHS. J. B; Exact Space-Times in Einstein's General Relativity