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Concerning then the Einstein Field Equations, today we have a plethora of solutions $[1]$, $[2]$, $[3]$. 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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$[1]$ STEPHANI.H; et al; Exact Solutions of Einstein’s Field Equations

$[2]$ MULLER.T; GRAVE.F; Catalogue of Spacetimes

$[3]$ PODOLSKI.J; GRIFFITHS. J. B; Exact Space-Times in Einstein's General Relativity

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    $\begingroup$ What exactly is the doubt about the equations?!!! $\endgroup$ – MBN May 18 '20 at 11:29
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The distinction is physical, not mathematical.

Consider your standard electromagnetism textbook. It contains many solutions to Maxwell's equations, given certain charges, currents, and boundary conditions. Some of these solutions are called "capacitors". Others are called "inductors". Yet others are called "waveguides" or "transmission lines".

What is the mathematical criterion for a solution to Maxwell's equations to describe an waveguide? There is no formal criterion; that's the wrong question to ask. Waveguides are real objects found everywhere across the world. A mathematical setup is called an "waveguide" if it can be used as a (possibly simplified) model to describe a real waveguide.

Similarly, a spacetime is called a "cosmology" if it has certain features in common with the universe we actually observe (allowing for simplifications and variations). Such features may or may not include homogeneity, isotropy, and expansion.

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Theoretically, they all describe possible cosmologies for they are are full spacetime solutions to the Einstein equations.

However, none of the vacuum solutions (such as the Kerr or Schwarzschild) can describe the real world cosmology because the real world has matter and energy. The solutions to the Einstein equations that are considered as realistic candidates for describing the actual cosmology are the ones obtained by plugging in what we think is a realistic stress energy tensor (such as the one we use to get the FRW metric).

Now, coming to the second question. Why are some of these solutions nonetheless useful to describe astrophysical objects? Well, because of the fact that the metric of the blackhole solutions, for example, becomes trivial if you go far away from the black holes. Thus, such solutions can describe "local" structures of spacetime as if they were the only structure in spacetime. One can argue that there shouldn't be any such possibility because we are already in the business of finding a metric that describes the cosmology of the real world which covers the whole of spacetime, so we shouldn't be able to deploy these extra black holes solutions at some spots because there is no such freedom on top of describing cosmology. This is obviously not true because cosmologies are described at a coarse grained level. This leaves open the freedom of finding all kinds of spacetime structures at (cosmologically speaking) micro grained scales. This is similar to saying that the Earth is as such round, but we can obviously speak of mountains and valleys locally.

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  • $\begingroup$ The funny thing is that, the concept of locally here isn't the "mathematical locally" provided by manifold theory, I think. The locally here is in fact the intuitive notion of "locally". Like: "hey we have a large thing (universe) and locally other minor things (black holes), but all of them satisfy the same master equation". $\endgroup$ – M.N.Raia May 18 '20 at 10:36
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    $\begingroup$ @M.N.Raia Yes, it is not mathematical locality in the sense of being applied to an open ball of an infinitesimal radius. But my usage of the word is pretty close to that in spirit, and in science, it is always applied in the sense I am applying it here. A point in the FRW metric consists of clusters and clusters of galaxies. And when we apply calculus in such an FRW setting, an infinitesimal distance would literally mean an astrophysically large distance. The idea of what is small and large is always scale and context-dependent in science. $\endgroup$ – Dvij D.C. May 18 '20 at 10:45

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