This discussion is beginning to shed some light on my questions from the thread I started on infinite versus finite space. Very interesting and illuminating.

Does this spacetime model (viewing spacetime as a single 4-dimensional, non-evolving geometry) apply only to the universe as a whole, or can it be used for a subset of the whole universe (ie. the observable universe)?

I don't claim to understand all of this but let me ask a couple questions.

If there are multiple solutions to the Einstein Field Equations that give you a model of spacetime that works for each, what is the justification for using the FRW metric as opposed to some other metric?

Is this idea of splitting up space and time into arbitrary components similar to the idea of no absolute reference frame in relativity? The mathematical models still account for the observational data and make accurate predictions by choosing an arbitrary way to slice spacetime, but you are left with artifacts that only correspond uniquely to your choice of how to slice it?

In principle you don't have to choose an arbitrary way to slice spacetime in order to make predictions; you can formulate everything in terms of coordinate-independent quantities. In practice choosing coordinates (i.e., choosing an arbitrary way to slice up spacetime into space and time) often makes it much easier to make predictions, which is why it's so often done.

A lot depends on what you mean by "different solution". If you have some solution to EInstein's field equations, and you derive a new solution from the old solution by a mathematical transformation (in this case, the appropriate technical name for the appropriate transformation would be a diffeomorphism), do you regard the solution as "different"?

I suspect from your question that you do, but I regard the solutions as equivalent, and I'd describe changing the coordinates via a transformation to a new set of coordinates gives a different representation of the same solution, not a different solution.

To borrow an analogy, suppose you have a map of some section of land. And you rotate the map by some angle. Is it a "different map" after you rotate it, or is it "the same map, rotated"?

Some thought about what the "observations are" is helpful. On the map analogy, "observables" might be the length of trips (curves) that we draw on the map. Then the mathematical point is that rotating the map doesn't change the length of any curve, of any trip.

I say "borrow an analogy" because the original inspiration for this is a section called "The Parable of the Surveyor" from Taylor & Wheeler's "Space-time physics". Note that a "change in reference frame" in special relativity is called a Lorentz boost (it's also a diffeomorphism, like the others, a specific example that applies to SR), and it's mathematically quite similar to the mathematics that describe rotating a map, which is the original point of the analogy.

Thanks, most of this is above my current knowledge but I think I understand what you are saying about derivations of prior equations not being "new" ones. And I think I can understand qualitatively about changing coordinate systems through transformations not altering invariant quantities.

Let me start with some more basic questions.

Are there solutions to the EFEs that are logically and mathematically consistent, yet no one uses them?

If by "uses them" you mean uses for practical modeling of something we expect to actually observe, sure: the Godel metric is the first example that comes to my mind. I'm sure there are many others that nobody has bothered to discover--after all, there are an infinite number of possible solutions, mathematically speaking.

Einstein's field equations relate the geometry of spacetime to the matter and energy distribution. You specify how the matter behaves and where it is at some point. You feed that into the EFEs and get sixteen simultaneous non-linear differential equations. If you've chosen a friendly setup many of those equations turn out to be 0=0 and you may be able to solve the rest. If not, you need a powerful computer.

Either way you end up with a metric tensor, which describes the geometry of spacetime given that distribution of matter. Schwarzschild started with a universe empty except for a spherically symmetric mass and derived the Schwarzschild metric, which works well for things like the solar system where 99%-ish of the mass is the Sun. Friedmann, Lemaitre, Robertson and Walker started with a universe completely filled with a same-everywhere mass distribution and came up with the FRW or FLRW metric, which works well on the scale where galaxies are dust grains.

There are a fair few known solutions, but an awful lot of interesting stuff (e.g. the binary mergers LIGO detects) can only be done numerically.

I'm not sure the question has been answered with all the pushback. I thought it took two massive objects moving near each other to produce gravitational waves. Is that correct? If you say it only takes one, can you prove that with observed data?

So far we have three data points, all of which are for two bodies. However - join the two bodies by a string. Then you've got one body that isn't massively different from a system we've spotted emitting gravitational waves.

More seriously, the source term for gravitational waves is a time varying quadrupole moment - so in principle a dumbell or a rod spinning will produce them. In practice, to produce detectable quantities of gravitational waves you're going to need stellar masses, and we don't have any materials strong enough to avoid simply collapsing into a near-sphere under their own weight at that scale. So I think all detectable sources are likely to be two-body sources for some time to come.