I started a reply which was getting too long and technical, with the aim of looking at three prongs. Post-Newtonian elements are at work, and it is easy to fool oneself thinking of the underlying one: metric gravitation and how it manifests within extended objects. (It does; we can see that from the figure of the Earth and its internal structure, and from other roundish celestial bodies.) Where self-fooling is easiest is in thinking (1) in terms of forces, (2) using too large a set of local coordinates or inappropriate coordinates, and (3) studying an object that extends beyond the boundaries of "local".
I have some things to do but will return to your comment and its parent in a few hours with a more detailed comment that hopefully won't stray off into the weeds of technicalities. :D
In typical vacuum solutions of the Einstein Field Equations the metric is what determines the available geodesics[0]. Putting an isolated object into a vacuum solution causes it to "select" an appropriate geodesic from the ones available, and if left alone, that object will at every time be somewhere on that geodesic. A "test object" is pointlike and massless, so adding it to the vacuum does not change the background metric at all. But if we make it heavier, or bigger and rotating or oscillating, then the spacetime is no longer exactly modelled by the vacuum solution. At that point one might use perturbation theory, and consider the vacuum metric as a background and the metric the not-quite-a-test-object itself generates as a perturbation field overlaid on that.
The algebra usually looks like g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} where the greek subscripts are the usual indices running 0,1,2,3 in 4-dimensional Lorentzian spacetimes (like ours, and most that you'll ever run into), \eta is the chosen background field, h is the perturbation field, and g is the "true" metric. One can also add perturbation fields or play around with higher-order contributions from self-interactions (say if the not-a-test-object is a black hole binary, or a solid fragment of a supernova (an Earth-massed blob of hot metal?) moving relativistically).
If we start with an expanding Robertson-Walker spacetime vacuum (no matter, not even dark matter), and add a single galaxy cluster, we need to figure out the metric to see how it evolves (or to see what geodesics are available for test particles we throw in as probes).
We can do this in a couple of ways: either using perturbative methods like above, e.g. g = {RW} + {galaxy}, or start with a single g for the whole spacetime from a known set of appropriate solutions like Schwarzschild-de Sitter or Kottler, or by engaging in "inside"/"outside" stitching together of metrics like I described earlier (swiss-cheese). (There are other approaches too!)
Each has advantages and drawbacks.
The middle option has the problem that a galaxy cluster will only look like a Schwarzschild source from so far away that it shrinks to a point. Closer observers will see (optically, even!) that it's lumpy, and if they have sufficiently sensitive gravimeters or a probe like Synge's "five-point curvature detector" they will see that Schwarzschild is not quite accurate. In particular, the geodesics the components of the gravimeter or probe "find" are not the ones that would be generated in the model metric.
The first option is hard to do and usually leads one into numerical relativity. That's not such a bad thing these days, thanks to modern codes and supercomputers. However, it's often hard to extract intuitions from numerical solutions, and even harder to boil down into an explanation for others ("just run this code and you'll see" is not very satisfying).
The last option has been around for decades and is a bit half-way. Its advantage is that you generally can say intelligent things about what's going on far from the thin shell boundary, and can develop a good map between the evolution of the configurations of matter within the "inside" metric and that outside. Its disadvantage is that it is a painful amount of work that computers aren't very good at helping with yet.
Which one chooses will depend on some combination of the problem being studied, personal preference, and trade-offs between manual work, accuracy, and comprehensibility. One can combine the perturbation approach with the inside/outside approach, so that one has (dropping the indices) g_{outside} = \eta_{outside} + h_{outside} and g_{inside} = \eta_{inside} + h_{inside} and an Israel-Darmois junction between g_{inside} and g_{outside} which is at least in principle something one can think about functionally (a test object crossing the junction will have its momentum altered; if it just drifts across, the junction is something like a "kink" linking two otherwise smooth geodesics).
Now, back to the standard cosmology, using the "inside"/"outside" approach. In the "outside", the Robertson-Walker vacuum does just what you say:
> [as the] universe [is] empty and expanding, then some light non-interacting test dust would indeed get pulled apart everywhere and at all length scales
and then
> But when we add matter and let it collapse
... we can no longer describe the whole spacetime as a Robertson-Walker vacuum that we're simply probing. Our matter perturbs the RW metric, so the true metric g must be the combination of RW and whatever metric the introduced mass generates. Since the introduced mass is collapsing, it's probably approximately described some sort of Tolman dust. So we select e.g. the Lemaître-Tolman-Bondi metric and perturb that in turn, so g = RW + LTB + O(h) + O(h^2) + ... where h is the deviation of the matter we introduced from exact LTB, and h^n is higher-order perturbations.
This g, like any other metric, will generate a set of geodesics throughout the spacetime. If our collapsing matter was a dust, then each of the individual particles will select its own metric. Flashes of light will find their own null geodesics determined by the metric, and so forth. The results are different from Robertson-Walker, at least close to the matter that we introduced. At even fairly short distances the higher order terms in h fall into irrelevance, and at large distances h itself becomes negligible.
I want to write a bit about extended objects, but that'll have to wait until a bit later. The key feature there is that the individual components of such objects are stuck together, so they don't find their own individual geodesics. The behaviour of extended objects in curved spacetime is fun!
> may eventually reach a kind of equilibrium where the effects of the expansion and the collapsing matter cancel out locally leading to a non-expanding region
You can move the thin shell of the junction around to explore that kind of thing. Or taking a purely perturbational approach you can look for regions of spacetime where geodesics have the characteristics you want (drop down test objects near point in spacetime near a suspected "cancelling-out" region and see if the inertially-moving test objects eventually collide or separate).
How you represent what's happening there is up to you. You could use your colloquial expression, a line from the Karen Masters link or something like it, or talk about the mathematical structures. The theory underdetermines the precise mathematical expression (there are lots of exactly equivalent ways of writing it down, and lots of so close it doesn't matter approximations); the English has it even worse. :/
However, the key thing is that exact vacuum solutions stop being exact when you add matter. Matter rarely cooperates with exact non-vacuum solutions (they typically are maximally symmetric: spherical arrangements of matter of uniform density, with no multipole moments from internal motions, etc. which is far from what we see measuring Earth's or Moon's gravitational fields with e.g. GOCE and GRAIL, and solar prominences big and obvious, galaxies can be spiral, a tiny black hole orbiting a large one will induce tides on the large one, and so on.)
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[0] I'm deliberately going to avoid talking about non-geodesic trajectories through spacetime for now. May come back to that when discussing extended objects in a followup.
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ETA: I should add that the metric expansion is not really an expansion of space, but rather an extra term tacked onto the metric g. So following the perturbation notation above we could start with flat spacetime \eta and say g = \eta + {expansion} and end up with g = {expanding Robertson-Walker}.
This is quite common: the "true" background is Minkowski flat-space, and then the vacuum eternal spherically-symmetric black hole is a perturbation on that, and then there may be other perturbations. (Instead of starting with the "true" background of Schwarzschild.) This kind of thing leads to some insights about asymptotic flatness.
So for an expanding cosmos, we can start with a perfect flat non-expanding one and perturb that, expanding the metric.
(This is being really loose with terminology, but I think gets the point across reasonably. It's the metric expansion of space because there is a time parameter in the line-element of the metric; spatial distances are determined by when (coordinate-time) you are. g = \eta + f(t), where f is a function taking a time coordinate and generating a perturbation field; maybe better to write g(t) = \eta(t) + f(t) where g(t) is the metric's spatial distance functions for all events on a slice of space that where everything is at the same time coordinate t; but this dives into 3+1 spacetime splittings which is another big can of worms...).
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ETA2: the Overview section of this puts it well in a different way. (No claims about the other sections, I haven't read them yet.) https://www.cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/m/Metric...
(2) below is also mostly for me, I think.
1. Thanks, Michael Weiss. This is essentially the same argument I'm making, only terser. http://math.ucr.edu/home/baez/physics/Relativity/GR/expandin...
2. One important thing about the expansion of the universe is that people hear "expansion of space" and reasonably think space has physical properties. https://arxiv.org/abs/0707.0380 is a good rant about that, and https://arxiv.org/abs/0809.4573 is another interesting take. There's a sort of Betteridge effect going on, in that none of the authors think that the concept of expanding space is awful, but they're instead capturing the "gotchas" that befall even working relativists stray into what philosophers call manifold substantivalism. (I hope that doesn't attract house philosophers, no offence. I don't think my text is all honey!)
In the EFE, if we move the cosmological constant factor to the right handside, where the stress–energy tensor is, then stress–energy tensor and the cosmological constant factor have opposite signs and stress–energy tensor can overcome the cosmological constant factor.
When we say EM-forces/Strong/Weak forces "cancel out" the expansion of the universe, it isn't because of those forces are holding matters together. But they contribute to the stress–energy tensor and "offset-ed" the effect of the cosmological constant factor.
Even without those forces, as long as the stress–energy tensor is greater than the cosmological constant factor, the solution of EFE will be non expanding.
Does this sound right to you? Thanks.
More theoretically, the LHS doesn't care about how you arrange the RHS at all. More finely, the metric does not care how you compose the metric. If you do it perturbatively, the leading term could be Minkowski or Gödel for all it cares, as long as the sub-leading terms then present a useful background for small perturbations that remain small (or better, vanish) rather than explode into large perturbations. Inside the solar system, Schwarzschild gets that right, and nobody has shown you need to correct for the global behaviour of the cosmic expansion.
> ... if we move the cosmological constant factor to the right handside ..
Before the 1920s (!) there was a discussion about where to put an everywhere isotropic tension in the EFEs. https://arxiv.org/abs/1211.6338
In section 3, the author quotes Einstein's response about how to determine the density. The part after, "Later, he finishes with..." seems especially germane, and cf. section 4 especially its last sentence (wherein "dark energy" in that context means a source field with a potential and dynamics and a coupling to the other source fields and a set of initial values).
[Heh, typo in the second sentence of the last paragraph of §3, "vale" instead of "value".]
> (matter forces) contribute to the stress-energy tensor
Ok, but if you hold that the CC acts isotropically everywhere you can think about decomposing the curvature tensor into its Ricci (and Weyl) components. Weyl tensor encodes the stretch-squash astigmatism: a spherical volume in the presence of a nonvanishing Weyl start to look more like a U.S. football. Ricci scalar encodes the isotropic change in volume. For a volume, nonvanishing Weyl with a vanishing Ricci is volume-preserving; a vanishing Weyl and a nonvanishing Ricci is a contraction or expansion.
The cosmological expansion at cosmological scales clearly is Ricci scalar rather than Weyl, since matter in the large remains isotropic and homogeneous.
Shuffling \Lambda over to the source side does not change that observation: we don't see broken cosmological-scale isotropy, so we must still have a vanishing Weyl and a nonvanishing Ricci scalar on the LHS. How does that happen? Presumably in the metric, since it's the obvious choice of dynamic field (although of course you could insist on making the RHS as complicated as you like, and there have been cases where having the metric maximally static with gravitational dynamics shuffled into "fake" matter is useful -- the original Hawking Radiation paper did just that, for instance; and of course you might want a quintessence-style field instead of a CC, and bite the bullet on recovering observations through that field's behaviour).
Ultimately what we care about is that we have the correct EOMs, and for test particles the EOMs in our labs and in our experimental platforms elsewhere in the solar system, and observed for astrophysical objects throughout our galaxy and the cluster it's in, are nowhere near the EOMs of FLRW. They are much nearer to Schwarzschild.
And as I am already typing, I will just add the first question I got when reading your response because I already typed it out. In the second paragraph - not counting the first sentence - you write »[...] \eta is the chosen background field, h is the perturbation field, and g is the "true" metric.« When you say »chosen«, are you only referring to the choice between signatures (1,3) and (3,1) or do you want to allow eta to be something other than the Minkowski metric there? Probably not to important overall but the first thing that I marked.
I mean that neither nature nor theory requires one to choose Minkowski as the background spacetime, although it is the most popular choice in perturbative General Relativity.
I've been avoiding sign conventions and coordinates deliberately; I think they would be more distracting here, and of course the choices made have no physical implications.
(Using \eta without it necessarily meaning Minkowski spacetime is arguably an abuse of notation, if that's what you mean. :D)
However, I made a point about the expansion being in the curvature scalar, and Carroll has a lecture which supports this view for lambdavacuum (eqn 15 at https://ned.ipac.caltech.edu/level5/Carroll2/Carroll1_3.html ), which convinces me that I am not too addle-brained.
(The curvature scalar is the Ricci scalar, g^{\alpha\beta}R_{\alpha\beta}, written as R in the Carroll reference in the previous paragraph. Since the metric is Lorentzian a positive scalar curvature still leads to diverging parallel geodesics. Consequently we can say things about the positive energy theorem, and we don't have to introduce things like negative mass (and specify active, passive, or inertial mass when we do). By comparison, if we have a positive definite metric (+,+,+,+ signature) parallel geodesics converge in the presence of a positive scalar curvature. The sign difference for the timelike coordinate in our Lorentzian spacetime makes all the difference!)
Why does this matter? Because, at the Carroll reference, we can add terms for matter to the action integral, and following the logic in Carroll's subsequent paragraph, the scalar curvature is influenced by the matter. But note that by adding matter to eqn 15 we are making R even more positive. But in our universe with known matter, we can't make R go below zero. That's the conceptual keystone for me: quantum foobar doesn't lead to any energy state lower than that of vacuum any more than classical dynamics can, so there doesn't seem to be any room for expansion in local quantum systems or extended classical objects. (I guess the Newtonian-limit argument then goes that in vacuum if \nabla^{2}\Phi = 0 then you can't get an expansion force by adding matter.)
Finally, in part just because I love the image here, http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/n2ext.h... It's related to the first paragraph: if you spin an object this way it will shed gravitational waves (the metric isn't spherically symmetrical at any time) and at some level that must couple to the restorative forces that keep the wrench looking wrench-like rather than deforming during its spin. Turn the metal wrench into plasticine, for example, and it will deform while spinning, and the new shape will generate yet another different metric. If we put a fairly heavy weight on top of the metal wrench when it's on the ground, the metal wrench stays wrench-shaped; but the plasticine wrench would deform (and also heat!). Capturing all of this in general curved spacetime is not so easy, and for just weak Schwarzschild (as in the diagram, presuming the ground is our Earth) or even that and sliced de Sitter I don't trust the results to be general, so I kinda give up the moment. :/
My total guess though was that if the CC acts in the solar system than everything in the solar system, including metal and plasticine wrenches, need a tiny restoration force to avoid deforming. Could the notional equivalent of the spinning plasticine wrench in a comoving Cavendish appratus directly measure an expansion force within the plasticine? Likewise, does the CC affect the temperature of the deforming weighted grounded plasticine wrench? Not sure. I'm attracted by a comparison with a binary NS or BH system, treating BIGns----smallNS : Bigend--handle where the --- has been thinned out to almost nothing. We can extract local expansion from NSNS gravitational wave astronomy in principle. (A test system would probably galaxy-cluster scales in spatial extent or mass, though, see below.)
Undoubtedly these ideas will resurface in some future conversation.
Lastly, Cooperstock et al. https://arxiv.org/abs/astro-ph/9803097v1 grinds up some numbers; they take seriously cosmological correction to local EOMs at several sub-cosmological scales, and find out they're extremely extremely small, and compatible with not existing. Note that we've had 20 years of numerical relativity and physical cosmology not reflected in that paper.
I can see where this model might be misleading or go wrong, it requires some friction between the marbles and the rubber sheet because all massive things are bound together at least by a tiny force and without friction the weakest imaginable spring would still prevent the marbles from separating. I have to think about this more carefully and try to figure out how badly this affects the model, maybe it's actually a really bad model.
But as far as I can remember, whenever raattgift said I am wrong, I was wrong. So I will reserve some time this evening for trying to work through his answer. I also skimmed the Ask Ethan article and at least at first it sound like what I intended to say, especially the following paragraph. On the other hand I did not notice any of raattgift's points in this article on a first quick pass.
The reason for this is subtle, and is related to the fact that the expansion itself isn’t a force, but rather a rate. Space is really still expanding on all scales, but the expansion only affects things cumulatively. There’s a certain speed that space will expand at between any two points, but if that speed is less than the escape velocity between those two objects — if there’s a force binding them — there’s no increase in the distance between them. And if there’s no increase in distance, that impetus to expand has no effect. At any instant, it’s more than counteracted, and so it never gets the additive effect that shows up between the unbound objects. As a result, stable, bound objects can survive unchanged for eternity in an expanding Universe.