So there is meaning to the previous persons question which is what the thought experiments were meant to show but obviously that's something you can't imagine.

No, it requires energy to be added to the system from outside the system. Which is precisely what you cannot do with the universe as a whole. That's what makes such thought experiments meaningless for the universe as a whole.

Yes that is exactly what energy that doesn't 'exist' means.

Yes correct, that's why that energy doesn't 'exist' because it's outside of the universe.

If you disagree, please show me the consistent model on which your scenario is based.

Yes that's correct in the general accepted sense for the term 'universe'.

Edit: question if the initial thought experiment was posed as 'observable universe' would it make a difference to you?

It would address my "can't operate on it from the outside" objection, yes. But it still wouldn't make the thought experiment meaningful, for the reasons I gave in response to wyager elsewhere in the thread.

Original thought experiment said matter you're talking about total energy in your response. We ought to take matter to mean ordinary every day matter, mass being the property that all regular ordinary matter posses. Since that's normal everyday language.

Yes in a closed universe the net energy is 0, nothing in; nothing out, so you wouldn't be able to magically 'pop' mass into one place without taking from somewhere else.

Now if mass A is 100 meters away from mass B it would take X amount of energy to push/pull them together. If they are now 150 meters apart now the energy required to bring them together is now higher.

So if galaxies are pulling away and away from each other over time then overtime the energy you would need to bring them closer together again increases.

Your original thought experiment asked about the energy required to push all the matter to one place. But to even try to formulate such a question in GR, "matter" has to mean "whatever stress-energy is present in the universe". And "energy" has to mean the same thing, because stress-energy is what "does stuff" in GR, not "mass".

> that's normal everyday language

You can't do physics in normal everyday language.

> in a closed universe the net energy is 0, nothing in; nothing out, so you wouldn't be able to magically 'pop' mass into one place without taking from somewhere else.

The fact that you can't "magically pop mass into one place" is true in any spacetime in GR, not just a closed universe; it's a consequence of the Einstein Field Equation and the Bianchi identities. You also can't magically "take" mass from some place, for the same reason.

> if mass A is 100 meters away from mass B it would take X amount of energy to push/pull them together. If they are now 150 meters apart now the energy required to bring them together is now higher.

These statements are only valid in the particular cases I have already listed: an asymptotically flat spacetime, or a stationary spacetime. An expanding universe is neither.

I understand that you don't get this; that's because you are using ordinary language, but, as I said above, you can't do physics in ordinary language. Your reasoning looks OK to you because you don't understand that, except in the special cases I described, the ordinary language you are using does not correspond to any valid physics. (I have explained why in other subthreads in this discussion.) I know it looks to you like it ought to; but it doesn't.

If you can't put something into simple everyday language then it's a pretty good indication of your lack of understanding.

Edit: my bad I misread your initial comment.

The focusing theorem basically says that an initially converging congruence converges more quickly in the future, and an initially diverging congruence diverges less quickly in the future.

The Raychaudhuri picture for your pair of masses is fairly straightfoward given that the small distances can effectively wash out any coupling to the metric expansion of space. If the masses are small (i.e., we're not talking about two black holes) the background gravitational metric is effectively flat, perturbed only by mass A and B. This lets us determine exactly the factors opposing recollapse of your initially-diverging mass A and mass B. We can also be confident that an initial impulse could drive the separation, with the focusing theorem behind the eventual collision between A and B.

At larger length scales the metric expansion of space becomes important, so the background gravitational metric is instead Friedmann-Lemaître-Robertson-Walker (FLRW) or similar. We can still talk about Raychaudhri focusing in that context, but we take "... will diverge less quickly in future" slightly differently depending on whether the galaxies are flying apart because of an initial impulse (i.e., the expansion is inertial), or whether there is also a cosmological constant (i.e., the expansion is accelerating).

In the inertial case, the ultimate recollapse depends on a critical value in the (average) energy-density of the whole universe. Returning to your mass A and mass B picture, in the previous case with a flat background spacetime where A and B colliding is inevitable, in the inertially-expanding spacetime the initial impulse on A and B can lead them to never meet again, even given infinite time.

In the case of a small positive cosmological constant (which best explains what we see in redshift surveys), heavy nearby galaxies will take their time to separate compared to lower-mass galaxies with similar separation, or the same heavy galaxies which formed with a larger separation. The Raychaudhuri focusing theorem adapted to this setting tells us whether the galaxies will merge or not, and this is useful in understanding the spatial extents and masses of galaxy clusters. Returning to mass A and mass B, given a positive cosmological constant, no initial impulse is necessary; the CC alone can determine whether they will be separate forever, or whether they will eventually collide.

Returning to your final paragraph, the (average) energy-density must be over some critical value for distant galaxies to come into contact with each other, and that critical value for all practical purposes only depends on the value of the cosmological constant (if any).

So far, with decades of trying to measure the average energy-density and critical value, it's safer to say that distant galaxies will lose contact with each other. There is no known mechanism to change that for arbitrary sets of galaxies.

Note that I have not discussed energy above except as energy-density (which is energy per unit volume, particularly where the volume is very very small) averaged across the entire cosmos, and the invariant masses of your A and B and of galaxy clusters.

Finally, for completeness, clusters of galaxies are known to move strikingly against the cosmological coordinates, leading to some famous cluster-cluster collisions (e.g. the Bullet Cluster). Why they don't just float at the same cosmological coordinate (remember the coordinates expand) like the overwhelming majority of galaxy clusters is a topic of active research. However, perhaps some late-time impulse could drive initially diverging galaxies (carried for all practical purposes only by the cosmological constant) into a collision in a way that matches your final paragraph. (Alternatively these collided galaxies might never have been diverging in the first place).

No. A thought experiment cannot be based on a contradiction.

> A cat can't be alive and death at the same time, either.

And the Schrodinger's Cat thought experiment does not claim that it is, even though many pop science discussions try to claim otherwise. The Schrodinger's Cat thought experiment is based on the math of QM, i.e., on a consistent underlying model. It simply points out consequences of that model that might not be obvious to many people.

The "thought experiment" I have been objecting to in this thread, by contrast, is not based on any consistent mathematical model. The operation it is proposing to do on the universe as a whole is inconsistent with GR, which is the only consistent model we have of the universe as a whole. That means it's not a valid thought experiment. "Thought experiment" does not mean you can make up whatever you want.

As actual physicists actually use thought experiments, they cannot make up whatever they want. Thought experiments involve taking a known consistent model and working out consequences of it that were not previously obvious or well known, and seeing where that leads. You can only do that usefully if you constrain the thought experiment by the model, i.e., if you do not allow yourself to make up whatever you want.

Since the discussion here is about an article on physics, it seems appropriate to treat proposed thought experiments the way actual physicists would treat them. That is what I have been doing.

Then you are not talking about the thought experiment that I was responding to, but about a different one. I have no objection to talking about the different thought experiment that you propose (and I'll do that below), but nothing in any such discussion is relevant to the objection I made to the original thought experiment, which was about the entire universe, not just some portion of it.

> you can calculate the energy in the volume

Actually, no, you can't. There is no known invariant in GR that corresponds to "the total energy inside this volume" for an expanding universe. There are only two cases in GR where we have known invariants that correspond to "the total energy inside this volume": (1) an asymptotically flat spacetime, where we can define the ADM energy and the Bondi energy; and a stationary spacetime, where we can define the Komar energy. An expanding universe does not fall into either of these categories.

You will find claims in the literature that a "total energy" for cases like an expanding universe can be calculated using so-called "pseudo tensors". However, such claims are not accepted by many physicists, and even physicists who do accept that "pseudo-tensors" are physically meaningful don't all agree on which pseudo-tensors those are.

You can, of course, choose some set of coordinates (such as the standard FRW coordinates used in cosmology), and integrate energy density over some spatial volume in a 3-surface of constant coordinate time. (It is not clear that this is a correct way to get "total energy", because in GR the source of gravity is the total stress-energy tensor, which includes momentum, pressure, and stresses as well as energy density, but we'll leave that aside for now.) But the result of any such computation is not an invariant; it depends on your choice of coordinates. The energies I referred to above (ADM, Bondi, Komar) do not. That is why they are accepted as physically meaningful by all physicists.

> The question upthread is clearly "does the energy in this volume increase due to expansion?"

It's not at all clear to me that that is the question being asked upthread (for one thing, that poster, in another subthread, has explicitly said the "energy" they are thinking of adding comes from outside the universe). But even if we assume it is, the question is still meaningless because it assumes there is such a thing as "the energy in this volume", which, as above, there isn't.

A couple questions come to mind:

1. In the latter case, where we use e.g. FRW coordinates to define our volume, can we use the usual hack for defining an invariant energy of defining the center of our coordinate system to be the center of mass of the volume? I'm willing to believe the answer is "no"; I'm just not sure where it would fall apart.

2. If we leave aside the notion of defining volumes entirely, can we meaningfully ask questions like "you have a toy universe with two gravitationally bound masses; does expansion increase the energy of this system in the center of mass reference frame?" I guess this is probably just equivalent to ADM/Bondi, since the spacetime is asymptotically flat.

A volume by itself doesn't have a center of mass. If you are talking about a standard FRW model where the energy density and pressure are constant in any given spacelike slice of constant FRW coordinate time, then you can pick a particular sub-volume of a spacelike slice and define the spatial center of FRW coordinates to be the geometric center of the sub-volume, and that point will also be the center of mass (or more properly the center of energy-momentum) of the stress-energy in the sub-volume.

Since all of the stress-energy is comoving in this model, you can pick out the set of comoving worldlines that are in the sub-volume at the instant of FRW coordinate time that you chose, and treat them as a "system", whose center of energy-momentum will be the comoving worldline at the spatial origin of FRW coordinates, and that will be true for all time. The issue comes with trying to define a "total energy" for this "system"; you still run up against the same issues I described.

> can we meaningfully ask questions like "you have a toy universe with two gravitationally bound masses

There is no known exact solution that describes this case, so the only way to treat it would be by numerical simulation. Astronomers do do this, for example to model binary pulsar systems (as in the Hulse-Taylor binary pulsar observations that won them the Nobel Prize). However--

> does expansion increase the energy of this system in the center of mass reference frame?"

Such a "universe", in the numerical simulations, will not be expanding. It will be asymptotically flat, and will slowly emit gravitational waves and become more tightly bound (this was the prediction that Hulse and Taylor's observations over many years verified). In short, this "toy universe" has nothing useful in common with our actual expanding universe.

In terms of energy, the ADM energy of such a system will be constant. The Bondi energy will slowly decrease with time as gravitational waves escape to infinity. But again, this system is not expanding, so these things tell you nothing useful about an expanding universe.

> I guess this is probably just equivalent to ADM/Bondi, since the spacetime is asymptotically flat.

You guess correctly. See above.

> A volume by itself doesn't have a center of mass.

Why not? This seems like something we could calculate in an invariant way (I have not actually tried coming up with an expression; this is a solicitation for context, not a claim). Also, to be clear, I am talking about a volume with some non-homogenous mass distribution. Maybe you draw a boundary around a solar system or something. Can we not come up with an invariant expression for the CoM of everything within that boundary?

> Such a "universe", in the numerical simulations, will not be expanding.

OK, this seems important. I never made it much past SR in undergrad, so this is a hole in my comprehension. Is the expansion of the universe directly deducible from GR? My understanding was that an expanding universe was one of the admissible solutions under GR, but is it the only admissible model for a universe that looks like ours? If so, what's the relevant difference between our universe and the toy model I mentioned, that causes GR to predict that our universe expands and the toy one doesn't?

Because "volume" is a geometric concept, not a physical thing. It has a geometric center, but not a center of mass. What you appear to be thinking of when you talk about calculating the center is the geometric center, not the center of mass. (And note that, unless an actual physical system has a high degree of symmetry, the geometric center defined by its spatial volume will not be the same as its physical center of mass.)

> Is the expansion of the universe directly deducible from GR?

The fact that the spacetime describing the universe cannot be stationary (i.e., that it must be either expanding or contracting) is deducible from the original 1915 Einstein Field Equation (i.e., without a cosmological constant) plus the assumptions of homogeneity and isotropy--roughly speaking, that the universe looks the same at all spatial locations and in all directions. We then pick out the "expanding" option as the one describing our actual universe based on observations.

Einstein actually discovered this in 1917, and he was bothered by it, because he believed (as did most physicists and astronomers at that time) that the universe was static--that it did not change with time on large scales. So he added the cosmological constant to his field equation to allow it to have a static solution that could describe a homogeneous and isotropic universe. Then, about ten years later, when evidence began to mount for the expansion of the universe, Einstein called adding the cosmological constant "the biggest blunder of my life"--because if he had trusted his original field equation, he could have predicted the expansion of the universe a decade before it was discovered.

Today, we believe that there is in fact a nonzero cosmological constant (our best current value for it is small and positive), but we also understand, what Einstein did not explore very thoroughly, that the Einstein Static Universe is an unstable solution, like a pencil balanced on its point: any small perturbation will cause it to either expand forever or collapse to a Big Crunch. So this solution is not considered a viable candidate to describe our actual universe. And we also know that there are no other static solutions that describe a homogeneous and isotropic universe.

No, I mean something along the lines of $Integral_V x*m(x) dx / Integral_V m(x)dx$ where $m$ is the mass-energy density function. The usual way of finding the center-of-momentum frame of a system that people mean when they say "invariant mass".

Also, the integral you describe will not, in general, be invariant; it will depend on your choice of coordinates, because you are integrating over a spacelike surface of constant coordinate time, and which surfaces those are depends on your choice of coordinates.

Your intuition about the "usual" invariant mass is based on the special cases where the integral you describe can be equated to one of the known invariants, the ADM mass, the Bondi mass, or the Komar mass. (Strictly speaking, even the Komar case is problematic, because the integral in question in a general stationary spacetime does not necessarily converge. In cases where it does converge, AFAIK the spacetime must be asymptotically flat and the Komar mass is equal to the ADM mass.) But an expanding universe is not one of those cases.