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.