Um, what? I can operate on an ordinary volume (say a beaker in my lab or a planet that I am in a distant orbit around) from the outside. I can't operate on the universe as a whole from the outside. How is this not an obvious difference?
If you fix a sub-volume of the universe where the boundaries of the volume are subject to the expansion of space, you can calculate the energy in the volume. The question upthread is clearly "does the energy in this volume increase due to expansion?". I'm not sure why you're so focused on integrating over the entire universe; that wasn't an important part of the question upthread. You are being very vague. If you have a coherent mathematical objection that you are trying to explain indirectly, please just say the mathematical objection.
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.
Excellent, thank you so much for the detailed response!
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.
> 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?
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.
> 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?