[thx-2718]

Wouldn't there be meaning in saying it would take this much energy to push all the matter in the universe to one place?

Which that amount should be increasing as space-time expands.

Thought experiment, if you could place a mass of an arbitrary amount at any one point in space, how much mass would you need such that all the mass of the universe is now falling towards it.

Or could you bend space-time to a point that all mass falls into it.

[pdonis]

> Wouldn't there be meaning in saying it would take this much energy to push all the matter in the universe to one place?

No. The universe is not an isolated system that we can operate on from the outside. You can't treat it as though it is. So your thought experiments aren't meaningful.

[thx-2718]

Obviously the thought experiment requires energy that doesn't 'exist' or doesn't have meaning in the sense that it could happen literally. It's a what-if and that does have a number and that does have meaning.

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.

[pdonis]

> Obviously the thought experiment requires energy that doesn't 'exist'

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.

[thx-2718]

> it requires energy to be added to the system from outside the system.

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

[pdonis]

But you can't do that for the universe as a whole. Asking what would be the case if you could is meaningless; it's like asking what would be the case if 2 + 2 were 5. No consistent model exists of such a situation, so the question is meaningless.

[thx-2718]

> you can't do that for the universe as a whole.

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

[faangsticle]

Now take off your nitpick hat and ignore that detail. Its what makes it a thought experiment.

[pdonis]

You can't ignore it because it's not a "detail"--it's a crucial feature of the thought experiment that doesn't work for the universe as a whole. What you're suggesting is like saying, in my thought experiment I assume that 2 + 2 = 5, just ignore the fact that 2 + 2 is actually 4.

[faangsticle]

Yes, that's what a thought experiment is. A cat can't be alive and death at the same time, either.

[wyager]

How is your objection not a useless and unfalsifiable catch-all that applies to calculating any quantity over a volume?

[pdonis]

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?

[wyager]

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.

[pdonis]

> If you fix a sub-volume of the universe

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.

[wyager]

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.