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This question isn't fundamentally new, but I haven't found one that elicits a clear answer on the conservation of energy problem.

Another question asks: "How Does Hubble's Expansion Affect Two Rope-Tied Galaxies?", and answers, including one from accomplished physicist Luboš Motl, focus on the issue of the scenario being physically invalid.

The rope would eventually break, and maybe it would be slowing the galaxies motion if it were a really tight rope (you can't get rope with the required rigidity to stop the motion of galaxies in Nature) [...] They want to move along the natural trajectories - those we observe - so any rope trying to prevent them from doing so will be stretched by the force of inertia of these galaxies. - Luboš Motl

A rope tethering two galaxies is not only physically complex, but impossible. Furthermore, the answers refer to the rope breaking, and the asker doesn't push them to give a straight answer on whether that's a conservation of energy problem. Motl refers to inertia of the two galaxies being a factor which sidesteps the core issue of the problem, which is the expansion of the two ends of the rope seeming to break the conservation of energy.

I'd like to simplify this question to focus on an extreme, but simpler and more physically valid scenario: A very long rope, by itself.

As space expands, galaxies move away from each-other - atoms in one place move farther away from atoms in another place. So it would stand to reason that the atoms at the two ends of a many trillion lightyear-long rope would move apart, pull away from each-other. In other words, would a force be exerted on the atoms within the rope? Wouldn't that equate to energy created from nothing?

This would seem to break our laws of conservation of energy. If not, why? How would the expansion of space affect a very long rope? Or do we not understand the phenomena well enough to answer?

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    $\begingroup$ Perhaps of interest is that many models with expansion of spacetime do not conserve energy globally. $\endgroup$ – Cort Ammon Jan 16 at 20:23
  • $\begingroup$ @CortAmmon would that mean that the universe is not an isolated system, as related to the wording in the law of conservation of energy? This answer seems to claim otherwise: physics.stackexchange.com/a/11705/75876 $\endgroup$ – J.Todd Jan 16 at 20:27
  • $\begingroup$ @CortAmmon Furthermore, even though space expansion doesn't affect things of small scale noticeably, it certainly must not "activate" at some range - in other words it must affect even quantum level particles to some incredibly small degree. Wouldn't that mean that any of those models are claiming there are no isolated systems in the universe (since everything is, of course, within space)? Which would in turn mean the law of conservation of energy can never be truly observed, and thus not proven (assuming one of the models you're referring to turns out to be true)? $\endgroup$ – J.Todd Jan 16 at 20:33
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    $\begingroup$ @J.Todd In GR there's no need for energy (when it can actually be defined) to be conserved. Energy-momentum must be covariantly conserved, but this isn't the same thing. It also really depends on how you define energy and what you include in that definition. Take a look at this blog post, which precisely addresses your question preposterousuniverse.com/blog/2010/02/22/… $\endgroup$ – Eletie Jan 16 at 20:52
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First off, there's no good way to define a conserved energy in general relativity. This is a big problem in quantum gravity since quantum mechanics seems to crucially depend on a global conserved energy. So you probably won't get a definitive answer to this question, unless against all odds we figure out quantum gravity.

However you can still talk about practical ability to extract useful work from the cosmos by various particular means, with the caveat that I don't think I can prove that there's no other way that I overlooked.

Case 1: An open $Λ=0$ universe.

As Lawrence B. Crowell said, you can extract energy from the relative motion of galaxies, just as you can from the relative motion of anything else. Eventually, they'll come to relative rest. Then they'll start moving toward each other under their mutual gravitation and the gravitation of whatever matter is between them. At that point you can attach rigid rods instead of ropes to them and extract more energy by the same means until they collide.

There's a limit to how much mass-energy you can accumulate before you get a black hole (= localized big crunch). A commonly asked question is why an open universe doesn't collapse into a black hole despite being inevitably smaller than its Schwarzschild radius if you go out far enough, and the answer is essentially "because it's expanding". When you stop the expansion locally, the Schwarzschild limit absolutely does apply. The location of the event horizon would be predictable, so I suppose you could hightail it out of there and find a new set of galaxies to destroy, and since there are infinitely many galaxies, and no cosmological horizons, I suppose you can continue doing this indefinitely. In some sense, the total kinetic energy of the whole universe is infinite, but it's "compartmentalized". Don't ask me to formalize what I mean by that.

Case 2: A $Λ>0$ near-vacuum, like the apparent future of our universe.

In this case a sufficiently long and light rope will be pulled toward the cosmological horizon, and you can extract energy by the same means. However, this is really just a conservative force whose potential happens to be maximal near the center and minimal near the horizon, and you can extract energy from it only because you started out with some matter near the center. Eventually you'll run out of matter.

I suppose that classically you can extract unlimited energy by lowering the end of the rope ever closer to the cosmological horizon, but this is no different from extracting unlimited energy from a pair of point particles with a $-1/r$ potential. Presumably some quantum gravitational effect prevents this in real life.

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