Dark energy = cosmological constant, any problems with that?

[PeterDonis]

I goes up as the masses separate, and L is conserved

Also, if you are extracting the energy and doing something else with it, is L still conserved? (Hint: electromagnetic fields, for example, can carry angular momentum.)

 

[kurros]

So what happens when all of that rotational energy is extracted?

Then you're done and no more can be extracted. What about it?

 

[kurros]

Also, if you are extracting the energy and doing something else with it, is L still conserved? (Hint: electromagnetic fields, for example, can carry angular momentum.)

It's conserved so long as nothing leaves the system, which there is no reason why it must for the purposes of this thought experiment.

 

[PeterDonis]

Then you're done and no more can be extracted.

Ok, good, so you agree that there is only a finite amount of rotational energy to be extracted. What state will the system be in when all of that energy is extracted?

It's conserved so long as nothing leaves the system

If nothing leaves the system, where does the energy you extracted from the rotational kinetic energy go?

 

[kurros]

Not in the Earth case, which is the case in which I agree work will be done. In the Earth case, what does the free body diagram of the cable's suspension point look like?

Or put it this way: suppose that both ends of the cable were free falling towards the Earth (instead of the cable being suspended by a structure), with a mass at each end. Could you extract work by paying out the cable?

No, because there is no tension in the cable in that scenario.

Take a step back to the Earth case, and modify it as I said above, to remove the obvious difference (as compared to the two galaxy case) of there being an extra force on the suspension point due to the Earth (i.e., not due to the tension in the cable). What happens?

Nothing happens, but now it isn't analogous anymore because both ends of the cable are just freely falling.

 

[kurros]

Ok, good, so you agree that there is only a finite amount of rotational energy to be extracted. What state will the system be in when all of that energy is extracted?

I never claimed we could extract infinite energy from any of this. And this system will have the masses at infinite separation when kinetic energy goes to zero. Though you will get less and less energy per unit change in separation since the tension in the cable gradually decreases. It is interesting though that the acceleration actually *increases* with separation in the dark energy case. I don't know what's up with that or where the energy comes from in that case. I blame it on some GR weirdness :). Maybe the rest of the universe experiences a slight decrease in accelerated expansion or something.

If nothing leaves the system, where does the energy you extracted from the rotational kinetic energy go?

It doesn't have to go anywhere. We could just turn on a lamp inside an insulated box or something, in which case the interior of the box just heats up.

 

[PeterDonis]

now it isn't analogous anymore because both ends of the cable are just freely falling

It's certainly not analogous to the original Earth case, no. But at least you agree that, if both ends of the cable are freely falling, no work can be extracted. Ok.

Now, suppose we start out with two masses and a cable connected to each one, but the cable is slack, and the two masses are being subjected to some tidal gravity. The masses will start to separate, and the cable will gradually get less slack until it reaches its natural unstressed length. Up to that point, will any work be extracted by this process?

 

[PeterDonis]

It doesn't have to go anywhere. We could just turn on a lamp inside an insulated box or something, in which case the interior of the box just heats up.

Ok, so from the standpoint of the center of mass inertial frame, you're transferring rotational kinetic energy to heat energy. Ok so far.

When you pay out a little bit of the cable and increase the separation of the masses, again from the standpoint of the center of mass inertial frame, does the angular velocity increase, decrease, or stay the same?

 

[kurros]

It's certainly not analogous to the original Earth case, no. But at least you agree that, if both ends of the cable are freely falling, no work can be extracted. Ok.

Now, suppose we start out with two masses and a cable connected to each one, but the cable is slack, and the two masses are being subjected to some tidal gravity. The masses will start to separate, and the cable will gradually get less slack until it reaches its natural unstressed length. Up to that point, will any work be extracted by this process?

No, because the masses are just moving on geodesics up to that point.

 

[PeterDonis]

Though you will get less and less energy per unit change in separation since the tension in the cable gradually decreases.

Agreed.

It is interesting though that the acceleration actually *increases* with separation in the dark energy case. I don't know what's up with that or where the energy comes from in that case.

Bingo! You just stated a crucial difference. The reason you are confused is that you are not seeing that this difference means that, while work is being extracted in the rotating case, work has to be input in the dark energy case--in order to increase the tension in the cable. (And similarly, if you have two masses connected by a cable inside a spaceship that is in a free-fall orbit in the Earth's tidal gravity, you would have to input energy to the two mass-cable system to extend the cable, because its tension will have to increase.)

Also, this illustrates the key difference between the dark energy case and the mass suspended on Earth case. In the mass suspended on Earth case, as the mass is lowered, the potential energy of the system decreases. But as the two galaxies separate in an expanding universe, the potential energy between them increases. (This is actually not a rigorous argument, because there isn't really a well-defined potential energy in a non-stationary spacetime like an expanding universe--whether dark energy is present or not. But as a heuristic it should serve.)

 

[kurros]

Ok, so from the standpoint of the center of mass inertial frame, you're transferring rotational kinetic energy to heat energy. Ok so far.

When you pay out a little bit of the cable and increase the separation of the masses, again from the standpoint of the center of mass inertial frame, does the angular velocity increase, decrease, or stay the same?

It goes down because L=I w, and L is fixed while I increases, so w decreases.

 

[PeterDonis]

It goes down because L=I w, and L is fixed while I increases, so w decreases.

Yes, agreed. But, per my previous post (#70), you've stated the key fact about this scenario already, so I don't think we need to analyze it further.

No, because the masses are just moving on geodesics up to that point.

Ok, good. Now, once the cable starts to go under tension, the system will reach an equilibrium in which the cable tension is just sufficient to hold the masses at constant separation. Compared to the point at which the cable was at its natural unstressed length, has energy been added to this system, or removed?

 

[kurros]

Agreed.

Bingo! You just stated a crucial difference. The reason you are confused is that you are not seeing that this difference means that, while work is being extracted in the rotating case, work has to be input in the dark energy case--in order to increase the tension in the cable. (And similarly, if you have two masses connected by a cable inside a spaceship that is in a free-fall orbit in the Earth's tidal gravity, you would have to input energy to the two mass-cable system to extend the cable, because its tension will have to increase.)

This cannot be correct. Consider the rope inside the ship case. You certainly do not *need* energy for the masses to separate, since this is what they will do just by following their freely falling paths if you cut the rope.

Also, this illustrates the key difference between the dark energy case and the mass suspended on Earth case. In the mass suspended on Earth case, as the mass is lowered, the potential energy of the system decreases. But as the two galaxies separate in an expanding universe, the potential energy between them increases. (This is actually not a rigorous argument, because there isn't really a well-defined potential energy in a non-stationary spacetime like an expanding universe--whether dark energy is present or not. But as a heuristic it should serve.)

I think you are wrong about this too. I think their potential energy effectively decreases as they separate, since we are in a regime where dark energy is dominating over "ordinary" attractive gravity.

 

[kurros]

Ok, good. Now, once the cable starts to go under tension, the system will reach an equilibrium in which the cable tension is just sufficient to hold the masses at constant separation. Compared to the point at which the cable was at its natural unstressed length, has energy been added to this system, or removed?

Energy has been added. And it will continue to get added if we let the cable extend while pulling against some turbine, until the tidal forces goes to zero because the separation is too large. That doesn't happen with dark energy though, the tidal force gets larger with separation rather than smaller, so we can just keep turning the turbine until we run out of rope.

You could instead picture an elastic band instead of a rope turning a turbine. The tidal forces will stretch the band further and further, storing more and more energy as elastic potential energy, until the elastic band tension matches the tidal force. So we are definitely putting energy into the system. Well, ignoring the loss of energy from whatever gravitational potential is associated with the tidal force. I assume we are counting that as "outside" the system.

I'm afraid that our conversation is making me more convinced that this works rather than less :). If you have a solid objection to any of this then I'd really like to hear it.

 

[PeterDonis]

You certainly do not *need* energy for the masses to separate, since this is what they will do just by following their freely falling paths if you cut the rope.

They will not separate immediately; if they are at rest when you cut the rope, they don't instantaneously acquire a nonzero velocity.

But cutting the rope is not what you were proposing; you were proposing paying out the rope. So what happens if you do that when the masses are at rest relative to each other and the system is in equilibrium?

I think their potential energy effectively decreases as they separate, since we are in a regime where dark energy is dominating over "ordinary" attractive gravity.

Hm. I'll have to look at how this works in de Sitter spacetime, which has zero stress-energy except for the "dark energy" of the cosmological constant, unlike the FRW spacetime in our current best-fit model, which has both ordinary matter and dark energy. The lack of the latter in de Sitter means it has a timelike Killing vector field, which means that a potential energy can be rigorously defined, unlike the hand-waving I was doing.

Energy has been added.

Are you sure? Consider: at the point where the cable was at its natural unstressed length, the masses were moving outward relative to each other. At the point where the cable is under tension and we are in equilibrium, the masses are at rest relative to each other. So where did their kinetic energy go?

until the tidal forces goes to zero because the separation is too large

Huh? Tidal acceleration increases with separation for ordinary tidal gravity like the Earth's. Tidal acceleration between two masses separated by a distance $L$ and in a circular orbit of radius $R$ around the Earth goes like $a \approx GM L / R^3$.

 

[PeterDonis]

If you have a solid objection to any of this then I'd really like to hear it.

Let me take a step back and try to make explicit the intuitions that lie behind what I've been saying.

Consider two test masses that are both moving on geodesics, which are diverging because of tidal gravity. Can those masses do work? It seems obvious to me that, if they are moving on geodesics, they can't do any work, because to do work, they would have to exert a force on something, which means they would have to transfer momentum, and by conservation of momentum, that would have to change their own momentum, which means they would have nonzero proper acceleration and could not be moving on geodesics.

Now, in the scenario under discussion (the original one, with two galaxies connected by a rope, but I'd rather view the "galaxies" as two test masses to avoid any complications from their own gravity), the masses were not claimed to be moving on geodesics; they are connected by a rope, and the rope has some tension, and that tension is restraining the motion of the masses. But we also said that the tension in the rope increases as the masses get farther apart. As you pointed out, that creates the question of where the energy is coming from. But even more basic than that: if the tension in the rope is increasing, then how can work be extracted from the system? It seems like work would have to be input into the system to increase the energy.

It is possible that your suggestion about potential energy could resolve at least some of this; as I said, I need to look at how the math actually would work in de Sitter spacetime, where the presence of a timelike KVF makes it possible to just turn the crank and derive a potential energy formula.

It's also worth noting that there is a key difference between "ordinary" tidal gravity, like that in the vacuum region around the Earth, and the "tidal gravity" produced by dark energy. The former is Weyl curvature; the latter is Ricci curvature. You can make them look similar by restricting attention in the Earth case to radial separations, where the tidal gravity causes geodesics to diverge (and I have implicitly been doing that in this discussion). But the difference is there.

 

[PeterDonis]

Let me take a step back and try to make explicit the intuitions that lie behind what I've been saying.

Following on from my previous post, here is a comparison of the "mass suspended from a rope on Earth" scenario (which I'll call A) and the "masses connected by a rope in free fall under tidal gravity" scenario (which I'll call B).

(1) Tension in the rope.

A: The tension is constant, and points upward (i.e., it pulls the mass upward, and the suspension point pulls the rope upwards).

B: The tension changes sign: it points inward at both ends (i.e., it pulls each mass inward), and decreases from each end towards the center of the rope, where it is zero.

(2) Work extracted.

A: Work can be extracted at the suspension point because the constant tension in the rope transmits the "force" of the mass descending.

B: Work cannot be extracted at one end based on the relative motion of the mass at the other end, because the tension in the rope is zero at the center, so the rope cannot transmit force from one end to the other. This is also evident from the fact that the tension at the two ends of the rope points in opposite directions.

This does raise the question: could work be extracted at the center of the rope, by paying out rope at equal rates in both directions? (It would have to be equal in both directions in order for the center of the rope to move on a geodesic, since the net force there must be zero.) I'll defer that until I've had a chance to look at the potential energy math.

 

[PeterDonis]

I'll defer that until I've had a chance to look at the potential energy math.

Still pondering this overall, but I wanted to go ahead and post results for potential energy in de Sitter spacetime. We use static coordinates since they are the ones adapted to the timelike KVF (i.e., in which all the metric coefficients are independent of the time coordinate). In these coordinates, the metric is

$$
d\tau^2 = \left( 1 - \frac{\Lambda}{3} r^2 \right) dt^2 - \frac{1}{1 - \frac{\Lambda}{3} r^2} dr^2 + r^2 d\Omega^2
$$

where $\Lambda$ is the cosmological constant and $d\Omega^2$ is the standard angular coordinate metric on a 2-sphere (we will be ignoring the angular coordinates here so I don't need to write that part of the metric down in detail).

By a procedure analogous to how the effective potential is derived in Schwarzschild spacetime, we can write the radial equation of motion in the standard form

$$
\frac{1}{2} \left( \frac{dr}{d\tau} \right)^2 = \frac{1}{2} \left( E^2 - 1 \right) + V(r)
$$

where $E$ is the energy per unit mass (a constant of geodesic motion) and the potential $V(r)$ is given by

$$
V(r) = - \frac{\Lambda}{6} r^2
$$

Notice that this potential decreases as $r$ increases and has a maximum at $r = 0$, so it is a "potential hill" instead of a "potential well"; contrast with Schwarzschild spacetime where the potential increases as $r$ increases, and has a maximum as $r \rightarrow \infty$ (note that the potential here diverges quadratically as $r \rightarrow \infty$).

We can also derive the proper acceleration of a static worldline at radius $r$, again by a proceudre analogous to that used in Schwarzschild spacetime; the result is

$$
a = - \frac{\Lambda r}{3 \sqrt{1 - \frac{\Lambda}{3} r^2}}
$$

This acceleration points inward, towards $r = 0$, and means that objects must be held static (constant $r$ coordinate) by pushing them inward (to keep them from rolling down the potential hill); again, contrast with Schwarzschild spacetime, where static objects must be pushed outward.

To adapt this to the "two masses connected by a rope" scenario, the center of the rope would be at $r = 0$ (where the proper acceleration is zero and the center of the rope remains at this location while free-falling on a geodesic), and the two masses would both be at the same positive $r$, but in opposite directions (the directions are captured by the angular coordinates, which I have left out in the above).

 

[alantheastronomer]

To expand a little on Orodruin's response, dark energy doesn't cause the expansion at all. It modifies the rate of expansion, making it higher in the late universe than would otherwise be the case.

Yes, I understand that the discovery of an accelerating expansion has prompted Adam Reiss to revive interest in the cosmological constant and point to "dark energy" as the source of the accelerated expansion, but by tailoring the cosmological constant to fit the observations however accurately, is still just a mathematical description of our observations, and doesn't address what is the underlying physical mechanism that causes the expansion, even if it had turned out to be decelerating instead.

 

[JMz]

...tailoring the cosmological constant to fit the observations however accurately, is still just a mathematical description of our observations, and doesn't address what is the underlying physical mechanism that causes the expansion, even if it had turned out to be decelerating instead.

How is this different from the rest of GR (or other parts of physics)? Maybe I'm misinterpreting here, but it seems like, if I wrote the Maxwell equations, they would be subject to the same objection: that the equations are just math descriptions without an underlying physical mechanism. Or likewise the Einstein equations for the case that Λ = 0: The equations are just descriptive of reality -- and that's no mean feat.

 

[Orodruin]

(1) Tension in the rope.

A: The tension is constant, and points upward (i.e., it pulls the mass upward, and the suspension point pulls the rope upwards).

Wait, what? Not sure I get this part of your argument. Tension is a scalar (or rank 2 rensor if you prefer). It does not point.

(i.e., it pulls the mass upward, and the suspension point pulls the rope upwards).

You are here looking at different objects. At both the ends, it pulls what is attached to that end towards the other end.

B: The tension changes sign: it points inward at both ends (i.e., it pulls each mass inward), and decreases from each end towards the center of the rope, where it is zero.

But now you moved the goal post. In A the ripe pulls the mass towards the branch and the branch towards the mass. In B it also pulls each galaxy towards the other. I also do not see why the tension would be zero in the middle.

 

[Haelfix]

I'm a little perplexed as well.. Even talking about the word 'tension' in the GR context is decidedly problematic. Firstly b/c of the obvious unphysical nature of what we're asking the rope to do (the sound speed of the Phonons of the material would exceed the speed of light and the tether would break, long before we could ever measure any sort of tiny effects due to a cosmological term).

Second, what sort of scales are we talking about here. The way I would model such a roped galaxy effect would be similar to the finger of god phenomena, where certain galaxies acquire large peculiar velocities. So we could say that a roped galaxy departs from the Hubble flow and is held in place by a local tension, and then ask questions about whether there is a difference between that sort of thing and a normal gravitationally bound galaxy (again the presence of the tension seems to me to be indistinguishable from normal stress energy and the usual locally bound gravitational effects). But then, note that Hubble's law (between redshift and distance) doesn't really deviate much from linearity until you get to very large scales (where the extra terms in the concordance model will enter) and it seems to me that it would only be there where you could measure the presence of a cosmological term. But note that at that sort of scale, I don't think it's strictly speaking valid to use 'Geodesic deviation' equations in the first place, which really require much more of a local analysis.

The best I think one could do is to argue that FRW GR limits in some sense to Newtonian cosmology (with a cosmological constant term) and there you could probably do some sort of analysis that would pick up such effects. To which I would foresee a lot of coordinate dependent ambiguities when descending from GR.

Anyway, all this to say I don't think the thought experiment is terribly enlightening.

 

[kurros]

Ah lots of posts since I checked last, lol, I'll try to catch up...

They will not separate immediately; if they are at rest when you cut the rope, they don't instantaneously acquire a nonzero velocity.

Sure they do. Well pretty much, anyway. There is instantly non-zero acceleration, and so also non zero velocity at t>0.

But cutting the rope is not what you were proposing; you were proposing paying out the rope. So what happens if you do that when the masses are at rest relative to each other and the system is in equilibrium?

Nothing, because the tension matches the tidal acceleration. But if you release the brakes on your turbine then the tension decreases, so the masses are pulled outward again until they are moving at some constant velocity where the internal friction in the turbine makes up the difference. And that friction can generate electricity, say.

Hm. I'll have to look at how this works in de Sitter spacetime, which has zero stress-energy except for the "dark energy" of the cosmological constant, unlike the FRW spacetime in our current best-fit model, which has both ordinary matter and dark energy. The lack of the latter in de Sitter means it has a timelike Killing vector field, which means that a potential energy can be rigorously defined, unlike the hand-waving I was doing.

Sure, but if we want to talk of a potential then it must surely decrease as stuff moves apart. Otherwise they would not accelerate away from each other.

Are you sure? Consider: at the point where the cable was at its natural unstressed length, the masses were moving outward relative to each other. At the point where the cable is under tension and we are in equilibrium, the masses are at rest relative to each other. So where did their kinetic energy go?

Into elastic potential energy in the cable. Same as if you were to catch a falling mass with a spring on Earth. It is very much the same. We could make one of our masses in the dark energy case really big and then it would be even more similar. The huge mass would hardly move off its freely falling path, while the smaller mass would be dragged way off that path by the tether. The small mass would "hang" outward from the large one.

Huh? Tidal acceleration increases with separation for ordinary tidal gravity like the Earth's. Tidal acceleration between two masses separated by a distance $L$ and in a circular orbit of radius $R$ around the Earth goes like $a \approx GM L / R^3$.

Ah sorry you are right, I spoke too quickly. The tidal force in orbit question is actually kind of complicated since a dumbbell mass in orbit will rotate once per orbit due to tidal forces. I'll think about a little more and answer again.

 

[kurros]

Let me take a step back and try to make explicit the intuitions that lie behind what I've been saying.

Consider two test masses that are both moving on geodesics, which are diverging because of tidal gravity. Can those masses do work? It seems obvious to me that, if they are moving on geodesics, they can't do any work, because to do work, they would have to exert a force on something, which means they would have to transfer momentum, and by conservation of momentum, that would have to change their own momentum, which means they would have nonzero proper acceleration and could not be moving on geodesics.

Now, in the scenario under discussion (the original one, with two galaxies connected by a rope, but I'd rather view the "galaxies" as two test masses to avoid any complications from their own gravity), the masses were not claimed to be moving on geodesics; they are connected by a rope, and the rope has some tension, and that tension is restraining the motion of the masses. But we also said that the tension in the rope increases as the masses get farther apart. As you pointed out, that creates the question of where the energy is coming from. But even more basic than that: if the tension in the rope is increasing, then how can work be extracted from the system? It seems like work would have to be input into the system to increase the energy.

It is possible that your suggestion about potential energy could resolve at least some of this; as I said, I need to look at how the math actually would work in de Sitter spacetime, where the presence of a timelike KVF makes it possible to just turn the crank and derive a potential energy formula.

It's also worth noting that there is a key difference between "ordinary" tidal gravity, like that in the vacuum region around the Earth, and the "tidal gravity" produced by dark energy. The former is Weyl curvature; the latter is Ricci curvature. You can make them look similar by restricting attention in the Earth case to radial separations, where the tidal gravity causes geodesics to diverge (and I have implicitly been doing that in this discussion). But the difference is there.

I agree with all of that. Work IS input to the system, that's how we always generate power. We have to do work moving some internal mechanism of our generator. We can then transfer that energy somewhere else and do something with it.

I agree also that it isn't clear where the energy comes from, but I think the existence of tension in the tether makes it crystal clear that energy is indeed able to come from somewhere. I suspect that the something along the lines of the weird gravitational potential shape is the answer. The energy comes from the big bang, essentially, in the initial configuration of matter we happen to have.

As for the subtleties regarding types of curvature, it has been too long since I studied GR to remember how that works. But if it somehow breaks my logic I'd be keen to know how.

 

[kurros]

Following on from my previous post, here is a comparison of the "mass suspended from a rope on Earth" scenario (which I'll call A) and the "masses connected by a rope in free fall under tidal gravity" scenario (which I'll call B).

(1) Tension in the rope.

A: The tension is constant, and points upward (i.e., it pulls the mass upward, and the suspension point pulls the rope upwards).

B: The tension changes sign: it points inward at both ends (i.e., it pulls each mass inward), and decreases from each end towards the center of the rope, where it is zero.

(2) Work extracted.

A: Work can be extracted at the suspension point because the constant tension in the rope transmits the "force" of the mass descending.

B: Work cannot be extracted at one end based on the relative motion of the mass at the other end, because the tension in the rope is zero at the center, so the rope cannot transmit force from one end to the other. This is also evident from the fact that the tension at the two ends of the rope points in opposite directions.

This does raise the question: could work be extracted at the center of the rope, by paying out rope at equal rates in both directions? (It would have to be equal in both directions in order for the center of the rope to move on a geodesic, since the net force there must be zero.) I'll defer that until I've had a chance to look at the potential energy math.

You could absolutely put your generator in the center. Just attach the spool correctly so that the cable going up and the cable going down are turning it in the same direction. Should be no problem at all. In fact I have some headphones that work like this, they attach to a spool with a spring in the center which automatically winds up the cable from the middle. You pull on both ends to unwind it, which stores energy in the spool spring. Could generate electricity just as well.

Edit: like these: https://www.amazon.com/dp/B000W8GMNY/?tag=pfamazon01-20

 

[JMz]

The underlying question for me about all the issues with the rope: In the presence of non-zero Λ, is there a theorem that states that something is conserved that we agree should be called "energy"?

This isn't subject to experiment, nor appropriate to 19th-Century math physics, because we're arguing about certain 20th-C. equations here.

 

[Orodruin]

The underlying question for me about all the issues with the rope: In the presence of non-zero Λ, is there a theorem that states that something is conserved that we agree should be called "energy"?

The theorem related to conserved quantities is Noether’s theorem. What we typically call ”energy” is the Noether charge related to time translation invariance. Since an expanding universe is not time-translation invariant, energy is generally not globally conserved. Of course, this does not violate local conservation of the stress-energy tensor.

 

[JMz]

The theorem related to conserved quantities is Noether’s theorem. What we typically call ”energy” is the Noether charge related to time translation invariance. Since an expanding universe is not time-translation invariant, energy is generally not globally conserved. Of course, this does not violate local conservation of the stress-energy tensor.

Yes, Noether's is obviously relevant. But the problem this thread seems to be focusing on is the question of whether there is something that would be conserved if Λ=0, that is (somehow) energy-like, but that isn't when Λ≠0.

Of course, this is partly a statement about physics and partly a statement about us: what "energy-like" could mean, that we might reach consensus on. For example, would this be true of (local) energy density everywhere, at least in a bound & finite universe?

 

[PeterDonis]

Tension is a scalar (or rank 2 rensor if you prefer). It does not point.

Yes, "tension" is actually not a good word for what I mean. I'm talking about the net force on an infinitesimal element of the rope--or, if you like the mass of that element times the proper acceleration of the element. For a rope of constant mass per unit length, we can ignore the mass and just look at the proper acceleration.

 

[PeterDonis]

I think the existence of tension in the tether makes it crystal clear that energy is indeed able to come from somewhere.

I think the potential energy in de Sitter spacetime makes it clearer where it's coming from: the center of the rope is at the top of the "potential hill" (since we put it at $r = 0$), and the two masses are each at the same lower elevation on opposite sides of the hill. Paying out a little bit of rope lowers each mass on the "hill", and transfers some of the potential energy in each mass to tension in the rope.

The question is how you extract any of that energy to do something else; it seems to me that the change in potential energy has to equal the change in energy stored in the rope due to the increase in tension, so there's nothing left over to transfer. But I haven't looked at the details of the math for that yet.

(Note, though, that again there is an obvious difference between this case and the case of the mass suspended on Earth. In the Earth case, the suspension point is not at the "top" of the potential well--that's at infinity. And there can't be any mass on the "other side"--there is no other side. The tension in the rope can be constant, or at least its change can be much smaller than the change in potential energy of the mass, so there is energy left over to be converted into work. At least I think so, but again, I haven't looked at the details of the math yet.)