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This question is a more concrete reincarnation of an old question about energy conservation in GR.

Are there mechanisms to extract energy from the cosmic rate of expansion? putting some extremely long ropes across the universe and extract work from their increase of separation?

assuming this is possible ( i don't see why it would not ), what form would take the back-reaction force? in all physics, every time you extract work from some system by some force, this system will exert a force back. Can we slow down cosmic expansion by variations of these mechanisms?

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Yes, I think you could extract energy from the expansion. If you take masses far enough apart that their gravitational interaction isn't significant and tie them together then wait, a tension will develop in the rope. You could then use this tension to do work.

When you tie the masses together you're locally changing the spacetime curvature. Assuming an FLRW metric with $\Omega = 0$, if you prevent masses from moving with spacetime you'll make your local region overdense and the area around it underdense.

In spite of thinking long and hard about your question "Can we slow down cosmic expansion" I haven't reached any conclusion. The trouble is that the universe is (at least effectively) infinite so global changes are impossible by definition. When you tie masses together I don't think you'd be doing anything very different from when you gather them together in a ball the mass of the Sun (or the mass of a galaxy etc). As long as there is a length scale where matter is approximately uniformly distributed you'll still get an FLRW metric.

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I have a conceptual problem here. We know that where the 4 forces hold masses together there is no expansion, otherwise all the basic constants of the interactions would be changing. Now when you tie two masses together with some matter, you are using presumably electromagnetic and gravitational interactions to keep them tied. The same condition should hold as it holds for a lump of matter, which does not expand with the expansion, not? Unless I have misunderstood something. –  anna v Jun 16 '12 at 14:22
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The expansion of spacetime produces a small force on everything, including you and me, but on scales smaller than galaxies the force is so small that it's negligable compared to say EM forces. Any piece of string is in principle under tension from the expansion, but the string would have to be as long as the distance between galaxies before you could measure it. –  John Rennie Jun 16 '12 at 15:03
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