Well consider the following thought experiment: We have a tool to extract/transform "nearly all" potential gravitational energy into other forms of energy. We can theoretically make such an object (think about a very efficient sling, or a slide with minimal drag resistance).

Now we can also consider a two body system one heavy central body and a smaller secondary body, which are initially at "rest" compared to each other. (gravitational potential stays equal). We can also have such an object theoretically.

Now our system would at some "point" release the secondary body allowing it to "fall", reducing the gravitational energy and increasing its velocity (which could in some way be converted to other energy).

This is all perfectly working in our classical system and doesn't violate any laws. We can get close to 100% efficiency theoretically. (unrealistic, sure but physically possible).

However add the expansion of space to this system: and it becomes messy. On a small scale it still "works", however since gravity is a force that works without boundaries, the system should still work on the large scale.

Say the two bodies are really far apart, gravity still works in this two body system and it will "fall down". Now we can deduce that space has increased between the bodies between us "lifting it up there" and "releasing it". Thus when it is at it's periapsis (or ground) the kinetic energy can be higher than the original energy put into the system. This additional energy could be "removed" before the object goes back up for another revolution. (and it would reach the original apoapsis again, but due to the time & distance that again would be "further away").

This would imply a perputual motion: of course it cannot work.

Now there are a few things that make this impossible in practice, most notably we consider our universe nearly homogeneous, thus when we travel far away a two body system can no longer be consider a two body system as there is equal mass in every direction. However this would imply that expansion of space would only occur in systems where mass is homogenously distributed: an N-body system could never have expansion of space.

Another idea is that space expansion only occurs between object not interacting with each other. (this fits the idea that exansion of space only works in big systems and not inside the solar system) But since gravity acts without bounds this can of course not be the case.

So these can't be the reasons the above perputal motion system won't work: what am I missing? Other things could of course be that the gravitational constant changes when space changes (but that seems illogical with all other things I know). Or even that matter (mass) itself changes while space expands, but that also seems to alien and not in line with other experiences.

I almost feel silly for considering a perpetual motion system, but one has to understand why it wouldn't work. So can that be explained?

  • $\begingroup$ You are missing the FLRW metric which is derived using general relativity. $\endgroup$ – StephenG Dec 22 '18 at 17:05

Energy is not globally conserved in general relativity. It's conserved locally, and you can also construct conserved measures of energy in certain special cases such as asymptotically flat spacetimes, but it's simply not globally conserved in a cosmological spacetime. The reason we don't believe in perpetual motion machines is that they violate conservation of energy locally.

By the way, general relativity also doesn't have any concept of gravitational potential energy.

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  • $\begingroup$ If it doesn't hold globally how can it hold "locally"? What determines when you go from global to local systems? (As per question it would be a global-perpetual-motion system). $\endgroup$ – paul23 Dec 22 '18 at 18:17
  • $\begingroup$ Also, the system doesn't require the use of "gravitational potential energy": any two body system with a very long revolution period and very large difference between distance to apogee and perigee would show this "behaviour", and could be used. This would mean that two body systems are invalidated in the "long scale": but that would mean that gravity works differently on large distances. $\endgroup$ – paul23 Dec 22 '18 at 18:26
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    $\begingroup$ If it doesn't hold globally how can it hold "locally"? The precise statement of local conservation is that the divergence of the stress-energy tensor is zero. This is interpreted as local conservation of energy-momentum. This doesn't lead to a global conservation law because energy-momentum is a four-vector, and you can't compare four-vectors at different places without ambiguities due to parallel transport. For more on this topic, see physics.stackexchange.com/questions/2838/… . $\endgroup$ – user4552 Dec 22 '18 at 21:40

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