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Considering the information found from the LIGO experiment. The gravitational wave causes a contraction(In horizontal direction) and expansion(In vertical direction). Can this be a source of energy. I am thinking in Planck Scale(tho there may be a smaller scale that may be a constant length), what happens to these constant units of length? Maybe everything is compressible?

To make an analogy to better explain my line of questioning, lets think of the smallest indivisible constituent of matter and relate that to the smallest discrete unit of length. I am presupposing the amount of energy to compress these either would be not feasible when it comes to scaling. So what happens to this unit of length, is it loss or is it truly compressible? What it suggest to me that gravity leans more to an Entropic principle.

Qmechanic
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Yes, they do! And this was actually one of the strongest arguments used to convince people that gravitational waves should actually exist, and should carry energy (called the 'sticky bead argument'). The total amount of energy in gravitational waves, permeating the universe, is quite small (compared to, e.g. the energy in the CMB, etc).

It's not clear how you're going from gravitational waves to small length scales... Because gravity is such a 'weak' force, the expansion/contraction has little effect on 'rigid' bodies (like atoms, molecules, solids) and more so on the spaces between them. Perhaps these cosmology questions on where expanding space comes from and what is it exactly that expands might be helpful.