Non-locality of gravitational energy

Question

Gravitational energy is non-local which is essentially because of the equivalence principle. The equivalence principle says that you can always transform your frame so that you feel like in a Minkowski space-time locally. Mathematically, there is no tensor-like definition for gravitational energy in General Relativity. All energy-momentum tensor for gravitational energy must be pseudo-tensor, namely frame-dependent tensor. About the non-locality of gravitational energy I have two questions:

1. Where does the energy of gravitational waves come from which seems way local?
2. How can the non-locality of gravitational energy be implemented in String theory where, for example, gravitons are simply zero modes of closed strings and strings are explicitly local (of course except for the resolution of strings which, as I see, is different from the non-locality of gravitational energy)?

Comments

I think this is a really really interesting question. I remember when I studied gravitational waves, it was shown (section 8.5) that one can in fact define an energy tensor for the gravitational wave, but it was only well-defined (read: gauge invariant) if one averages out its value over a region considerably larger than the wavelength of the wave. As those lecture notes point out this can be quite large: 

This might be a rather large region: the LIGO detector looks for waves with frequency around 100Hz, corresponding to a wavelength λ ∼ 3000km.

This then confronts us with a real puzzle: the theory can only define the energy of the wave in a region of 3000 km, while on a more physical level the LIGO detector is clearly interacting (and hence extracting energy) from the wave at a scale of a few kilometers. Does nature have a better way of defining energy than us? Looking foward to the answers here :)

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@RubenVerresen Thanks for your comment.

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I only can give a reference on a paper by L.D. Faddeev in a Russian journal: http://www.physics-online.ru/MessageFiles/7268/gravity-faddeev-ufn.pdf

It should be translated in the West, but I was too lazy to search.

EDIT: I found this reference: http://iopscience.iop.org/article/10.1070/PU1982v025n03ABEH004517/pdf

http://mr.crossref.org/iPage?doi=10.1070%2FPU1982v025n03ABEH004517

http://www.turpion.org/php/paper.phtml?journal_id=pu&paper_id=4517

Anyway, it is said that the gravitational energy is not localizable because there are many equivalent asymptotically flat metrics. Concerning the total energy (gravity+matter), it is proven to be positive (E. Witten and others). L.D. Fddeev uses an analogy with Classical Electrodynamics considering its proof "ideal". I have, however, a reservation about this. Briefly, if one manipulates variables formally and implies existence of physical solutions, then it looks OK.

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@VladimirKalitvianski: Please also give a standard reference to the paper in English transcription, so that one can search for it?

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I am not an expert in general relativity but can't one define a local energy-momentum tensor assuming an asymptotically flat universe. Isn't that enough?