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Massless light beams produce a surrounding gravity field. A massive particle will move towards it and at the same time be dragged along in the direction of the beam's velocity.
But what will happen around a gravity "beam"? Will spacetime get distorted in the same way as around a light beam with comparable energy? In the light of general relativity, the wave contains no particles (it's just a distortion of spacetime traveling along), so how can it contain energy? Or is the energy contained in the curvature of spacetime itself?
So maybe my question should be, does empty (classical) curved spacetime contain energy?

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To answer only the second part of your question: yes, curved 'empty' spacetime has energy-momentum. This follows immediately from the field equations of GR. Indeed this is reasonably intuitively obvious when you think about gravitational waves, which are curvature of 'empty' spacetime. For instance the gravitational radiation produced by GW150914 was about $5\times 10^{47}\,\mathrm{J}$ – about 3 solar masses worth (or about $10^{32}$ megaton nuclear-weapons worth). All of that energy was carried away in spacetime curvature.

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