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Imagine a photon passes by a stationary atom in a large void of empty space. We know that the gravity of the atom will bend the path of the photon. It will also blue shift as it approaches the atom and redshift again as it leaves. Light is energy and according to general relativity it too bends spaces creating a gravitational tug on the atom. One would expect this to set the atom moving slightly in the direction of the photon. Because gravity propagates at the same speed as the photon, lightspeed, the atom would not experience any tug until the photon passed. The photon must have given the atom this energy and it stands to reason it would mean the exit color would be redder than it's initial color.

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The basic result you have found --- the photon being redshifted --- does indeed exist, and it is a crucial element of Compton scattering.

The effect of the outgoing photon having lower energy than the incoming one is independent of the source of the scattering, so it should occur even for gravitationally-mediated scattering; it follows from four-momentum conservation only.

We should, however, note that the gravitational cross-sections of these kinds of processes are very much negligible compared to the electromagnetic cross-sections: regular Compton scattering has a cross-section of the order of 1 barn ($= 10^{-28} \mathrm{m}^2$), while the cross-sections for gravitational processes are of the order of the Planck length squared, around $10^{-42}$ barn.

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