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Nov 9, 2018 at 5:43 comment added S. McGrew A photon does not have a "position"; it's actually described as a wave function that is spread out over all space. So the question really translates to "would the wave functions of two photons ever ever get concentrated, due to mutual gravitational interaction, into a small region in 4-space corresponding to a world line?". I think the answer is probably "no", but don't know for sure. A knowledgeable answer would depend on a good theory that combines general relativity and quantum mechanics; and there is not yet such a theory. Maybe you will develop that theory in 10 years. I hope so!
Nov 8, 2018 at 21:14 comment added user212343 If they attract at all, would this attraction be capable of, on a theoretical infinite path, pulling the two closer until their position/direction is the same. Please don’t hold back with the mathematical principle, as my knowledge on the subject is purely based on general ideas, and I would like to expand that. Thank you
Nov 8, 2018 at 20:55 history answered S. McGrew CC BY-SA 4.0