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I know massive objects can curve light, can we capture the image of past Earth by lights from Earth that curved by massive objects?

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Light can be made to do this around black holes, sort of. At a certain distance from a black hole you have what's called a photon sphere. At this distance, determined by the mass of the black hole, photons travel in orbits because of the space-time curvature.

So to answer your question, maybe.

Firstly, a massive entity would need to be located at some point in space in order to influence the photons into an orbit - okay yes, possible. Secondly, this massive object would need to have a mechanism of instantaneously disappearing from space-time in order for the photos to tangentially come out of orbit and back towards earth.

I read on a previous question on Physics.SE that there is no such solution to the Einstein Field Equations which allows for a massive object to just disappear - the closest thing would be for the density distribution of the object to change but I can't say if that would yield the result you are asking about.

Rob Jeffries mentioned that images do exist which suggest that photons experience such a change in trajectory - this could happen on paths which are very close to a photon sphere-like one, but deviate enough from it to allow photos to escape.

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    $\begingroup$ I am not sure this is correct at all and have seen a number of images suggesting that photons can follow trajectories that come close to a black hole and end up on a trajectory back towards the object that emitted them. A vanishingly small fraction of photons may be following these paths, but that is a different answer to no. Perhaps you could provide some proof or references. $\endgroup$
    – ProfRob
    Oct 9, 2015 at 17:36
  • $\begingroup$ No references to add to this, unfortunately. Answer edited to reflect this. $\endgroup$
    – Phizzy
    Oct 12, 2015 at 10:34

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