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My professor in his lectures has stated that a consequence of Einstein Equivalence Principle (EEP) alone is that light bends in a gravitational field.

The proof went as follow: Imagine to be a free falling observer (so an inertial observer in GR by definition) in a box, and imagine to shoot a beam of light through a hole in your box. Since you are a free falling observer locally laws of Special Relativity (SR) apply to you (as in fact stated by EEP) and so if the beam and the hole are aligned the light goes right through the hole in the box (because of course in SR light has straight trajectories).
Ok, now immagine to be an observer on the ground, you see the box (let's imagine a glass box so you can see through) falling and the light exiting the hole, but of course in the mean time that the light crossed the length of the box the same box has fallen a bit, so the light must take a curved trajectory to be able to exit the box through the hole.

Ok, the thought experiment seems to show that indeed light must take a curved trajectory for an observer on the ground. But: my problem lies in the initial statement: "a consequence of Einstein Equivalence Principle (EEP) alone"; in fact we are not only using the EEP, instead seems to me that we are assuming that both observer should agree on the outcome of the events. In other worlds: both observer must see the light passing through the hole, it cannot be that one sees light traversing the hole and the other sees light not traversing the hole.

This seems reasonable to postulate but is surely a principle not equivalent to the EEP; am I right?

If so, what is the name of the principle?

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It is a fundamental supposition in physics: if you keep a detector satellite up there to detect the light, and if it odes detect light coming out from the hole, then that will be true for all observers; they will all see that the detector has detected something. I don't know the name of the principle, but you may call it something like 'universally agreed event'. Now, this is a fundamental supposition, so it can be always neglected as 'assumed': there is no need to specify it. It is sort of like, if you prove the sin-cosine law using the Pythagoras theorem, you can assume that Euclid's axioms have to be followed and mention the proof as being done only by the Pythagorean Theorem.

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