Question

What happens if you perform a double slit experiment near an event horizon, if one of the slits is outside, one is inside the event horizon?

Answer 1

This is just a little bit subtle, because at first it looks like nobody has causal access to both light rays, which is required for the existence of interference pattern. On the other hand the principle of equivalence seems to assure us that local experiments done by a freely falling observer would not be affected by the presence of the horizon. There is a sense in which this is true, as follows.

For an observer freely falling into the horizon of a large black hole while conducting the double slit experiment, there will not be any change in the results of the experiment. Gravity is weak and nothing special happens when they cross the horizon, they will only discover they fell into a black hole sometime in the future when tidal forces will make life very uncomfortable. Any experiment localized within space and time will be the same up to tiny corrections due to weak gravity. This is true also if one of the slits (and therefore necessarily the light source) happens to be outside the horizon at the time of the experiment. Of course, for the freely falling observer that statement does not mean much, only the outside observer will be able to make meaningful distinction between inside and outside the horizon.

On the other hand, for an observer staying outside the horizon, one of the slits is invisible, they don't have access to all the light rays, and they will not see any interference. In fact they will not even be aware that there is a double slit experiment going on. There is no problem there because they are not freely falling observers, and there is nothing that states that the result of their experiment is identical to the same experiment made in flat space.

The fact that infalling and outside observers describe the same phenomena so differently lead to many black hole paradoxes, where quantum mechanics and GR (or the principle of equivalence) just barely co-exist without contradiction. Lenny Susskind's popular book on black holes has a good discussion, I think.

Edit: If you jump into the black hole you can have access to the interference pattern after the fact. So, it looks like you can generate another paradox, in addition to the original one. Look at arxiv.org/abs/0808.2096 for the resolution of that one. Roughly speaking, after you jump into the black hole you have only finite time to make measurements before you encounter the singularity. This time is insufficient to get more information than you are entitled to by the rules of quantum mechanics. This statement requires a detailed calculation which is in that paper.