What happens if you place one at the event horizon where time stops and one out in deep space?

What happens if the one particle crosses the even horizon?

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Dunno either. Not for sure, but here's my best guess...

The event horizon is not like some sort of force-field or barrier. It's simply just the spherical area bound by all trajectories or orbits where the escape velocity of the singularity at the center exceeds that of the speed of light. If you went to a really supermassive black hole, on the order of millions... maybe billions of solar masses, you can have a very large event horizon. And due to the large size, the gravitational gradient which would give you effects like destructive tidal forces, and severe differences in time dialations aren't as immediately pronounced.

Also, "above" the event horizon, I dunno, and probably within it, if the black hole is rotating, there is a region called the ergosphere, where the black hole's spin is actually dragging space-time around it in a circular motion, in a poor analogy, kind of like how the atmosphere of a planet (wind and weather notwithstanding) is dragged along with the rotation of the planet itself.

So I think the size/mass of the black hole will have some bearing on what happens and when to your quantum entanglement experiment. At least insofar as if you parked an observer inside the event horizon. Because in a really big black hole, they'd survive for awhile from their reference frame at least. But they're never ever getting out to tell you about it either.

Also, since time runs slower in gravity wells, and at the singularity, the curvature of space/gravitational gradient is infinite, technically, all the stuff that's fallen past the event horizon is falling forever, and actually hasn't reached the singularity yet. I'm not even sure if the initial mass that formed the black hole in the supernova explosion or whatever has reached the singularity yet either... From the standpoint of an external observer, anything crossing the event horizon seems to take an infinite amount of time to cross it anyway. So in another weird way of looking at it, a black hole's event horizon is a sort of hologram of everything that's fallen in. However, AFAIK quantum entanglement changes are instantaneous, so time dilation may not come into play at all.

(brain 'sploded, pushing some stuff back in... owie.)

One thing though about quantum entanglement is that it can't convey information. You can't send a meaningful message with it. Nor can you really use it to beat the speed of light. Because to set up an "entanglement message" (which is sort of a worthless message anyway) the particle pair you're using had to get to the two places at light speed (or less) first anyway... Think of quantum entanglement as ordering two pies with a friend. One's cherry, the other pumpkin. You cut the pies in half, and each pan has 1/2 of each pie. You don't know which side, left or right of either pie is is the cherry or the pumpkin, you're both blindfolded, and the lady at the pie store mixed them up for you. So until you look it's "both". (and in quantum mechanics it really is "both" until you look)

So then you look at your pie, and you see your right half is cherry, your left is pumpkin, which means your friend's pie, his right half must be pumpkin, and his left half must be the cherry. The "spooky" thing about quantum entanglement is no matter how far apart you move the pies, their wave function, spin, or other quantum attribute "collapses" to each other's complimentary state once you finally do something to check which yours happens to be. So if you now toss one of the pies into a black hole before you "look", when you do finally look, you still get a result, and you now know what the state of the opposite half-n-half pie you chucked into the black hole is/was.

So in reality, if quantum entanglement doesn't convey information, chucking it into a black hole doesn't tell you anything either. You can't get inside the black hole to check if the other pie really flipped to it's opposite state from yours once you checked, or if it's flipping more slowly due to time dilation or what. Because you or anything that goes into the event horizon to check is never coming out.

I guess the more appropriate question is then, "Does tossing an entangled particle into a black hole somehow collapse it's state into one of the possible choices, making your particle you're holding outside collapse too?

I'm not sure you can even test that. Because when you look to see if your particle has collapsed to one fixed state by it's twin in the black hole being collapsed, how do you know you're the one who didn't collapse it first by checking?

I'm not a physicist, but I THINK the answer is, FUBAR "Kobiashi Maru" No-Win. Tossing an entangled particle into a black hole can't stop the one you're holding outside the black hole from flipping. But that's all you'll ever know. And you can't even tell if you're the one that flipped it to a specific state by observing it, or if it was already flipped by something in the black hole.