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As we know, particles have finite size, if a particle, a part is inside the event horizon and the remain part is outside the event horizon, is it still considered as "unable to escape from black holes"?

Or in other words, when do particles considered as unable to escape from black holes? When it touches event horizon? A half of mass inside event horizon? Or totally inside event horizon?

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    $\begingroup$ see this relevant question physics.stackexchange.com/q/160276 . Elementary particles do not have extent, they are points. Composites, including protons, would probably suffer the fate described in the answers. $\endgroup$
    – anna v
    Feb 22, 2016 at 6:28

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If your particle is some macroscopic object then it will simply be torn in half at the horizon. If we calculate the acceleration required to remain stationary at a distance $r$ from the black hole we get:

$$ a = \frac{GM}{r^2} \frac{1}{\sqrt{1 - \frac{r_s}{r}}} $$

This goes to infinity as $r \rightarrow r_s$ so no matter how tough your material is it cannot resist the forces acting as we approach the horizon.

But I would guess that you're asking about an elementary particle e.g. an electron, and if so we simply don't know what happens. Elementary particles don't have a size, but in general they are described by a wavefunction that is delocalised over a region of space. So to this extent we can talk about them having a size.

The trouble is that we don't know what an event horizon looks like at quantum scales. The behaviour of quantum particles is described by quantum field theory, and this is normally formulated in flat spacetime. It is possible to do QFT calculations in a curved spacetime, for example this was how hawking originally discovered Hawking radiation. However in extreme cases like at an event horizon I don't think anyone knows for sure exactly what happens.

If you consider a slightly different case of two entangled particles, which are still a single quantum object i.e. described by a single wavefunction, then the calculations have lead some authors to very strange conclusions such as the existance of a firewall.

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