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I've read that even light cannot escape from black holes.
The speed of light is theoretically predicted to be the escape velocity of a black hole.
I've also read that the general notion that nothing can escape from a black hole is wrong. Even black holes emit particles and radiation known as the Hawking radiation.
My question is: How can black holes emit any kind of radiation? If they even do emit, wouldn't the speed of the radiation be less than or equal to velocity of light?
If the escape velocity of a black hole is c, then how can the Hawking radiation even escape the black hole? Wouldn't it be pulled back into the black hole? Then, how is it even emitting anything?

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The escape velocity is c at the event horizon of a black hole. Above the event horizon the escape velocity is below c. Escaping particles of the Hawking radiation form above the event horizon; that's why they can escape, if they are pointing towards a sufficiently narrow angle to vertical upward, and if they are sufficiently energetic.

Escaping particles form as virtual particle-antiparticle pairs in the "infalling" coordinate system: One of the two particles forms outside the event horizon; the counterpart forms below the event horizon. Thus the originally virtual particles cannot annihilate, and therefore become real particles; one particle can escape; the counterpart falls toward the singularity.

The energy needed to form the escaping particle, and its remaining kinetic energy after escape, is subtracted from the mass of the black hole.

The described mechanism probably works also for both particles forming very close above the event horizon if the tidal forces are high enough to separate the virtual particle pair, before it can annihilate.

Formation of virtual particles is due to the Heisenberg uncertainty applied to time and energy: Very short time intervals require energy uncertainty, leading to short-lived particle-antiparticle pairs.

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But, when the Hawking radiation is on the inside, everything else coming form inside, how can a particle, even if it is virtual, form on the outside? Also, how exactly are they 'virtual'? –  user748 Jan 7 at 13:47
    
Virtual particles form spontaneously and usually live for a very short time, less than about 1e-22 seconds, details see en.wikipedia.org/wiki/Virtual_particle. Only half of the radiation is in the inside, the outside (anti-)particle can become Hawking radiation. A particle looking virtual from an infalling coordiante system can look real from outside, because of extreme time dilation near the event horizon. –  Gerald Jan 7 at 17:07
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