Does a black hole have enough time to actually form a singularity?

Question

I am trying to wrap my head around black holes, singularties and hawking radiation. Physics.se contains many intresting questions and answers, but from none I could so far read about the interaction between formation of a singularity and hawking radiation (most seem to talk about the point at which matter plunges into a black hole that already has a singularity).

From what I understand, from the perspective of a star collapsing into a singularity, the perceived time is just like in classical mechanics: the mass is accelerated towards the center of gravity, and when "everything" is there, it is a singularity. Now this takes a finite time, and the matter itself perceives just that. Part of the question is: how long is this time? I am assuming in the magnitude of milliseconds.

Now as I understand hawking radiation, as soon as there is an event horizon, it will start emitting negative mass/energy into the direction of the singularity. Time dilation gets infinitely stronger as we approach the singularity, so I would assume that from the perspective of those negative particles, they would all the time reduce their perceived distance to the particles of positive energy/mass that are already on their way to the singularity.

Since approaching the singularity, time dilation becomes kind of infinite, even the $10^{100}$ years or so it is supposed to take for hawking radiation to evaporate a black hole seem to be enough to send the necessary amount of negative energy/mass particles down to the singularity so that they can "chase" the positive energy/mass paticles very closely.

But will they reach them before a singularity forms, so that they cancel out each other to the point there is not enough gravitational force?

If this is just a matter of calculating "$n$s until matter reaches singularity" and "$m$s until negative mass/energy reaches singularity" and calculation shows that always $n<m$ then I would be intrested in the calculation (if possible presented in a way that doesn't require too exotic concepts).

If this is a qualitative matter, in that some of my ideas are fundamentally wrong, please point those out (again if possible in a way that doesn't require too exotic concepts).

Answer 1

Well, the escape velocity of a black hole is the speed of light, c.

So to exit the black hole you have to move faster than light.

It is not actually prohibited in nature to move faster than light. You just cannot transmit information at this speed.

Even electrons or light rays can occasionally move faster than light. This is because of Heisenberg uncertainty principle.

As you know, for a short time $\Delta t$ the total energy of the closed system can diverge from its prevuous value no more than by $\Delta E$ such that $\Delta t \Delta E \le \hbar$.

This means that even in pure vacuum sometimes for a short time can appear virtual particles.

Now a particle (say, negatively charged) moves in the right-hand direction encounters with such formation of a virtual particle and antiparticle from vacuum:

$E^-------><---E^+---*---E^---->$

The original particle annihilates with the virtual anti-particle, and the virtual particle which moves to the right continues propagation. Since it does not differ from the original particle, one can say that it was the original particle who moved faster than its mean speed.

The same can happen with photons as well. So the creation of a virtual photon-antiphoton pair allows the photon to travel faster than light and escape the black hole. The number of antiparticles which form in this process cannot be greater than the number of the non-antiparticles. If they do not meet with the real particle, they annihilate with their virtual pair and disappear.

Since the anti-particle of a photon is also photon, it turns out that this anti-particle simply represents the electro-magnetic field in the opposite phase.

Answer 2

Time dilation gets infinitely stronger as we approach the singularity, so I would assume that from the perspective of those negative particles, they would all the time reduce their perceived distance to the particles of positive energy/mass that are already on their way to the singularity.

I am no physicist, so take this with a grain of salt (or more like a pound) but I believe the above quote highlights where your assumption is wrong. The perception of time dilation changes as you move closer to the singularity. Let me give you an example that I am more familiar with (after all, nobody really knows what happens inside the horizon), and replace singularity with event horizon: an observer looking at an object falling into a black hole will perceive that object kinda stalling out near the event horizon. But once the observer gets closer to the horizon, and eventually crosses it, that perception changes as the observer himself comes under the effects of time dilation. The observer will never be able to "catch" the object that he observed in this way.

Or putting it differently: objects closer to a singularity feel a stronger gravitational force than objects further away. As such, they move faster towards the singularity the closer they are. Which means that if you were inside the event horizon, you would perceive everything moving away from you (things closer to the attractor move faster than you, and you move faster than things further away from it than you are).