Exploring Hawking Radiation Emission Process

[mivanit]

So I realize that I'm probably wrong about this, but it seems to me that Hawking radiation cannot be emitted only at the event horizon. If we make the (albeit almost certainly wrong) assumption that the quantity of emitted particles is directly proportional to the potential for gravitational tidal forces between two particles, we find that the radiation emitted becomes constant and independent of the radius, which is not the case.
So here is my question: how can things like time dilation differences between the virtual particles, probabilities of the relative velocities of the particles, etc. be accounted for when calculating the amount of particle splits, and also do these splits occur in just a set region around the black hole, or are they just most probably very close to the black hole?

(sorry for any breaking of etiquette/bad english/bad physics, this is my first post and I'm not exactly very knowledgeable about physics or anything else)

 

[stedwards]

Well, your premise needs adjusting. A gravitational potential gradient
$\nabla \phi$, results in tidal forces.

 

[mivanit]

I think that is what I meant, difference between forces on two particles due to gravity
(sorry for bad english)

 

[PeterDonis]

If we make the (albeit almost certainly wrong) assumption that the quantity of emitted particles is directly proportional to the potential for gravitational tidal forces between two particles, we find that the radiation emitted becomes constant and independent of the radius

I don't understand how you're reaching this result. (It's not really germane to the main question anyway, as you'll see below, but since you admit you're not very knowledgeable about physics, I'm curious how you arrived at this result.)

how can things like time dilation differences between the virtual particles, probabilities of the relative velocities of the particles, etc. be accounted for when calculating the amount of particle splits, and also do these splits occur in just a set region around the black hole, or are they just most probably very close to the black hole?

The picture of Hawking radiation being produced by pairs of virtual particles separated by tidal gravity, so one falls into the hole while the other flies away, although it is commonly found in pop science presentations, is only a heuristic picture, and doesn't really work if you're trying to gain insight into why the radiation is emitted near the horizon (except for the obvious point that, for one of the particles to fall into the hole, the pair can't be produced too far from the horizon). Unfortunately, there isn't really a simple way to explain what's going on that works better than the heuristic picture for this (at least, I don't know of one); you would have to delve into the actual quantum field theory calculation, which is quite a bit more complicated than the heuristic picture suggests.

do these splits occur in just a set region around the black hole, or are they just most probably very close to the black hole?

There is no hard "cutoff" to the region where Hawking radiation is emitted; like all quantum processes, it is really a matter of probability. But the probability of emission drops off very steeply as the distance from the horizon increases.

 

[mivanit]

Thank you so much for helping me!
as for how I "arrived" at it being directly proportional, I just assumed it would be related for trying to calculate power. sorry for bad wording on my part.
After trying several possibilities for area where radiation occurs i figure out pretty fast it is not directly proportional to tidal force haha
thank you again so much!