|MadSci Network: Physics|
That's a good question. To be honest, I am not 100% sure, but I think the answer is No. The actual number, or energy density, of "vacuum fluctuations" should be pretty much the same everywhere. This is definitely true in places where gravity is weak; according to the principle of General Relativity, there should not be ANY observable differences in the microscopic laws of physics - including the creation of vacuum particle-antiparticle pairs - in different regions with different gravitational fields. (But this is not be area of expertise - it is possible that something strange happens in extremely strong gravity) In any case, it is not necessary to have more vacuum pairs, in order to have more Hawking radiation near small black holes. Let me explain:
So, you've probably heard that black holes emit Hawking radiation; this occurs when a particle-antiparticle pair pops out of the vacuum near a black hole. This happens all over the Universe - not just near black holes - but ordinarily, the pair has to meet and annihilate. (Otherwise, the law of conservation of energy is violated). Near a black hole, there is of course very a very strong gravitational field - moreover, the field changes rapidly when you get closer to the hole. If one of the new particles is a bit closer to the hole than the other, it can be ripped away and fall into event horizon. If the other particle escapes this fate, perhaps by virtue of starting an extra nanometer or two away from the horizon, it flies off into the distance. We call it "Hawking radiation", and it appears to be a particle created "out of nowhere". It didn't come out of the black hole's horizon, but somehow it stole some energy from it. But in order for one particle to escape and one to fall in, it is necessary to have that rapidly-changing field. You need the closer of the two particles to accelerate faster than its partner, in order for one to escape and the other not.
We experience gravity gradients on Earth, but the effect is very weak: if your feet are feeling a gravitational acceleration of 9.8 m/s, your head should only be feeling (say) 9.799995 m/s. Weird! So, strictly speaking, if you created a particle near your head and its antiparticle near your feet, and let them free-fall towards the center of the Earth, they would be pulled apart gently as they fell, by this extra 0.0000005 m/s of acceleration. But the separation would be very gentle.
If you imagine standing on a very dense asteroid - let's say it is 1m across, and lighter than the Earth so that the surface gravity is the same - then having 9.8 m/s of accleration at your feet, your head (two meters away, if you're pretty tall) will feel only 1.1 m/s of acceleration. So the differential force between your heat and feet - or between any particle/antiparticle pair in the vacuum - gets larger and larger, the closer you get to an object.
It is this differential gravity that rips particles and antiparticles out of the vacuum. That is why there is more Hawking radiation from a smaller black hole - not because there are more particle-antiparticle pairs there, but because they are ripped apart more easily when you get close to the object. Now, although a black hole itself - the actual massive bit - may be infinitesilmally small, the event horizon (where Hawking radiation takes place) is closer to a smaller, less-massive hole. Thus the gravity gradient is larger at the horizon of a lighter hole; thus the Hawking radiation is stronger.
Good question! Hope you are enjoying thinking about black holes. If you have not read Steven Hawking's books ("A Brief History of Time", and "Black Holes and Baby Universes") I am sure you would enjoy them.
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