If a schwartzchild radius of an object is calculated to be below atomic size, can it exist?

Answer 1

It can certainly exist. However, consider what this really means:

A black hole with a schwartzchild radius equal to the size of an atom, (let's pick the radius of a hydrogen atom,) will have a radius of:

r = 5.3 * 10^-11 m

Given it's radius we can calculate it's mass at:

m = 3.6 * 10^16 kg

This is relatively small, cosmologically: The earth's mass is on the order of 10^24 kg.

Now, here's where it gets interesting:

A black hole radiates a small amount of energy, called Hawking Radiation. Oddly, the LARGER the black hole gets, the LESS hawking radiation it emits! So, as it gets smaller, it radiates away it's mass faster and faster, until POP! Nobody really knows how it avoids an infinity at small enough masses, but it's a black hole: We're already dealing with infinities. The infinity doesn't matter that much. We can integrate the rate at which it loses mass (from the Hawking radtiation,) as a function of mass, and determine how long it has to live:

t = 1.2 * 10^26 years

Still a long time to be sure, so yes: It can still exist: The age of the universe is only on the order of about 1.3 * 10^10 years!

If you get a sufficiently small black hole, however, say, at one millionth of the radius of a proton, (~1.2 * 10^-21 m,) then things get really interesting: At that mass, (800,000 kg,) then the black hole is radiating A LOT of energy! In fact, at that small size it's got a luminosity of:

l = 5.5 * 10^20 Watts

That's a millionth of the luminosity of the sun, (which is still A LOT.) At that rate, (which increases even further as the thing evaporates faster,) the black hole only has another 45 SECONDS to live! By comparison, the first black hole we looked at had a feeble luminosity of 0.3 watts. (Snore.)

Even better, when the black hole has a mass of 1,000 kilograms, (which happens 44.9999999 seconds after what we just observed,) it's luminosity is so great that it's emitting more energy than the SUN ITSELF! Of course, at that point it's only got another 7*10^-8 seconds to live.)

So the short answer is "yes." But the more interesting answer, well, you just got it. There's a terrific website that allows you to plug numbers into these calculations, it's referenced below. The second reference explains why there's even such a thing as "Hawking radiation" at all! (After all, aren't black holes supposed to be, well, black?)

Answer 2

Yes. There is nothing special about the atom that defines any threshold for existance in terms of size. Such a black hole would give off a great deal of Hawking radiation and decay, though.