Smallest possible black hole containing any information?

Question

My apologies for the meandering nature of this question. Using some back-of-the-envelope numbers:

Planck length ≈ $10^{-35}$ m
Planck area ≈ $10^{-70}$ m
Radius of a black hole with a surface area of one planck area ≈ $10\sqrt{area}$ ≈ $10^{-34}$ m

Proton mass = $10^{-27}$ kg
Schwarzschild radius = $2GM/c^2$ = $10^{-27}$ m/kg * M
Radius of a Schwarzschild black hole with one proton mass = $10^{-54}$ m

Given the notion that every planck area on the surface of a black hole encodes one bit of information, it appears to me that compressing a proton into a black hole would mean it has zero information, which seems to violate the conservation of information.

Working backwards, a black hole with one bit of information would have a radius of about a planck length, so...

$10^{-35}$ m = $10^{-27}$ m/kg * M --> M = $10^{-8}$ kg = 10 micrograms

This indicates anything with a mass less than 10 micrograms couldn't become a black hole. That corresponds to a rest energy of $10^{27}$ eV.

There was speculation that the LHC might have produced microscopic black holes, but since the most energetic particles it produced were on the order of $10^{12}$ eV, it doesn't seem remotely possible that this could ever happen.

All this raises a few questions:
- Is every modern physics textbook that talks about the schwarzschild radius of a proton describing something that's truly theoretically impossible?
- Is a black hole smaller than this possible? If so, how much information could it have?
- If a black hole this size WAS possible, what happened to all the extra information? 10 micrograms of mass would need something like $10^{17}$ protons, and even more if it was made of electrons or neutrinos. However much information was in all those particles, only one bit remains. Is the solution in assuming the particles are traveling at relativistic speeds?
- How much information does one particle contain? Is this even a reasonable question to ask? Even if I were considering just its position, it seems like this could require any arbitrary number of bits depending on the desired precision and the coordinate system, which may betray my thinking of quantum information in the same way I think about computer data.

One more thought. Combining the schwarzschild radius with the amount of information encoded on the surface of the black hole, you can work out an information mass density as a function of radius. Depending on the energy of the particles that made up this black hole, there must be a break-even point where each particle would have one bit of information associated with it. Any less than that and it seems like you'd HAVE to lose information...but this is dependent on the nature of the matter that made up the black hole in the first place, which would violate the no-hair theorem.

Any fallacies in my thinking here?

Answer 1

Thanks for your interesting question and for your calculations. There is nothing fundamentally wrong with any of your calculations, EXCEPT (and that is a big except) that you are trying to apply classical General Relativity solutions (such as the Schwartzchild black hole solutions) to a situation where a full quantum mechanical gravitational theory would need to be used. For example, the mass you calculated of $10\ \mu gm$ is exactly a Planck mass (within a factor of 2).

The regime where a full quantum theory of gravity is needed is when approximately a Planck mass is confined to a Planck volume. Unfortunately, a full quantum theory of gravity is not yet available, so no good answers to your questions can be given.