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Comment by jimhefferon | Hacker News Reader

"Since a finite volume of space can't contain more than a finite amount of information"

Is that perfectly clear? Not so, to me.

In physicist's ontology, as you surely know, that's just an appeal to Bekenstein bound.

Not really being a mathematical result, it is a consequence of thermodynamics and classical (continuous) gravity. Einstein equations can be recovered from thermodynamics and Bekenstein bound as an assumption and that's all there really is to it.

It's a known result in thermodynamics. The amount of information that can be contained within a sphere of radius r is bounded by O(r^2), surprisingly. An object can attain this bound if and only if it is a Black Hole.

This result is related to the fact that black holes evaporate. If black holes can contain information, it must be possible to read that information, or else it's observationally equivalent to the information having been destroyed. The evaporation of black holes leaks the information contained inside, avoiding a paradox.

> The evaporation of black holes leaks the information contained inside, avoiding a paradox.

Has blackhole evaporation been observed, or is this just a result of the equations?

This stuck out to me too. Fractals teach us that infinite length can be fitted into finite area, so I'm dubious about this claim. Though I guess if we're in the realm of physics, the Planck constant determines a kind of minimum length, so the original statement may be true?

Pendant here: the Planck constant is simply a convenient length for all things subatomic, like a hectare or a fortnight. No minimum whatsoever implied there. Many, many things are smaller than the Planck constant.

"Many, many things are smaller than the Planck constant."

Do we have a proof of that? Honest question. I know strings are often postulated to be smaller (by many orders of magnitude in some cases) but we are not yet particularly confident they exist. I'm not aware of us being reasonably confident that anything is smaller than Planck's size. ("Smaller than Planck's size" isn't even all that easy to define.)

Event horizon of a black hole is many times smaller for instance.

The mass of a blackhole determines the size of the event horizon.

To have an event horizon smaller than the planck constant, the mass have to be also small. While theoretically possible, i don't believe this has ever been considered possible in reality - see https://en.wikipedia.org/wiki/Kugelblitz_(astrophysics)

"Smaller" in what sense? Surely not diameter. There's a known formula that gives you the diameter as a function of the BH's mass.

> Pendant here: the Planck constant is simply a convenient length...

You tipped your hand by misspelling "pedant."

You mean the Planck length (related to the Planck constant, the speed of light and the gravitational constant).

> the Planck constant is simply a convenient length

m^2 * kg / s

Ideal fractals also don't exist. :-)

I'm also curious. Is it true for position/momentum/energy/spin/... of any particle?

AFAIK energy levels of electrons orbiting around a proton are discrete quanta but are all other things above also discrete, or can some be continuous?

What about the hidden quantum state of things?

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