0
$\begingroup$

I'm curious about a possible connection between the Berkenstein Bound and the physical nature of information, specifically in reference to the fact that the amount of information that can be written or erased increases as the temperature of the storage medium decreases. I started trying to crunch the numbers with photons and found that if a photon's wavelength of the peak of the black body radiation curve, then the photon should have a bit value of around 7.16 bits of information. This would apply to any photon, even one emitted from the surface of last scattering and collected today as a photon of the CMB.

So then I tried to calculate the informational content of an electron... And of course, since the electron has a rest mass, it turns out that the bit value of an electron varies greatly with the temperature, completely unlike the photon. And when the electron gets really, really cold, this informational content starts going up very quickly.

Then, tonight, I realized that, at a certain point, if the radius and rest mass of a massive particle remain constant, the information content of that particle could potentially exceed the Berkenstein Bound. At that critical temperature, a particle should spontaneously form an extremely small black hole. This critical temperature for a lead atom came out to around 1.1662e-29 K and the result I got for a carbon atom was about 3.88742e-29 K.

Now, that's pretty darn cold (well below the current world record). And, the surface temperature of a black hole that small would be considerably higher. So, to me, that implies that the newly formed black hole would evaporate almost instantly, releasing all of that information (4.8182416807235103e+42 bits) as (11187895819 eV) of energy in the form of photons.

I always hear about heat death or big rip or big crunch or vacuum fluctuation powered cyclic whatevers, but I don't recall ever hearing anyone mention the possibility that massive particles could, at extremely low temperatures, undergo spontaneous conversion (via black hole formation and subsequent Hawking radiation emission) into massless, gauge invariant bosons (i.e. photons).

Am I way off base here? Does this line of inquiry have any merit?


* EDIT *


Oops! I found several errors in my earlier calculations, so those were totally wrong, but I think the same question still applies. Here are the revised estimates and a more precise explanation of how I arrived at them.

Proton

Predicted Black Hole Transition Temperature ≈ 1.6989×10^-27 K

Mass: 1.672621898e-27 kg

Energy Equivalence: 1.5032775928961053e-10 J

Radius: 8.768000000000001e-16 m

Radius in Planck Units: 54249738124981060000 lp

Surface Area in Planck Units: 3.698325706327235e+40 lp^2

Max Information: 9.245814265818087e+39 bits

Energy required to read/erase 1 bit: 1.6258353426200812e-50 J

Proton Bit Value: 9.246185966615349e+39 bits

Neutron

Predicted Black Hole Transition Temperature ≈ 1.7013×10^-27 K

Mass: 1.674927471e-27 kg

Energy Equivalence: 1.5053497385698108e-10 J

Radius: 8.768000000000001e-16 m

Radius in Planck Units: 54249738124981060000 lp

Surface Area in Planck Units: 3.698325706327235e+40 lp^2

Max Information: 9.245814265818087e+39 bits

Energy required to read/erase 1 bit: 1.6281321257281445e-50 J

Neutron Bit Value: 9.24586963663393e+39 bits

Carbon Atom

Predicted Black Hole Transition Temperature ≈ 6.9218×10^-37 K

Mass: 1.9944234999999996e-26 kg

Energy Equivalence: 1.7924984493060003e-9 J

Radius: 1.5e-10 m

Radius in Planck Units: 9.280863046016375e+24 lp

Surface Area in Planck Units: 1.082397030284868e+51 lp^2

Max Information: 2.70599257571217e+50 bits

Energy required to read/erase 1 bit: 6.624113882245969e-60 J

Carbon Atom Bit Value: 2.706019976664768e+50 bits

Lead Atom

Predicted Black Hole Transition Temperature ≈ 3.5092×10^-35 K

Mass: 3.4406365999999995e-25 kg

Energy Equivalence: 3.0922899625500006e-8 J

Radius: 8.75e-11 m

Radius in Planck Units: 5.413836776842886e+24 lp

Surface Area in Planck Units: 3.68315656138601e+50 lp^2

Max Information: 9.207891403465025e+49 bits

Energy required to read/erase 1 bit: 3.358279701172753e-58 J

Lead Atom Bit Value: 9.207958352814193e+49 bits

$\endgroup$
  • 1
    $\begingroup$ I think you should show us the equations you used to produce those numbers. Bear in mind that this question may be closed as off-topic if it looks like what you're doing contradicts mainstream physics. $\endgroup$ – PM 2Ring Jan 24 at 9:15
  • $\begingroup$ What kind of radius did you use for the electron? I think a problem here is that changing temperature for particles also changes their wavelengths, so as you reduce it to zero their position becomes indeterminate and they hence escape the bound if you count their "radius" as their wavelength. A more fundamental problem is that the bound applies to spacetime regions and fields inside them, not individual particles. $\endgroup$ – Anders Sandberg Jan 24 at 9:31
  • $\begingroup$ I haven't calculated the "transition temperature" of the electron yet. Just the carbon atom and the lead atom. For the carbon atom I used a diameter of 0.3nm and a mass of 1.9944235e-26 kg and for the lead atom I used a diameter of 0.175 nm and a mass of 3.4406366e-25 kg. The energy of each was found by converting the aforementioned masses into joules and the "information content" for a given temperature was then calculated using Landauer's principle (kT ln 2). $\endgroup$ – Thor Jan 24 at 13:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.