Here's what for me was a fascinating (and straight forward) description of How Jacob Beckenstein concluded information is displayed on the horizon of the black hole. The formulas are so simple and the results so profound I wanted to post them here...I'd never seen this before...(adsbygoogle = window.adsbygoogle || []).push({});

This is excerpted from THE BLACK HOLE WAR 2008, by Leonard Susskind, most of which appears around pages 151 to 155...

The most difficult conceptual idea for me is the starting point:

Beckenstein assumed a one bit photon falls into a black hole. To make the photon location as "uncertain" as possible, minimizing its information content, following Heinsenberg's uncertainty principle, he assumed the wavelength to about equal the Schwarzchchild radius, the longest feasible wavelength for capture to spread the photon to be as uncertain as possible. Susskind says longer wavelength photons would bounce off the horizon.

Beckenstein wondered how the size of a black hole would change if a single bit of information were dropped in...today we take for granted it's one Planck area, but how did he calculate that????? Here's the derivation:

To figure the increase in mass, let's figure the energy of the photon and then convert that to an equivalent mass.

Photon energy is E = hf and if wavelength is R_{s/SUB], from v = f x wavelength, frequency (f) is c/Rs/SUB] so E = hf becomes hc/Rs/SUB]. (1) From E =mc2, dividing energy by c2 gives mass, so the change in mass becomes h/Rsc. (2) The Schwarzschild radius is given by Rs - 2MG/c2 (3) and substituting the above change in mass for the photon, substituting (2) in (3) increases the radius 2hG/Rsc3.... So we have a photon adding a bit (pun) of mass to a black hole. Plugging in all the numbers gives a radius increase of 10-72 meters.... Finally the (spherical) horizon area of a stable black hole is given by A= 4(pi) Rs2 so if the radius increases by a power of -72, the area increases by a power of -70.... What a coincidence...that's Planck area!!!!! (the square of planck length of 10-35 meter) .... so adding one bit adds one Planck area to the horizon!!! It works for any size black hole. Susskind concludes: Somehow, hidden in the principles of quantum mechanics and the general theory of relativity there is a mysterious connection between indivisible bits of information and Planck sized bits (pieces) of area. If anyone can explain a bit more about the logic underlying one bit per photon in this example, I'd appreciate it.}

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# Black Hole Horizon Information

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