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"4.2 Black hole quantum mechanics

We have seen that one thought experiment, the string in a small box, was exceedingly productive. For quantum gravity, black holes turn out to be another valuable laboratory, leading to the entropy puzzle and the information paradox.

4.2.1 Black hole entropy

By considering a process of feeding quantum bits into a black hole, Bekenstein argued that black holes have a well-defined information-carrying capacity. By the uncertainty principle, it requires an energy hc/R to fit a quantum into a black hole of radius R. Using the mass- radius relation for a black hole, c2R/2G, one finds that a black hole of radius can contain of order c3R2/hbits of information.

This argument was reinforced by the discovery of Hawking radiation [48]. Black holes radiate like a hot body with a temperature of order hc/RkB. By thermodynamic relations this translates into an entropy

S/kB ~ R^2/4Lp^2 "

Surface energy density ~ T^4 we measure far away from apparent horizon is ~ hc/R^4

However, I allege a second low entropy but higher temperature hc/(RLp)^1/2

With entropy S'/kB ~ R/Lp  from quantum gravity metric fluctuations in radial position of horizon

Asymptotic red shifted energy density is ~ hc/Lp^2R^2  same as dark energy in cosmic de Sitter case

Horizons are heat engines.

These are spin 2 random black body gravity waves at the UV limit.

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