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Historically, Bekenstein estimated the entropy associated with a black hole in 1973, obtaining: $$ S_B = \frac{\ln(2)k_Bc^3}{8\pi\hbar G}A. $$ He already acknowledges in his article that his estimates are based on classical principles and that a quantum mechanical treatment will yield a different constant, though within a factor of order one the same. A year later Hawking derived: $$ S_H = \frac{k_Bc^3}{4G\hbar}A, $$ i.e. $S_B = (\ln(2)/2\pi) S_H$, such that $S_B<S_H$.

I am wondering if there could have been examples, showing that $S_B$ was not correct. So, without knowing Hawking's results, can we see that $S_B$ cannot be correct, maybe by giving a certain counter example, or using the fact that $S_B<S_H$?

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3 Answers3

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I recently went to a colloquium with the theme "98 years of black hole physics" by string theorist Jan de Boer from the university of Amsterdam. I asked him this question and he replied that there have been lattice computations for black hole thermodynamics, yielding precisely Hawking's factor of $1/4$. Furthermore the result has been obtained using different methods, strengthening its reliability.

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What about this derivation?

Is this derivation of Black Hole entropy viable?

In short, Bekenstein's entropy is integer amount of bits, which cannot be true for variable measured at Planck scale, where the fundamental unit of information is nat, that is $1/\ln 2 $ bit

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Anixx
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In informational terms, the relation between thermodynamic entropy $S$ and Shannon entropy $H$ is given by relation between $S$ & $H$:

$$ S=kH\ln(2)$$

whence

$$ H \le 2πRE/\hbar c\ln(2) $$

where $H$ is the Shannon entropy expressed in number of bits contained in the quantum states in the sphere.

as the ln 2 factor comes from defining the information as the logarithm to the base 2 of the number of quantum states

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