Reconciling the Black Hole Entropy with Classical Thermodynamics

Details Reconciling the Black Hole Entropy with Classical Thermodynamics Reconciling the Black Hole Entropy with Classical Thermodynamics

2012-02-09

arxiv.org/abs/1202.2154v1

Physics

Condensed Matter

Statistical Mechanics

7 pages, 2 figures

Scientific paper

As early as 1902, Gibbs pointed out that systems with long-range interactions, like gravitation, lie outside the validity of standard statistical mechanics. Nevertheless, the entropy of a black hole has been repeatedly calculated within Boltzmann-Gibbs concepts. Since the pioneering Bekenstein-Hawking results, it has become common to state that the black-hole "entropy" is proportional to its area. Similarly it exists the {\it area law}, so named because the "entropy" of a wide class of $d$-dimensional quantum systems is proportional to the $d$-dimensional area $A_d = L^{d-1}$ ($d>1$; $L$ is a characteristic length), instead of the $d$-dimensional volume $V_d = A_d^{d/(d-1)} = L^d$. These results violate the extensivity of the thermodynamical entropy. This inconsistency disappears if we realize that the entropies of such nonstandard systems must {\it not} be associated with the additive expression $S_{BG}=k_B\ln{W}$ but with appropriate nonadditive generalizations. Here we introduce a generalized form of entropy which solves the puzzle for the black hole and the area law.

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