Black hole thermodynamical entropy

Constantino Tsallis; Leonardo J. L. Cirto

doi:10.1140/epjc/s10052-013-2487-6

paper

Black hole thermodynamical entropy

arXiv:1202.2154 · doi:10.1140/epjc/s10052-013-2487-6

Abstract

As early as 1902, Gibbs pointed out that systems whose partition function diverges, e.g. gravitation, lie outside the validity of the Boltzmann-Gibbs (BG) theory. Consistently, since the pioneering Bekenstein-Hawking results, physically meaningful evidence (e.g., the holographic principle) has accumulated that the BG entropy $S_{B G}$ of a $(3 + 1)$ black hole is proportional to its area $L^{2}$ ( $L$ being a characteristic linear length), and not to its volume $L^{3}$ . Similarly it exists the \emph{area law}, so named because, for a wide class of strongly quantum-entangled $d$ -dimensional systems, $S_{B G}$ is proportional to $ln L$ if $d = 1$ , and to $L^{d - 1}$ if $d > 1$ , instead of being proportional to $L^{d}$ ( $d \geq 1$ ). These results violate the extensivity of the thermodynamical entropy of a $d$ -dimensional system. This thermodynamical inconsistency disappears if we realize that the thermodynamical entropy of such nonstandard systems is \emph{not} to be identified with the BG {\it additive} entropy but with appropriately generalized {\it nonadditive} entropies. Indeed, the celebrated usefulness of the BG entropy is founded on hypothesis such as relatively weak probabilistic correlations (and their connections to ergodicity, which by no means can be assumed as a general rule of nature). Here we introduce a generalized entropy which, for the Schwarzschild black hole and the area law, can solve the thermodynamic puzzle.

7 pages, 2 figures. Accepted for publication in EPJC

Black hole thermodynamical entropy

Abstract

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