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Black hole thermodynamical entropy

  • Constantino Tsallis
  • Leonardo J. L. Cirto
Regular Article - Theoretical Physics

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 BG 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 area law, so named because, for a wide class of strongly quantum-entangled d-dimensional systems, S BG is proportional to lnL if d=1, and to L d−1 if d>1, instead of being proportional to L d (d≥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 not to be identified with the BG additive entropy but with appropriately generalized 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.

Keywords

Entropy Black Hole Event Horizon Thermodynamical Entropy Dimensional Black Hole 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We acknowledge useful conversations with M. Jauregui. One of us (CT) also acknowledges (recent and old) conversations with L. Bergstrom, F. Caruso, H. Casini, A. Coniglio, E.M.F. Curado, M.J. Duff, A.S. Fokas, G. ’t Hooft, F.D. Nobre, N. Pinto Neto, G. Ruiz, H. Saida, I.D. Soares, L. Thorlacius and J. Zanelli. We have benefited from partial financial support from CNPq, Faperj and Capes (Brazilian agencies).

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Copyright information

© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Authors and Affiliations

  1. 1.Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex SystemsRio de JaneiroBrazil
  2. 2.Santa Fe InstituteSanta FeUSA

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