Skip to main content
SpringerLink
Log in

Black hole thermodynamical entropy

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal C Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Notice that Newtonian gravitation generates a double difficulty, namely at vanishing distances and at diverging distances. The mathematical difficulty at short distances, where the potential decreases without limit, is nowadays considered to be solved due to the quantum nature of physical laws; this was of course unknown by Gibbs at his time. The mathematical difficulty at long distances remains still today in what concerns the thermostatistical properties, in fact intensively studied nowadays.

  2. From the microscopic (classical) dynamical point of view, this anomaly is directly related to the fact that the entire Lyapunov spectrum vanishes in the N→∞ limit, which typically impeaches ergodicity (see [34, 35] and references therein). This type of difficulty is also present, sometimes in an even more subtle manner, in various quantum systems (the single hydrogen atom constitutes, among many others, an elementary such example; indeed its BG partition function diverges due to the accumulation of electronic energy levels just below the ionization energy).

References

  1. J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  2. J.D. Bekenstein, Phys. Rev. D 9, 3292 (1974)

    Article  ADS  Google Scholar 

  3. S.W. Hawking, Nature 248, 30 (1974)

    Article  ADS  Google Scholar 

  4. S.W. Hawking, Phys. Rev. D 13, 191 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  5. G. ’t Hooft, Nucl. Phys. B 256, 727 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  6. G. ’t Hooft, Nucl. Phys. B 355, 138 (1990), and references therein

    Article  MathSciNet  Google Scholar 

  7. L. Susskind, Phys. Rev. Lett. 71, 2367 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. J. Maddox, Nature 365, 103 (1993)

    Article  ADS  Google Scholar 

  9. M. Srednicki, Phys. Rev. Lett. 71, 666 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. A. Strominger, C. Vafa, Phys. Lett. B 379, 99 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  11. J. Maldacena, A. Strominger, J. High Energy Phys. 2, 014 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  12. S. Das, S. Shankaranarayanan, Phys. Rev. D 73, 121701(R) (2006)

    ADS  Google Scholar 

  13. R. Brustein, M.B. Einhorn, A. Yarom, J. High Energy Phys. 01, 098 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  14. L. Borsten, D. Dahanayake, M.J. Duff, H. Ebrahim, W. Rubens, Phys. Rep. 471, 113 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  15. T. Padmanabhan, arXiv:0910.0839 (2009)

  16. H. Casini, Phys. Rev. D 79, 024015 (2009)

    Article  ADS  Google Scholar 

  17. L. Borsten, D. Dahanayake, M.J. Duff, A. Marrani, W. Rubens, Phys. Rev. Lett. 105, 100507 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  18. C. Corda, J. High Energy Phys. 08, 101 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  19. S. Kolekar, T. Padmanabhan, Phys. Rev. D 83, 064034 (2011)

    Article  ADS  Google Scholar 

  20. H. Saida, Entropy 13, 1611 (2011)

    Article  MathSciNet  Google Scholar 

  21. H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd edn. (Wiley, New York, 1985)

    MATH  Google Scholar 

  22. P. Jund, S.G. Kim, C. Tsallis, Phys. Rev. B 52, 50 (1995)

    Article  ADS  Google Scholar 

  23. S.A. Cannas, F.A. Tamarit, Phys. Rev. B 54, R12661 (1996)

    Article  ADS  Google Scholar 

  24. L.C. Sampaio, M.P. de Albuquerque, F.S. de Menezes, Phys. Rev. B 55, 5611 (1997)

    Article  ADS  Google Scholar 

  25. C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998)

    Article  ADS  Google Scholar 

  26. J.R. Grigera, Phys. Lett. A 217, 47 (1996)

    Article  ADS  Google Scholar 

  27. S. Curilef, C. Tsallis, Phys. Lett. A 264, 270 (1999)

    Article  ADS  Google Scholar 

  28. R.F.S. Andrade, S.T.R. Pinho, Phys. Rev. E 71, 026126 (2005)

    ADS  Google Scholar 

  29. C.A. Condat, J. Rangel, P.W. Lamberti, Phys. Rev. E 65, 026138 (2002)

    MathSciNet  ADS  Google Scholar 

  30. H.H.A. Rego, L.S. Lucena, L.R. da Silva, C. Tsallis, Physica A 266, 42 (1999)

    Article  Google Scholar 

  31. U.L. Fulco, L.R. da Silva, F.D. Nobre, H.H.A. Rego, L.S. Lucena, Phys. Lett. A 312, 331 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. C. Tsallis, Introduction to Nonextensive Statistical Mechanics—Approaching a Complex World (Springer, New York, 2009)

    MATH  Google Scholar 

  33. J.W. Gibbs, Elementary Principles in Statistical Mechanics—Developed with Especial Reference to the Rational Foundation of Thermodynamics (C. Scribner’s Sons, New York, 1902) (Yale University Press, New Haven, 1948; OX Bow Press, Woodbridge, Connecticut, 1981), p. 35

    Book  MATH  Google Scholar 

  34. C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998)

    Article  ADS  Google Scholar 

  35. A. Campa, A. Giansanti, D. Moroni, C. Tsallis, Phys. Lett. A 286, 251 (2001)

    Article  ADS  MATH  Google Scholar 

  36. L.P. Hughston, K.P. Tod, An Introduction to General Relativity (Cambridge University Press, Cambridge, 1990)

    MATH  Google Scholar 

  37. E.F. Taylor, J.A. Wheeler, Exploring Black Holes: Introduction to General Relativity (Addison-Wesley, San Francisco, 2000)

    Google Scholar 

  38. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972)

    Google Scholar 

  39. J. Eisert, M. Cramer, M.B. Plenio, Rev. Mod. Phys. 82, 277 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. M. Banados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. S. Carlip, Class. Quantum Gravity 12, 2853 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  42. S. Carlip, Quantum Gravity in 2+1 Dimensions (Cambridge University Press, Cambridge, 1998)

    Book  MATH  Google Scholar 

  43. R. Hanel, S. Thurner, Europhys. Lett. 93, 20006 (2011)

    Article  ADS  Google Scholar 

  44. R. Hanel, S. Thurner, Europhys. Lett. 96, 50003 (2011)

    Article  ADS  Google Scholar 

  45. U. Tirnakli, C. Tsallis, M.L. Lyra, Eur. Phys. J. B 11, 309 (1999)

    ADS  Google Scholar 

  46. G. Ruiz, C. Tsallis, Eur. Phys. J. B 67, 577 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. F. Baldovin, A. Robledo, Europhys. Lett. 60, 518 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  48. G. Casati, C. Tsallis, F. Baldovin, Europhys. Lett. 72, 355 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  49. E.P. Borges, C. Tsallis, G.F.J. Ananos, P.M.C. de Oliveira, Phys. Rev. Lett. 89, 254103 (2002)

    Article  ADS  Google Scholar 

  50. G.F.J. Ananos, C. Tsallis, Phys. Rev. Lett. 93, 020601 (2004)

    Article  ADS  Google Scholar 

  51. F. Baldovin, A. Robledo, Phys. Rev. E 66, R045104 (2002)

    Article  ADS  Google Scholar 

  52. F. Baldovin, A. Robledo, Phys. Rev. E 69, 045202(R) (2004)

    Article  MathSciNet  ADS  Google Scholar 

  53. E. Mayoral, A. Robledo, Phys. Rev. E 72, 026209 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  54. E. Mayoral, A. Robledo, Physica A 340, 219 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  55. G. Ruiz, C. Tsallis, Phys. Lett. A 376, 2451 (2012)

    Article  ADS  MATH  Google Scholar 

  56. H. Touchette, Phys. Lett. A 377, 436 (2013)

    Article  ADS  Google Scholar 

  57. G. Ruiz, C. Tsallis, Phys. Lett. A 377, 491 (2013)

    Article  ADS  Google Scholar 

  58. C. Tsallis, J. Stat. Phys. 52, 479 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. C. Tsallis, in Nonextensive Entropy—Interdisciplinary Applications, ed. by M. Gell-Mann, C. Tsallis (Oxford University Press, New York, 2004), p. 4

    Google Scholar 

  60. C. Tsallis, M. Gell-Mann, Y. Sato, Proc. Natl. Acad. Sci. USA 102, 15377 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. O. Penrose, Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon, Oxford, 1970), p. 167

    MATH  Google Scholar 

  62. C.-Y. Wong, G. Wilk, Acta Phys. Pol. B 43, 2043 (2012)

    Google Scholar 

  63. C.-Y. Wong, G. Wilk, Phys. Rev. D 87, 114007 (2013)

    Article  ADS  Google Scholar 

  64. V. Khachatryan et al. (CMS Collaboration), Phys. Rev. Lett. 105, 022002 (2010)

    Article  ADS  Google Scholar 

  65. V. Khachatryan et al. (CMS Collaboration), J. High Energy Phys. 05, 064 (2011)

    Article  ADS  Google Scholar 

  66. K. Aamodt et al. (ALICE Collaboration), Eur. Phys. J. C 71, 1655 (2011)

    Article  ADS  Google Scholar 

  67. K. Aamodt et al. (ALICE Collaboration), Phys. Lett. B 693, 53 (2010)

    Article  ADS  Google Scholar 

  68. G. Aad et al. (ATLAS Collaboration), New J. Phys. 13, 053033 (2011)

    Article  ADS  Google Scholar 

  69. A. Adare et al. (PHENIX Collaboration), Phys. Rev. D 83, 052004 (2011)

    Article  ADS  Google Scholar 

  70. A. Adare et al. (PHENIX Collaboration), Phys. Rev. C 83, 064903 (2011)

    Article  ADS  Google Scholar 

  71. M. Shao et al., J. Phys. G, Nucl. Part. Phys. 37, 085104 (2010)

    Article  ADS  Google Scholar 

  72. R.M. Pickup, R. Cywinski, C. Pappas, B. Farago, P. Fouquet, Phys. Rev. Lett. 102, 097202 (2009)

    Article  ADS  Google Scholar 

  73. C. Tsallis, J.C. Anjos, E.P. Borges, Phys. Lett. A 310, 372 (2003)

    Article  ADS  Google Scholar 

  74. C. Anteneodo, C. Tsallis, J. Mol. Liq. 71, 255 (1997)

    Article  Google Scholar 

  75. F.A. Tamarit, S.A. Cannas, C. Tsallis, Eur. Phys. J. B 1, 545 (1998)

    Article  ADS  Google Scholar 

  76. P. Douglas, S. Bergamini, F. Renzoni, Phys. Rev. Lett. 96, 110601 (2006)

    Article  ADS  Google Scholar 

  77. R.G. DeVoe, Phys. Rev. Lett. 102, 063001 (2009)

    Article  ADS  Google Scholar 

  78. J.S. Andrade, G.F.T. da Silva, A.A. Moreira, F.D. Nobre, E.M.F. Curado, Phys. Rev. Lett. 105, 260601 (2010)

    Article  ADS  Google Scholar 

  79. Y. Levin, R. Pakter, Phys. Rev. Lett. 107, 088901 (2011)

    Article  ADS  Google Scholar 

  80. J.S. Andrade, G.F.T. da Silva, A.A. Moreira, F.D. Nobre, E.M.F. Curado, Phys. Rev. Lett. 107, 088902 (2011)

    Article  ADS  Google Scholar 

  81. M.S. Ribeiro, F.D. Nobre, E.M.F. Curado, Phys. Rev. E 85, 021146 (2012)

    Article  ADS  Google Scholar 

  82. M.S. Ribeiro, F.D. Nobre, E.M.F. Curado, Eur. Phys. J. B 85, 399 (2012)

    Article  ADS  Google Scholar 

  83. M.S. Ribeiro, F.D. Nobre, E.M.F. Curado, Entropy 13, 1928 (2011)

    Article  MathSciNet  Google Scholar 

  84. B. Liu, J. Goree, Phys. Rev. Lett. 100, 055003 (2008)

    Article  ADS  Google Scholar 

  85. L.F. Burlaga, A.F. Vinas, N.F. Ness, M.H. Acuna, Astrophys. J. 644, L83 (2006)

    Article  ADS  Google Scholar 

  86. L.F. Burlaga, N.F. Ness, Astrophys. J. 737, 35 (2011)

    Article  ADS  Google Scholar 

  87. A. Esquivel, A. Lazarian, Astrophys. J. 710, 125 (2010)

    Article  ADS  Google Scholar 

  88. F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Phys. Rev. Lett. 106, 140601 (2011)

    Article  ADS  Google Scholar 

  89. F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Europhys. Lett. 97, 41001 (2012)

    Article  ADS  Google Scholar 

  90. R.N. Costa Filho, M.P. Almeida, G.A. Farias, J.S. Andrade Jr., Phys. Rev. A 84, 050102(R) (2011)

    Article  ADS  Google Scholar 

  91. S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76, 307 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  92. S. Umarov, C. Tsallis, M. Gell-Mann, S. Steinberg, J. Math. Phys. 51, 033502 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  93. F. Caruso, C. Tsallis, Phys. Rev. E 78, 021102 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  94. A. Saguia, M.S. Sarandy, Phys. Lett. A 374, 3384 (2010)

    Article  ADS  MATH  Google Scholar 

  95. E.P. Borges, I. Roditi, Phys. Lett. A 246, 399 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  96. V. Schwammle, C. Tsallis, J. Math. Phys. 48, 113301 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  97. P. Tempesta, Phys. Rev. E 84, 021121 (2011)

    ADS  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil

    Constantino Tsallis & Leonardo J. L. Cirto

  2. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA

    Constantino Tsallis

Authors
  1. Constantino Tsallis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonardo J. L. Cirto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Constantino Tsallis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tsallis, C., Cirto, L.J.L. Black hole thermodynamical entropy. Eur. Phys. J. C 73, 2487 (2013). https://doi.org/10.1140/epjc/s10052-013-2487-6

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1140/epjc/s10052-013-2487-6

Keywords

Search

Navigation