Advertisement

The European Physical Journal C

, 72:2152 | Cite as

Condensation of an ideal gas with intermediate statistics on the horizon

  • Somayeh Zare
  • Zahra Raissi
  • Hosein MohammadzadehEmail author
  • Behrouz Mirza
Regular Article - Theoretical Physics

Abstract

We consider a boson gas on the stretched horizon of the Schwartzschild and Kerr black holes. It is shown that the gas is in a Bose–Einstein condensed state with the Hawking temperature T c =T H if the particle number of the system be equal to the number of quantum bits of space-time \(N \simeq{A}/{l_{p}^{2}}\). Entropy of the gas is proportional to the area of the horizon (A) by construction. For a more realistic model of quantum degrees of freedom on the horizon, we should presumably consider interacting bosons (gravitons). An ideal gas with intermediate statistics could be considered as an effective theory for interacting bosons. This analysis shows that we may obtain a correct entropy just by a suitable choice of parameter in the intermediate statistics.

Keywords

Black Hole Black Hole Entropy Helmholtz Free Energy Holographic Screen Newman 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 would like to thank R. Casadio for useful comments. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/2359.

References

  1. 1.
    S.W. Hawking, Phys. Rev. Lett. 26, 1344 (1971) ADSCrossRefGoogle Scholar
  2. 2.
    J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phys. 31, 161 (1973) MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973) MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    S.W. Hawking, Nature 248, 5443 (1974) CrossRefGoogle Scholar
  5. 5.
    S.W. Hawking, Commun. Math. Phys. 43, 199 (1975) MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    S.W. Hawking, Phys. Rev. D 14, 2460 (1976) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    A. Strominger, C. Vafa, Phys. Lett. B 379, 2460 (1996) MathSciNetGoogle Scholar
  8. 8.
    C. Rovelli, Phys. Rev. Lett. 77, 3288 (1996) MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    A. Ashtekar, A. Corichi, K. Krasnov, Adv. Theor. Math. Phys. 3, 419 (2000) MathSciNetGoogle Scholar
  10. 10.
    A. Ashtekar, J. Baez, K. Krasnov, Adv. Theor. Math. Phys. 4, 1 (2000) MathSciNetzbMATHGoogle Scholar
  11. 11.
    G. ’t Hooft, Nucl. Phys. B 256, 727 (1985) MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    L. Susskind, Phys. Rev. D 49, 6606 (1994) MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    L. Susskind, Phys. Rev. Lett. 71, 2367 (1993) MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    W.H. Zurek, K.S. Thorne, Phys. Rev. Lett. 54, 2171 (1985) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    L. Bombelli, R.K. Koul, J. Lee, R.D. Sorkin, Phys. Rev. D 34, 373 (1986) MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    M. Srednicki, Phys. Rev. Lett. 71, 666 (1993) MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    C. Callan, F. Wilczek, Phys. Lett. B 333, 55 (1994) MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    U.H. Gerlach, Phys. Rev. D 14, 1479 (1976) ADSCrossRefGoogle Scholar
  19. 19.
    J.W. York, Phys. Rev. D 28, 2929 (1983) MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    K.S. Thorne, R.H. Price, D.A. Macdonald, Black Holes: The Membrane Paradigm (Yale University Press, New Haven, 1986) Google Scholar
  21. 21.
    L. Susskind, L. Thorlacius, J. Uglum, Phys. Rev. D 48, 3743 (1993) MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    N. Iizuka, D. Kabat, G. Lifschitz, D.A. Lowe, Phys. Rev. D 68, 084021 (2003) MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    S. Singh, R.K. Pathria, J. Phys. A 17, 2983 (1984) MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    L. Parker, Y. Zhang, Phys. Rev. D 44, 2421 (1991) MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    D.J. Toms, Phys. Rev. Lett. 69, 1152 (1992) MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    D.J. Toms, Phys. Rev. D 47, 2483 (1993) MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    X. Li, Z. Zhao, Phys. Rev. D 62, 104001 (2000) MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    A.M. Scarfone, P. Narayana Swamy, J. Phys. A, Math. Theor. 41, 275211 (2008) MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    A.M. Scarfone, P. Narayana Swamy, J. Stat. Mech. P02055 (2009) Google Scholar
  30. 30.
    D.V. Anghel, Fractional exclusion statistics—the method to describe interacting particle systems as ideal gas. arXiv:1207.6534
  31. 31.
    B. Mirza, H. Mohammadzadeh, Phys. Rev. E 82, 031137 (2010) ADSCrossRefGoogle Scholar
  32. 32.
    B. Mirza, H. Mohammadzadeh, J. Phys. A, Math. Theor. 44, 475003 (2011) MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    G. ’t Hooft, Dimensional reduction in quantum gravity. gr-qc/9310026
  34. 34.
    L. Susskind, J. Math. Phys. 36, 6377 (1995) MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. 35.
    T. Padmanabhan, Phys. Rev. D 81, 124040 (2010) ADSCrossRefGoogle Scholar
  36. 36.
    E. Verlinde, J. High Energy Phys. 1104, 029 (2011) MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    S. Kolekar, T. Padmanabhan, Phys. Rev. D 83, 064034 (2011) ADSCrossRefGoogle Scholar
  38. 38.
    R.K. Pathria, Statistical Mechanics (Elsevier, Amsterdam, 2005) Google Scholar
  39. 39.
    R. Hakim, Introduction to Relativistic Statistical Mechanics (World Scientific, Singapore, 2011) zbMATHCrossRefGoogle Scholar
  40. 40.
    S. Hod, Phys. Lett. B 695, 294 (2011) MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    W. Israel, S. Mukohyama, Phys. Rev. D 58, 104005 (1998) MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    R. Zhao, L. Zhang, H. Li, Y. Wu, Int. J. Theor. Phys. 47, 3083 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    F.D.M. Haldane, Phys. Rev. Lett. 67, 937 (1991) MathSciNetADSzbMATHCrossRefGoogle Scholar
  44. 44.
    G. Gentile, Nuovo Cimento 19, 109 (1942) CrossRefGoogle Scholar
  45. 45.
    A.P. Polychronakos, Phys. Lett. B 365, 202 (1996) MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    C.R. Lee, J.P. Yu, Phys. Lett. A 150, 63 (1990) MathSciNetADSCrossRefGoogle Scholar
  47. 47.
    Y.J. Ng, J. Phys. A 23, 1023 (1990) MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 48.
    M. Chaichian, R. Gozales Felipe, C. Montonen, J. Phys. A 26, 4017 (1993) MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    J.A. Tuszyński, J.L. Rubin, J. Meyer, K. Kibler, Phys. Lett. A 175, 173 (1993) MathSciNetADSCrossRefGoogle Scholar
  50. 50.
    A. Lavagno, P. Narayana Swamy, Phys. Rev. E 65, 036101 (2002) ADSCrossRefGoogle Scholar
  51. 51.
    Y. Tian, X.-N. Wu, J. High Energy Phys. 01, 150 (2011) MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    Y.-X. Liu, Y.-Q. Wang, S.-W. Wei, Class. Quantum Gravity 27, 185002 (2010) MathSciNetADSCrossRefGoogle Scholar
  53. 53.
    R.A. Konoplya, Eur. Phys. J. C 69, 555 (2010) ADSCrossRefGoogle Scholar
  54. 54.
    Y. Tian, X.-N. Wu, Phys. Rev. D 81, 104013 (2010) ADSCrossRefGoogle Scholar
  55. 55.
    Y.-X. Chen, J.-L. Li, Phys. Lett. B 700, 380 (2011) MathSciNetADSCrossRefGoogle Scholar
  56. 56.
    R.B. Mann, J.R. Mureika, Phys. Lett. B 703, 167 (2011) MathSciNetADSCrossRefGoogle Scholar
  57. 57.
    M. Visser, Conservative entropic forces. arXiv:1108.5240
  58. 58.
    R.M. Wald, General Relativity (The University of Chicago Press, Chicago, 1984) zbMATHGoogle Scholar
  59. 59.
    L.C.B. Crispino, A. Higuchi, G.E.A. Matsas, Rev. Mod. Phys. 80, 787 (2008) MathSciNetADSzbMATHCrossRefGoogle Scholar
  60. 60.
    R.B. Mann, S.N. Solodukhin, Nucl. Phys. B 523, 293–307 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar
  61. 61.
    M. Schiffer, J. Bekenstein, Phys. Rev. D 42, 3598 (1990) ADSCrossRefGoogle Scholar
  62. 62.
    A. Ghosh, P. Mitra, Phys. Rev. Lett. 73, 2521 (1994) MathSciNetADSzbMATHCrossRefGoogle Scholar
  63. 63.
    D. Fursaev, Phys. Rev. D 51, 5352 (1995) MathSciNetADSCrossRefGoogle Scholar
  64. 64.
    R. Emparan, J. High Energy Phys. 9906, 036 (1999) MathSciNetADSCrossRefGoogle Scholar
  65. 65.
    S. Carlip, Nucl. Phys. Proc. Suppl. 88, 10 (2000) MathSciNetADSCrossRefGoogle Scholar
  66. 66.
    A. Sen, Logarithmic corrections to Schwarzschild and other non-extremal black hole entropy in different dimensions. arXiv:1205.0971
  67. 67.
    Y. Sekino, L. Susskind, J. High Energy Phys. 0810, 065 (2008) ADSCrossRefGoogle Scholar
  68. 68.
    C. Vaz, L.C.R. Wijewardhana, Class. Quantum Gravity 27, 055009 (2010) MathSciNetADSCrossRefGoogle Scholar
  69. 69.
    G. Dvali, C. Gomez, Black holes as critical point of quantum phase transition. arXiv:1207.4059

Copyright information

© Springer-Verlag / Società Italiana di Fisica 2012

Authors and Affiliations

  • Somayeh Zare
    • 1
  • Zahra Raissi
    • 1
  • Hosein Mohammadzadeh
    • 2
    Email author
  • Behrouz Mirza
    • 1
    • 3
  1. 1.Department of PhysicsIsfahan University of TechnologyIsfahanIran
  2. 2.Department of PhysicsUniversity of Mohaghegh ArdabiliArdabilIran
  3. 3.Research Institute for Astronomy and Astrophysics of Maragha (RIAAM)MaraghaIran

Personalised recommendations