Advertisement

Higher order WKB corrections to black hole entropy in brick wall formalism

  • Wontae Kim
  • Shailesh KulkarniEmail author
Regular Article - Theoretical Physics

Abstract

We calculate the statistical entropy of a quantum field with an arbitrary spin propagating on the spherical symmetric black hole background by using the brick wall formalism at higher orders in the WKB approximation. For general spins, we find that the correction to the standard Bekenstein–Hawking entropy depends logarithmically on the area of the horizon. Furthermore, we apply this analysis to the Schwarzschild and Schwarzschild–AdS(dS) black holes and discuss our results.

Keywords

Black Hole Event Horizon Black Hole Entropy Statistical Entropy Logarithmic Correction 
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 E. Son and M. Eune for discussions. S. Kulkarni wish to thank L. Sriramkumar and S.K. Modak for their useful comments and suggestions. S. Kulkarni is being supported by National Research Foundation (NRF) grant funded by the Korea government (MEST) through the Center for Quantum Spacetime (CQUeST) of Sogang University with grant number 2005-0049409. W. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0008359).

References

  1. 1.
    J.D. Bekenstein, Lett. Nuovo Cimento 4, 737 (1972) ADSCrossRefGoogle Scholar
  2. 2.
    J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973) MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    S.W. Hawking, Nature (London) 248, 30 (1974) ADSCrossRefGoogle Scholar
  4. 4.
    S.W. Hawking, Commun. Math. Phys. 43, 199 (1975) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    G.W. Gibbons, S.W. Hawking, Phys. Rev. D 15, 2752 (1977) MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    J.W. York, Phys. Rev. D 33, 2092 (1986) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    H.W. Braden, J.D. Brown, B.F. Whiting, J.W. York, Phys. Rev. D 42, 3376 (1990) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    S.P. de Alwis, N. Ohta, hep-th/9412027
  9. 9.
    S.P. de Alwis, N. Ohta, Phys. Rev. D 52, 3529 (1995). hep-th/9504033 MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    A. Strominger, C. Vafa, Phys. Lett. B 379, 99 (1996). hep-th/9601029 MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    C.V. Johnson, P.R. Khuri, R.C. Meyers, Phys. Lett. B 378, 78 (1996). hep-th/9603061 MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    J.M. Maldacena, A. Strominger, Phys. Rev. Lett. 77, 428 (1996). hep-th/9603060 ADSCrossRefGoogle Scholar
  13. 13.
    S. Banerjee, R.K. Gupta, A. Sen, J. High Energy Phys. 1103, 147 (2011). arXiv:1005.3044 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    S. Banerjee, R.K. Gupta, I. Mandal, A. Sen, J. High Energy Phys. 1111, 143 (2011). arXiv:1106.0080 [hep-th] ADSCrossRefGoogle Scholar
  15. 15.
    C.G. Callan, J.M. Maldacena, Nucl. Phys. B 472, 591 (1996). hep-th/9602043 MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    G.T. Horowitz, A. Strominger, Phys. Rev. Lett. 77, 2368 (1996). hep-th/9602051 ADSCrossRefGoogle Scholar
  17. 17.
    A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, Phys. Rev. Lett. 80, 904 (1998). gr-qc/9710007 MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    R.K. Kaul, P. Majumdar, Phys. Rev. Lett. 84, 5255 (2000). gr-qc/0002040 MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    M. Srednicki, Phys. Rev. Lett. 71, 666 (1993). hep-th/9303048 MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    S. Das, S. Shankaranarayanan, S. Sur, Phys. Rev. D 77, 064013 (2008). arXiv:0705.2070 [gr-qc] ADSCrossRefGoogle Scholar
  21. 21.
    S. Solodukhin, Living Rev. Relativ. 14, 8 (2011). arXiv:1104.3712 ADSGoogle Scholar
  22. 22.
    S. Carlip, Phys. Rev. Lett. 88, 241301 (2002). gr-qc/0203001 MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    S. Carlip, Class. Quantum Gravity 17, 4175 (2000). gr-qc/0005017 MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    R.-G. Cai, L.-M. Cao, N. Ohta, J. High Energy Phys. 1004, 082 (2010). arXiv:0911.4379 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    K. Srinivasan, T. Padmanabhan, Phys. Rev. D 60, 024007 (1999). gr-qc/9812028 MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    R. Banerjee, B.R. Majhi, J. High Energy Phys. 0806, 095 (2008). arXiv:0805.2220 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    A. Sen, arXiv:1205.0971 [hep-th]
  28. 28.
    G. ’t Hooft, Nucl. Phys. B 256, 727 (1985) ADSCrossRefGoogle Scholar
  29. 29.
    L. Susskind, J. Uglum, Phys. Rev. D 50, 2700 (1994) MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    R.B. Mann, L. Tarasov, A. Zelnikov, Class. Quantum Gravity 9, 1487 (1992) MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    J.-G. Demers, R. Lafrance, R.C. Myers, Phys. Rev. D 52, 2245 (1995). gr-qc/9503003 MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    J.L.F. Barbon, R. Emparan, Phys. Rev. D 52, 4527 (1995). hep-th/9502155 MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    T. Padmanabhan, Phys. Lett. B 173, 43 (1986) MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    S. Mukohyama, W. Israel, Phys. Rev. D 58, 104005 (1998). gr-qc/9806012 MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    S. Das, P. Majumdar, R.K. Bhadury, Class. Quantum Gravity 19, 2355 (2002). hep-th/0111001 ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    R. Banerjee, S. Modak, J. High Energy Phys. 05, 063 (2009). arXiv:0903.3321 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    S. Sarkar, S. Shankaranarayanan, L. Sriramkumar, Phys. Rev. D 78, 024003 (2008). arXiv:0710.2013 [gr-qc] MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    A. Ghosh, P. Mitra, Phys. Lett. B 357, 295 (1995). hep-th/9411128 MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    S.W. Kim, W.T. Kim, Y.J. Park, H. Shin, Phys. Lett. B 392, 311 (1997). hep-th/9603043 MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    S.P. Kim, S.K. Kim, K.S. Soh, J.H. Yee, Phys. Rev. D 55, 2159 (1997). gr-qc/9608015 MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    J. Ho, W.T. Kim, Y.J. Park, H. Shin, Class. Quantum Gravity 14, 2617 (1997). gr-qc/9704032 MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    H.K. Jassal, L. Sriramkumar, Class. Quantum Gravity 24, 2589 (2007). gr-qc/0611102 MathSciNetADSzbMATHCrossRefGoogle Scholar
  43. 43.
    E.J. Son, M. Eune, W. Kim, Phys. Lett. B 706, 447 (2012). arXiv:1109.5486 [hep-th] ADSCrossRefGoogle Scholar
  44. 44.
    G. Cognola, P. Lecca, Phys. Rev. D 57, 1108 (1998). hep-th/9706065 MathSciNetADSCrossRefGoogle Scholar
  45. 45.
    Y.G. Shen, D.M. Chen, T.J. Zhang, Phys. Rev. D 56, 6698 (1997). hep-th/9603043 MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    S.Q. Wu, M.L. Yan, Phys. Rev. D 69, 044019 (2004). gr-qc/0303076 MathSciNetADSCrossRefGoogle Scholar
  47. 47.
    Y.G. Shen, Phys. Lett. B 537, 187 (2001) ADSGoogle Scholar
  48. 48.
    C. Ding, J. Jing, J. High Energy Phys. 0709, 067 (2007). arXiv:0709.1186 [gr-qc] MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    S.A. Tuekolsky, Astrophys. J. 185, 635 (1973) ADSCrossRefGoogle Scholar
  50. 50.
    S. Chandrasekhar, The Mathematical Theory of Black Holes (Clarendon, Oxford, 1983) zbMATHGoogle Scholar
  51. 51.
    V.P. Frolov, I.D. Novikov, Black Hole Physics (Kluwer Academic, Dordrecht, 1998) zbMATHCrossRefGoogle Scholar
  52. 52.
    C.M. Bender, S.A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978) zbMATHGoogle Scholar
  53. 53.
    T. Padmanabhan, Rep. Prog. Phys. 73, 046901 (2010). arXiv:0911.5004 [gr-qc] ADSCrossRefGoogle Scholar
  54. 54.
    T. Padmanabhan, Gravitation: Foundations and Frontiers (Cambridge University Press, Cambridge, 2010) zbMATHCrossRefGoogle Scholar
  55. 55.
    R. Brou, S. Massar, R. Parentani, Ph. Spindel, Phys. Rep. 260, 329 (1995). arXiv:0710.4345 [gr-qc] MathSciNetADSCrossRefGoogle Scholar
  56. 56.
    E. Winstanley, Phys. Rev. D 63, 084013 (2001). hep-th/0011176 MathSciNetADSCrossRefGoogle Scholar
  57. 57.
    R.A. Bruere, M.P. Ryan, S. Waller, Proc. R. Soc. Lond. A 358 (1977) Google Scholar
  58. 58.
    P.M. Morse, H. Fishbach, in Methods in Theoretical Physics, Vol. 2 (McGraw-Hill, New York, 1963) Google Scholar
  59. 59.
    S.N. Solodukhin, Phys. Rev. D 51, 618 (1995). hep-th/9408068 MathSciNetADSCrossRefGoogle Scholar
  60. 60.
    S.N. Solodukhin, Phys. Rev. D 54, 3900 (1996). hep-th/9601154 MathSciNetADSCrossRefGoogle Scholar
  61. 61.
    D.V. Fursaev, S.N. Solodukhin, Phys. Lett. B 365, 51 (1996). hep-th/9412020 MathSciNetADSCrossRefGoogle Scholar
  62. 62.
    V.P. Frolov, D.V. Fursaev, Class. Quantum Gravity 15, 2041 (1998). hep-th/9802010 MathSciNetADSzbMATHCrossRefGoogle Scholar
  63. 63.
    T. Shimomura, Phys. Lett. B 480, 207 (2000) ADSCrossRefGoogle Scholar
  64. 64.
    Y.S. Myung, Phys. Rev. D 77, 104007 (2008). arXiv:0712.3315 [gr-qc] MathSciNetADSCrossRefGoogle Scholar
  65. 65.
    S. Tadaki, S. Takagi, Prog. Theor. Phys. 83, 941 (1990) MathSciNetADSzbMATHCrossRefGoogle Scholar
  66. 66.
    S. Tadaki, S. Takagi, Prog. Theor. Phys. 83, 1126 (1990) MathSciNetADSzbMATHCrossRefGoogle Scholar
  67. 67.
    R.M. Wald, General Relativity (Chicago University Press, Chicago, 1994) Google Scholar
  68. 68.
    W.-Z. Chao, Int. J. Mod. Phys. D 7, 887 (1998). gr-qc/9712066 ADSzbMATHCrossRefGoogle Scholar
  69. 69.
    R. Banerjee, S.K. Modak, S. Samanta, Eur. Phys. J. C 70, 317 (2010). arXiv:1002.0466 [hep-th] ADSCrossRefGoogle Scholar
  70. 70.
    R. Banerjee, S.K. Modak, S. Samanta, Phys. Rev. D 84, 064024 (2011). arXiv:1005.4832 [hep-th] ADSCrossRefGoogle Scholar
  71. 71.
    R. Banerjee, S.K. Modak, D. Roychowdhury, arXiv:1106.3877 [gr-qc]

Copyright information

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

Authors and Affiliations

  1. 1.Center for Quantum Spacetime (CQUeST)Sogang UniversitySeoulSouth Korea
  2. 2.Department of PhysicsSogang UniversitySeoulSouth Korea

Personalised recommendations