Skip to main content
SpringerLink
Log in

Approximation of the naive black hole degeneracy

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

In 1996, Rovelli suggested a connection between black hole entropy and the area spectrum. Using this formalism and a theorem we prove in this paper, we briefly show the procedure to calculate the quantum corrections to the Bekenstein–Hawking entropy. One can do this by two steps. First, one can calculate the “naive” black hole degeneracy without the projection constraint (in case of the \(U(1)\) symmetry reduced framework) or the \(SU(2)\) invariant subspace constraint (in case of the fully \(SU(2)\) framework). Second, then one can impose the projection constraint or the \(SU(2)\) invariant subspace constraint, obtaining logarithmic corrections to the Bekenstein–Hawking entropy. In this paper, we focus on the first step and show that we obtain infinite relations between the area spectrum and the naive black hole degeneracy. Promoting the naive black hole degeneracy into its approximation, we obtain the full solution to the infinite relations.

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

Similar content being viewed by others

References

  1. Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D 7, 2333 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  2. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)

    Google Scholar 

  3. Kaul, R.K., Majumdar, P.: Logarithmic correction to the Bekenstein–Hawking entropy. Phys. Rev. Lett. 84, 5255 (2000). [gr-qc/0002040]

    Google Scholar 

  4. Ghosh, A., Mitra, P.: A bound on the log correction to the black hole area law. Phys. Rev. D 71, 027502 (2005)

    Google Scholar 

  5. Ghosh, A., Mitra, P.: An Improved lower bound on black hole entropy in the quantum geometry approach. Phys. Lett. B 616, 114 (2005)

    Google Scholar 

  6. Corichi, A., Diaz-Polo, J., Fernandez-Borja, E.: Quantum geometry and microscopic black hole entropy. Class. Quantum Gravity 24, 243 (2007) [gr-qc/0605014].

    Google Scholar 

  7. Sahlmann, H.: Entropy calculation for a toy black hole, Class. Quantum Gravity 25, 055004 (2008) [arXiv:0709.0076 [gr-qc]]

  8. Agullo, I., Fernando Barbero, G.J., Borja, E.F., Diaz-Polo, J., Villasenor, E.J.S.: The combinatorics of the SU(2) black hole entropy in loop quantum gravity. Phys. Rev. D 80, 084006 (2009) [arXiv:0906.4529 [gr-qc]]

  9. Meissner, K.A.: Black hole entropy in loop quantum gravity. Class. Quantum Gravity 21, 5245–5252 (2004). [gr-qc/0407052]

    Google Scholar 

  10. Rovelli, C.: Black hole entropy from loop quantum gravity. Phys. Rev. Lett. 77, 3288–3291 (1996). [gr-qc/9603063]

    Google Scholar 

  11. Rovelli, C., Smolin, L.: Discreteness of area and volume in quantum gravity. Nucl. Phys. B 442, 593–622 (1995). [gr-qc/9411005]

    Google Scholar 

  12. Frittelli, S., Lehner, L., Rovelli, C.: The Complete spectrum of the area from recoupling theory in loop quantum gravity. Class. Quantum Gravity 13, 2921–2932 (1996). [gr-qc/9608043]

    Google Scholar 

  13. Ashtekar, A., Lewandowski, J. Quantum theory of geometry. 1: area operators. Class. Quantum Gravity 14, A55–A82 (1997). [gr-qc/9602046]

  14. Domagala, M., Lewandowski J.: Black hole entropy from quantum geometry. Class. Quantum Gravity 21, 5233–5244 (2004). [gr-qc/0407051]

    Google Scholar 

  15. Tanaka, T., Tamaki, T.: Black hole entropy for the general area spectrum, arXiv:0808.4056 [hep-th]

  16. Ashtekar, A., Baez, J.C., Krasnov, K.: Quantum geometry of isolated horizons and black hole entropy. Adv. Theor. Math. Phys. 4, 1 (2000) [gr-qc/0005126]

    Google Scholar 

  17. Digital Library of Mathematical Functions. 2011–08-29. National Institute of Standards and Technology from http://dlmf.nist.gov chapter 26.11 integer partitions: compositions http://dlmf.nist.gov/26.11

  18. MacMahon, P.A.: Combinatory Analysis, vol. I, pp. 150–154. Cambridge University Press, (1915–1916)

  19. Livine, E.R., Terno, D.R.: Quantum black holes: entropy and entanglement on the horizon. Nucl. Phys. B 741, 131 (2006) [gr-qc/0508085]

    Google Scholar 

  20. Engle, J., Perez, A., Noui, K.: Black hole entropy and SU(2) Chern-Simons theory. Phys. Rev. Lett. 105, 031302 (2010) [arXiv:0905.3168 [gr-qc]]

    Google Scholar 

  21. Engle, J., Noui, K., Perez, A., Pranzetti, D.: Black hole entropy from an SU(2)-invariant formulation of type I isolated horizons. Phys. Rev. D 82, 044050 (2010) [arXiv:1006.0634 [gr-qc]]

    Google Scholar 

  22. Engle, J., Noui, K., Perez, A., Pranzetti, D.: The SU(2) black hole entropy revisited. JHEP 1105, 016 (2011) [arXiv:1103.2723 [gr-qc]]

  23. Yoon, Y.: Quantum corrections to Hawking radiation spectrum, to appear

Download references

Acknowledgments

We thank Jong-Hyun Baek for the helpful and crucial discussions. This work was supported by the National Research Foundation of Korea (NRF) grants 2012R1A1B3001085 and 2012R1A2A2A02046739.

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, Seoul National University, Seoul, 151-747, Korea

    Youngsub Yoon

Authors
  1. Youngsub Yoon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Youngsub Yoon.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yoon, Y. Approximation of the naive black hole degeneracy. Gen Relativ Gravit 45, 373–386 (2013). https://doi.org/10.1007/s10714-012-1475-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10714-012-1475-8

Keywords

Search

Navigation