Entropy of a Generic Null Surface from Its Associated Virasoro Algebra

  • Sumanta ChakrabortyEmail author
Part of the Springer Theses book series (Springer Theses)


Null surfaces act as one-way membranes, blocking information from those observers who do not cross them. These observers associate an entropy (and temperature) with the null surface. In this spirit the black hole entropy can be computed from the central charge of an appropriately defined, local, Virasoro algebra on the horizon. In this chapter, we demonstrate that one can extend these ideas to a general class of null surfaces, all of which possess a Virasoro algebra and a central charge, leading to an entropy density which is just (1/4). All the previously known results arise as special cases of this very general property of null surfaces and the result represented here provides the derivation of the entropy-area law in the most general context.


  1. 1.
    T. Padmanabhan, Gravitation: Foundations and Frontiers (Cambridge University Press, Cambridge, UK, 2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    J.D. Bekenstein, Black holes and entropy. Phys. Rev. D 7, 2333–2346 (1973)Google Scholar
  3. 3.
    J. D. Bekenstein, Generalized second law of thermodynamics in black hole physics. Phys. Rev. D 9, 3292–3300 (1974)Google Scholar
  4. 4.
    J.D. Bekenstein, Statistical black hole thermodynamics. Phys. Rev. D 12, 3077–3085 (1975)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Hawking, Particle creation by black holes. Commun. Math. Phys. 43, 199–220 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    P. Davies, Scalar particle production in Schwarzschild and Rindler metrics. J. Phys. A 8, 609–616 (1975)Google Scholar
  7. 7.
    W. Unruh, Notes on black hole evaporation. Phys. Rev. D 14, 870 (1976)ADSCrossRefGoogle Scholar
  8. 8.
    T. Padmanabhan, Thermodynamical aspects of gravity: new insights. Rept. Prog. Phys. 73, 046901 (2010). arXiv:0911.5004 [gr-qc]
  9. 9.
    T. Padmanabhan, General relativity from a thermodynamic perspective. Gen. Rel. Grav. 46, 1673 (2014). arXiv:1312.3253 [gr-qc]
  10. 10.
    B. Zwiebach, A first course in string theory, (Cambridge University Press, 2006),
  11. 11.
    J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories. Nucl. Phys. B 270, 186–204 (1986)Google Scholar
  12. 12.
    H.W.J. Bloete, J.L. Cardy, M.P. Nightingale, Conformal ivariance, the central charge, and universal finite size amplitudes at criticality. Phys. Rev. Lett. 56, 742–745 (1986)Google Scholar
  13. 13.
    S. Carlip, Entropy from conformal field theory at killing horizons. Class. Quant. Grav. 16, 3327–3348 (1999). arXiv:gr-qc/9906126 [gr-qc]
  14. 14.
    S. Carlip, Black hole thermodynamics. Int. J. Mod. Phys. D 23, 1430023 (2014). arXiv:1410.1486 [gr-qc]
  15. 15.
    O. Dreyer, A. Ghosh, J. Wisniewski, Black hole entropy calculations based on symmetries. Class. Quant. Grav. 18, 1929–1938 (2001). arXiv:hep-th/0101117 [hep-th]
  16. 16.
    S. Silva, Black hole entropy and thermodynamics from symmetries. Class. Quant. Grav. 19, 3947–3962 (2002). arXiv:hep-th/0204179 [hep-th]
  17. 17.
    S. Carlip, Symmetries, horizons, and black hole entropy. Gen. Rel. Grav. 39, 1519–1523 (2007). arXiv:0705.3024 [gr-qc]. [Int. J. Mod. Phys.D17,659(2008)]
  18. 18.
    O. Dreyer, A. Ghosh, A. Ghosh, Entropy from near-horizon geometries of Killing horizons. Phys. Rev. D 89(2), 024035 (2014). arXiv:1306.5063 [gr-qc]
  19. 19.
    G. Kang, J.-i. Koga, M.-I. Park, Near horizon conformal symmetry and black hole entropy in any dimension. Phys. Rev. D 70, 024005 (2004). arXiv:hep-th/0402113 [hep-th]
  20. 20.
    B.R. Majhi, T. Padmanabhan, Noether current, horizon virasoro algebra and entropy. Phys. Rev. D 85, 084040 (2012). arXiv:1111.1809 [gr-qc]
  21. 21.
    B.R. Majhi, T. Padmanabhan, Noether current from the surface term of gravitational action, virasoro algebra and horizon entropy. Phys. Rev. D 86, 101501 (2012). arXiv:1204.1422 [gr-qc]
  22. 22.
    B.R. Majhi, T. Padmanabhan, Thermality and heat content of horizons from infinitesimal coordinate transformations. arXiv:1302.1206 [gr-qc]
  23. 23.
    B.R. Majhi, Conformal transformation, near horizon symmetry, virasoro algebra and entropy. Phys. Rev. D 90(4), 044020 (2014). arXiv:1404.6930 [gr-qc]
  24. 24.
    B.R. Majhi, Thermodynamics of Sultana-Dyer black hole. JCAP 1405, 014 (2014). arXiv:1403.4058 [gr-qc]
  25. 25.
    S. Bhattacharya, A note on entropy of de Sitter black holes. Eur. Phys. J. C 76(3), 112 (2016). arXiv:1506.07809 [gr-qc]
  26. 26.
    S. Chakraborty, T. Padmanabhan, Thermodynamical interpretation of the geometrical variables associated with null surfaces. Phys. Rev. D 92(10), 104011 (2015). arXiv:1508.04060 [gr-qc]
  27. 27.
    K. Parattu, S. Chakraborty, B.R. Majhi, T. Padmanabhan, Null surfaces: counter-term for the action principle and the characterization of the gravitational degrees of freedom. arXiv:1501.01053 [gr-qc]

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsIndian Association for the Cultivation of ScienceJadavpur, KolkataIndia

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