Foundations of Physics

, Volume 33, Issue 2, pp 323–348 | Cite as

Horizon Entropy

  • Ted Jacobson
  • Renaud Parentani


Although the laws of thermodynamics are well established for black hole horizons, much less has been said in the literature to support the extension of these laws to more general settings such as an asymptotic de Sitter horizon or a Rindler horizon (the event horizon of an asymptotic uniformly accelerated observer). In the present paper we review the results that have been previously established and argue that the laws of black hole thermodynamics, as well as their underlying statistical mechanical content, extend quite generally to what we call here “causal horizons.” The root of this generalization is the local notion of horizon entropy density.

black hole thermodynamics entropy horizon 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Christodoulou, “Reversible and irreversible transformations in black-hole physics”, Phys. Rev. Lett. 25, 1596(1970).Google Scholar
  2. 2.
    R. Penrose and R. M. Floyd, “Extraction of rotational energy from a black hole”, Nature (Physical Science) 229, 177(1971).Google Scholar
  3. 3.
    S. W. Hawking, “Gravitational radiation from colliding black holes”, Phys. Rev. Lett. 26, 1344(1971).Google Scholar
  4. 4.
    J. D. Bekenstein, “Black holes and entropy”, Phys. Rev. D 7, 2333(1973).Google Scholar
  5. 5.
    J. D. Bekenstein, “Generalized second law of thermodynamics in black-hole physics”, Phys. Rev. D 9, 3292(1974).Google Scholar
  6. 6.
    S. W. Hawking, “Particle creation by black holes”, Commun. Math. Phys. 43, 199(1975).Google Scholar
  7. 7.
    J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics”, Commun. Math. Phys. 31, 161(1973).Google Scholar
  8. 8.
    J. D. Bekenstein, “Statistical black-hole thermodynamics”, Phys. Rev. D 12, 3077(1975).Google Scholar
  9. 9.
    S. W. Hawking, “Black holes and thermodynamics”, Phys. Rev. D 13, 191(1976).Google Scholar
  10. 10.
    W. H. Zurek and K. S. Thorne, “Statistical mechanical origin of the entropy of a rotating, charged black hole” Phys. Rev. Lett. 54, 2171(1985).Google Scholar
  11. 11.
    W. G. Unruh and R. M. Wald, “Acceleration radiation and the generalized second law of thermodynamics”, Phys. Rev. D 25, 942(1982)Google Scholar
  12. 12.
    V. P. Frolov and D. N. Page, “Proof of the generalized second law for quasistationary semiclassical black holes”, Phys. Rev. Lett. 71, 3902(1993).Google Scholar
  13. 13.
    G. W. Gibbons and S. W. Hawking, “Cosmological event horizons, thermodynamics, and particle creation,” Phys. Rev. D 15, 2738(1977).Google Scholar
  14. 14.
    G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantum gravity”, Phys. Rev. D 15, 2752(1977).Google Scholar
  15. 15.
    J. D. Bekenstein, “Extraction of energy and charge from a black hole”, Phys. Rev. D 7, 949(1973).Google Scholar
  16. 16.
    J. D. Bekenstein, “Do we understand black hole entropy?” in Proceedings of the Seventh Marcel Grossmann Meeting, R. T. Jantzen, G. M. Keiser, and R. Ruffini, eds. (World Scientific, Singapore, 1996).Google Scholar
  17. 17.
    T. Jacobson, “Thermodynamics of spacetime: The Einstein equation of state”, Phys. Rev. Lett. 75, 1260(1995).Google Scholar
  18. 18.
    P. Martinetti and C. Rovelli, “Diamonds's temperature: Unruh effect for bounded trajectories and thermal time hypothesis”, arXiv:gr-qc/0212074.Google Scholar
  19. 19.
    I. Racz and R. M. Wald, “Extensions of spacetimes with Killing horizons”, Class. Quantum Grav. 9, 2643(1992).Google Scholar
  20. 20.
    B. Carter, “Black hole equilibrium states”, in Black Holes, C. DeWitt and B. S. Dewitt, eds. (Gordon & Breach, New York, 1973).Google Scholar
  21. 21.
    I. Racz and R. M. Wald, “Global extensions of spacetimes describing asymptotic final states of black holes”, Class. Quantum Grav. 13, 2643(1996).Google Scholar
  22. 22.
    R. Penrose, “Gravitational collapse”, in Graviational Radiation and Gravitational Collapse, C. DeWitt-Morette, ed. (Reidel, Dordrecht, 1974)Google Scholar
  23. 23.
    S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, 1973).Google Scholar
  24. 24.
    R. M. Wald, General Relativity (University of Chicago Press, 1984).Google Scholar
  25. 25.
    P. T. Chruśsciel, E. Delay, G. J. Galloway, and R. Howard, “Regularity of horizons and the area theorem”, Annales Henri Poincaré 2, 109(2001).Google Scholar
  26. 26.
    P. C. W. Davies, “Cosmological horizons and the generalized second law of thermodynamics”, Class. Quant. Grav. 4, L225(1987)Google Scholar
  27. 27.
    S. A. Hayward, T. Shiromizu, and K. Nakao, “A cosmological constant limits the size of black holes”, Phys. Rev. D 49, 5080(1994).Google Scholar
  28. 28.
    T. Shiromizu, K. Nakao, H. Kodama, and K. Maeda, “Can large black holes collide in de Sitter space-time? An inflationary scenario of an inhomogeneous universe”, Phys. Rev. D 47, 3099(1993).Google Scholar
  29. 29.
    K. Maeda, T. Koike, M. Narita, and A. Ishibashi, “Upper bound for entropy in asymptotically de Sitter space-time”, Phys. Rev. D 57, 3503(1998).Google Scholar
  30. 30.
    R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994).Google Scholar
  31. 31.
    S. W. Hawking and J. B. Hartle, “Energy and angular momentum flow into a black hole”, Comm. Math. Phys. 27, 283(1972).Google Scholar
  32. 32.
    B. Carter, “The general theory of mechanical, electromagnetic and thermodynamic properties of black holes”, in General Relativity: An Einstein Centenary Survey, S. W. Hawking and W. Israel, eds. (Cambridge University Press, 1979).Google Scholar
  33. 33.
    S. W. Hawking, G. T. Horowitz, and S. F. Ross, “Entropy, area, and black hole pairs”, Phys. Rev. D 51, 4302(1995).Google Scholar
  34. 34.
    J. Pfautsch, cited in P. C. W. Davies, “Mining the universe,” Phys. Rev. D 30, 737(1984)Google Scholar
  35. 35.
    R. Price, K. Thorne, and D. A. MacDonald, eds., Black Holes: The Membrane Paradigm (Yale University Press, 1986), Sec. VII.E.1.Google Scholar
  36. 36.
    P. Candelas and D. W. Sciama, “Irreversible thermodynamics of black holes”, Phys. Rev. Lett. 38, 1372(1977).Google Scholar
  37. 37.
    D. W. Sciama, “Black holes and fluctuations of quantum particles: An Einstein synthesis,” in Relativity, Quanta, and Cosmology in the Development of the Scientific Thought of Albert Einstein, Vol. II, M. Pantaleo, dir., and F. De Finis, ed. (Giunti Barbèra, Firenze, 1979; reprinted by Johnson Reprint Corporation, Harcourt Brace Jovanovich subsidiary, 1979).Google Scholar
  38. 38.
    D. Garfinkle, S. B. Giddings, and A. Strominger, “Entropy in black hole pair production”, Phys. Rev. D 49, 958(1994).Google Scholar
  39. 39.
    S. Massar and R. Parentani, “Schwinger mechanism, Unruh effect and production of accelerated black holes”, Phys. Rev. D 55, 3603(1997).Google Scholar
  40. 40.
    R. B. Mann and S. F. Ross, “Cosmological production of charged black hole pairs”, Phys. Rev. D 52, 2254(1995).Google Scholar
  41. 41.
    S. W. Hawking and G. T. Horowitz, “The gravitational Hamiltonian, action, entropy and surface terms”, Class. Quant. Grav. 13, 1487(1996).Google Scholar
  42. 42.
    S. Massar and R. Parentani, “How the change in horizon area drives black hole evaporation”, Nucl. Phys. B 575, 333(2000).Google Scholar
  43. 43.
    J. B. Hartle and S. W. Hawking, “Path-integral derivation of black-hole radiance”, Phys. Rev. 13, 2188(1976).Google Scholar
  44. 44.
    W. G. Unruh, “Notes on black hole evaporation”, Phys. Rev. D 14, 870(1976).Google Scholar
  45. 45.
    R. Parentani, “The recoils of the accelerated detector and the decoherence of its fluxes”, Nucl. Phys. B 454, 227(1995).Google Scholar
  46. 46.
    S. Carlip and C. Teitelboim, “The off-shell black hole”, Class. Quant. Grav. 12, 1699(1995).Google Scholar
  47. 47.
    P. Kraus and F. Wilczek, “Self-interaction correction to black hole radiance”, Nucl. Phys. B 433, 403(1995).Google Scholar
  48. 48.
    E. Keski-Vakkuri and P. Kraus, “Microcanonical D-branes and back reaction”, Nucl. Phys. B 491, 249(1997).Google Scholar
  49. 49.
    R. Parentani, “Time-dependent perturbation theory in quantum cosmology”, Nucl. Phys. B 492, 501(1997).Google Scholar
  50. 50.
    R. Brout et al., “A primer for black hole quantum physics”, Phys. Rep. 260, 330(1995).Google Scholar
  51. 51.
    S. Carlip, “Entropy from conformal field theory at Killing horizons”, Class. Quant. Grav. 16, 3327(1999)Google Scholar
  52. 52.
    C. Rovelli, “Black hole entropy from loop quantum gravity,” Phys. Rev. Lett. 77, 3288(1996).Google Scholar
  53. 53.
    G. T. Horowitz, “Quantum states of black holes”, in Black Holes and Relativistic Stars, R. M. Wald, ed. (University of Chicago Press, 1998).Google Scholar
  54. 54.
    O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, “Large N field theories, string theory and gravity”, Phys. Rept. 323, 183(2000).Google Scholar
  55. 55.
    S. Hawking, J. Maldacena, and A. Strominger, “DeSitter entropy, quantum entanglement and AdS/CFT”, JHEP 0105, 001(2001).Google Scholar
  56. 56.
    S. R. Das and A. Zelnikov, “Unruh radiation, holography and boundary cosmology”, Phys. Rev. D 64, 104001(2001).Google Scholar
  57. 57.
    T. Jacobson, “On the nature of black hole entropy”, in General Relativity and Relativistic Astrophysics: Eighth Canadian Conference, AIP Conference Proceedings 493, C. Burgess and R. C. Myers, eds. (AIP Press, 1999).Google Scholar
  58. 58.
    R. Bousso, “The holographic principle”, Rev. Mod. Phys. 74, 825(2002).Google Scholar
  59. 59.
    L. Bombelli, R. K. Koul, J. H. Lee, and R. D. Sorkin, “A quantum source of entropy for black holes”, Phys. Rev. D 34, 373(1986).Google Scholar
  60. 60.
    R. D. Sorkin, “Toward a proof of entropy increase in the presence of quantum black holes”, Phys. Rev. Lett. 56, 1885(1986)Google Scholar
  61. 61.
    L. Susskind and J. Uglum, “Black hole entropy in canonical quantum gravity and super-string theory,” Phys. Rev. D 50, 2700(1994).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Ted Jacobson
    • 1
  • Renaud Parentani
    • 2
  1. 1.Department of PhysicsUniversity of MarylandCollege Park
  2. 2.Laboratoire de Mathématiques et Physique ThéoriqueUniversité de ToursToursFrance

Personalised recommendations