Abstract
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.
Similar content being viewed by others
REFERENCES
D. Christodoulou, “Reversible and irreversible transformations in black-hole physics”, Phys. Rev. Lett. 25, 1596(1970).
R. Penrose and R. M. Floyd, “Extraction of rotational energy from a black hole”, Nature (Physical Science) 229, 177(1971).
S. W. Hawking, “Gravitational radiation from colliding black holes”, Phys. Rev. Lett. 26, 1344(1971).
J. D. Bekenstein, “Black holes and entropy”, Phys. Rev. D 7, 2333(1973).
J. D. Bekenstein, “Generalized second law of thermodynamics in black-hole physics”, Phys. Rev. D 9, 3292(1974).
S. W. Hawking, “Particle creation by black holes”, Commun. Math. Phys. 43, 199(1975).
J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics”, Commun. Math. Phys. 31, 161(1973).
J. D. Bekenstein, “Statistical black-hole thermodynamics”, Phys. Rev. D 12, 3077(1975).
S. W. Hawking, “Black holes and thermodynamics”, Phys. Rev. D 13, 191(1976).
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).
W. G. Unruh and R. M. Wald, “Acceleration radiation and the generalized second law of thermodynamics”, Phys. Rev. D 25, 942(1982)
V. P. Frolov and D. N. Page, “Proof of the generalized second law for quasistationary semiclassical black holes”, Phys. Rev. Lett. 71, 3902(1993).
G. W. Gibbons and S. W. Hawking, “Cosmological event horizons, thermodynamics, and particle creation,” Phys. Rev. D 15, 2738(1977).
G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantum gravity”, Phys. Rev. D 15, 2752(1977).
J. D. Bekenstein, “Extraction of energy and charge from a black hole”, Phys. Rev. D 7, 949(1973).
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).
T. Jacobson, “Thermodynamics of spacetime: The Einstein equation of state”, Phys. Rev. Lett. 75, 1260(1995).
P. Martinetti and C. Rovelli, “Diamonds's temperature: Unruh effect for bounded trajectories and thermal time hypothesis”, arXiv:gr-qc/0212074.
I. Racz and R. M. Wald, “Extensions of spacetimes with Killing horizons”, Class. Quantum Grav. 9, 2643(1992).
B. Carter, “Black hole equilibrium states”, in Black Holes, C. DeWitt and B. S. Dewitt, eds. (Gordon & Breach, New York, 1973).
I. Racz and R. M. Wald, “Global extensions of spacetimes describing asymptotic final states of black holes”, Class. Quantum Grav. 13, 2643(1996).
R. Penrose, “Gravitational collapse”, in Graviational Radiation and Gravitational Collapse, C. DeWitt-Morette, ed. (Reidel, Dordrecht, 1974)
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, 1973).
R. M. Wald, General Relativity (University of Chicago Press, 1984).
P. T. Chruśsciel, E. Delay, G. J. Galloway, and R. Howard, “Regularity of horizons and the area theorem”, Annales Henri Poincaré 2, 109(2001).
P. C. W. Davies, “Cosmological horizons and the generalized second law of thermodynamics”, Class. Quant. Grav. 4, L225(1987)
S. A. Hayward, T. Shiromizu, and K. Nakao, “A cosmological constant limits the size of black holes”, Phys. Rev. D 49, 5080(1994).
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).
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).
R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994).
S. W. Hawking and J. B. Hartle, “Energy and angular momentum flow into a black hole”, Comm. Math. Phys. 27, 283(1972).
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).
S. W. Hawking, G. T. Horowitz, and S. F. Ross, “Entropy, area, and black hole pairs”, Phys. Rev. D 51, 4302(1995).
J. Pfautsch, cited in P. C. W. Davies, “Mining the universe,” Phys. Rev. D 30, 737(1984)
R. Price, K. Thorne, and D. A. MacDonald, eds., Black Holes: The Membrane Paradigm (Yale University Press, 1986), Sec. VII.E.1.
P. Candelas and D. W. Sciama, “Irreversible thermodynamics of black holes”, Phys. Rev. Lett. 38, 1372(1977).
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).
D. Garfinkle, S. B. Giddings, and A. Strominger, “Entropy in black hole pair production”, Phys. Rev. D 49, 958(1994).
S. Massar and R. Parentani, “Schwinger mechanism, Unruh effect and production of accelerated black holes”, Phys. Rev. D 55, 3603(1997).
R. B. Mann and S. F. Ross, “Cosmological production of charged black hole pairs”, Phys. Rev. D 52, 2254(1995).
S. W. Hawking and G. T. Horowitz, “The gravitational Hamiltonian, action, entropy and surface terms”, Class. Quant. Grav. 13, 1487(1996).
S. Massar and R. Parentani, “How the change in horizon area drives black hole evaporation”, Nucl. Phys. B 575, 333(2000).
J. B. Hartle and S. W. Hawking, “Path-integral derivation of black-hole radiance”, Phys. Rev. 13, 2188(1976).
W. G. Unruh, “Notes on black hole evaporation”, Phys. Rev. D 14, 870(1976).
R. Parentani, “The recoils of the accelerated detector and the decoherence of its fluxes”, Nucl. Phys. B 454, 227(1995).
S. Carlip and C. Teitelboim, “The off-shell black hole”, Class. Quant. Grav. 12, 1699(1995).
P. Kraus and F. Wilczek, “Self-interaction correction to black hole radiance”, Nucl. Phys. B 433, 403(1995).
E. Keski-Vakkuri and P. Kraus, “Microcanonical D-branes and back reaction”, Nucl. Phys. B 491, 249(1997).
R. Parentani, “Time-dependent perturbation theory in quantum cosmology”, Nucl. Phys. B 492, 501(1997).
R. Brout et al., “A primer for black hole quantum physics”, Phys. Rep. 260, 330(1995).
S. Carlip, “Entropy from conformal field theory at Killing horizons”, Class. Quant. Grav. 16, 3327(1999)
C. Rovelli, “Black hole entropy from loop quantum gravity,” Phys. Rev. Lett. 77, 3288(1996).
G. T. Horowitz, “Quantum states of black holes”, in Black Holes and Relativistic Stars, R. M. Wald, ed. (University of Chicago Press, 1998).
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).
S. Hawking, J. Maldacena, and A. Strominger, “DeSitter entropy, quantum entanglement and AdS/CFT”, JHEP 0105, 001(2001).
S. R. Das and A. Zelnikov, “Unruh radiation, holography and boundary cosmology”, Phys. Rev. D 64, 104001(2001).
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).
R. Bousso, “The holographic principle”, Rev. Mod. Phys. 74, 825(2002).
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).
R. D. Sorkin, “Toward a proof of entropy increase in the presence of quantum black holes”, Phys. Rev. Lett. 56, 1885(1986)
L. Susskind and J. Uglum, “Black hole entropy in canonical quantum gravity and super-string theory,” Phys. Rev. D 50, 2700(1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jacobson, T., Parentani, R. Horizon Entropy. Foundations of Physics 33, 323–348 (2003). https://doi.org/10.1023/A:1023785123428
Issue Date:
DOI: https://doi.org/10.1023/A:1023785123428