Communications in Mathematical Physics

, Volume 87, Issue 4, pp 577–588 | Cite as

Thermodynamics of black holes in anti-de Sitter space

  • S. W. Hawking
  • Don N. Page


The Einstein equations with a negative cosmological constant admit black hole solutions which are asymptotic to anti-de Sitter space. Like black holes in asymptotically flat space, these solutions have thermodynamic properties including a characteristic temperature and an intrinsic entropy equal to one quarter of the area of the event horizon in Planck units. There are however some important differences from the asymptotically flat case. A black hole in anti-de Sitter space has a minimum temperature which occurs when its size is of the order of the characteristic radius of the anti-de Sitter space. For larger black holes the red-shifted temperature measured at infinity is greater. This means that such black holes have positive specific heat and can be in stable equilibrium with thermal radiation at a fixed temperature. It also implies that the canonical ensemble exists for asymptotically anti-de Sitter space, unlike the case for asymptotically flat space. One can also consider the microcanonical ensemble. One can avoid the problem that arises in asymptotically flat space of having to put the system in a box with unphysical perfectly reflecting walls because the gravitational potential of anti-de Sitter space acts as a box of finite volume.


Black Hole Cosmological Constant Event Horizon Thermal Radiation Einstein Equation 
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  1. 1.
    Hawking, S.W.: Gravitational radiation from colliding black holes. Phys. Rev. Lett.26, 1344–1346 (1971)Google Scholar
  2. 2.
    Christodoulou, D.: Reversible and irreversible transformations in black-hole physics. Phys. Rev. Lett.25, 1596–1597 (1970)Google Scholar
  3. 3.
    Bardeen, J.M., Carter, B., Hawking, S.W.: The four laws of black hole mechanics. Commun. Math. Phys.31, 161–170 (1973)Google Scholar
  4. 4.
    Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D7, 2333–2346 (1973)Google Scholar
  5. 5.
    Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys.43, 199–220 (1975)Google Scholar
  6. 6.
    Hartle, J.B., Hawking, S.W.: Path-integral derivation of black-hole radiance. Phys. Rev. D13, 2188–2203 (1976)Google Scholar
  7. 7.
    Gibbons, G.W., Perry, M.J.: Black holes and thermal green functions. Proc. R. Soc. London A358, 467–494 (1978)Google Scholar
  8. 8.
    Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D15, 2752–2756 (1977)Google Scholar
  9. 9.
    Gibbons, G.W., Hawking, S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D15, 2738–2751 (1977)Google Scholar
  10. 10.
    Nariai, H.: On some static solutions of Einstein's gravitational field equations in a spherically symmetric case. Sci. Rep. Tôhoku Univ.34, 160–167 (1950); on a new cosmological solution of Einstein's field equations of gravitation. Sci. Rep. Tôhoku Univ.35, 62–67 (1951)Google Scholar
  11. 11.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973Google Scholar
  12. 12.
    Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys.80, 381–402 (1981)Google Scholar
  13. 13.
    Abbott, L.F., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys. B195, 76–96 (1982)Google Scholar
  14. 14.
    Gibbons, G.W., Hawking, S.W., Horowitz, G.W., Perry, M.J.: Positive mass theorems for black holes. Commun. Math. Phys. (to appear)Google Scholar
  15. 15.
    Breitenlohner, P., Freedman, D.Z.: Positive energy in anti-de sitter backgrounds and gauged extended supergravity. MIT preprint (1982); stability in gauged extended supergravity. MIT preprint (1982)Google Scholar
  16. 16.
    Gibbons, G.W., Hull, C.M., Warner, N.P.: The stability of gauged supergravity. DAMTP preprint (1982)Google Scholar
  17. 17.
    Hawking, S.W.: Black holes and thermodynamics. Phys. Rev. D13, 191–197 (1976)Google Scholar
  18. 18.
    Gross, D.J., Perry, M.J., Yaffe, L.G.: Instability of flat space at finite temperature. Phys. Rev. D25, 330–355 (1982)Google Scholar
  19. 19.
    Gibbons, G.W., Hawking, S.W., Perry, M.J.: Path integrals and the indefiniteness of the gravitational action. Nucl. Phys. B138, 141–150 (1978)Google Scholar
  20. 20.
    Gibbons, G.W., Perry, M.J.: Quantizing gravitational instantons. Nucl. Phys. B146, 90–108 (1978)Google Scholar
  21. 21.
    Page, D.N.: Positive-action conjecture. Phys. Rev. D18, 2733–2738 (1978)Google Scholar
  22. 22.
    Perry, M.J.: Instabilities in gravity and supergravity. In: Superspace and supergravity: Proceedings of the Nuffield Workshop, Cambridge, June 16 – July 12, 1980. Hawking, S.W., Roček, M. (eds.). Cambridge: Cambridge University Press 1981Google Scholar
  23. 23.
    Hawking, S.W.: Euclidean quantum gravity. In: Recent developments in gravitation: Cargèse 1978. NATO Advanced Study Institutes Series, Series B: Physics, Vol. 44. Lévy, M., Deser, S. (eds.). New York: Plenum Press 1979Google Scholar
  24. 24.
    Avis, S.J., Isham, C.J., Storey, D.: Quantum field theory in anti-de sitter space-time. Phys. Rev. D18, 3565–3576 (1978)Google Scholar
  25. 25.
    Page, D.N.: Thermodynamic paradoxes. Physics Today30, 11–15 (1977)Google Scholar
  26. 26.
    Page, D.N.: Black hole formation in a box. Gen. Rel. Grav.13, 1117–1126 (1981)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • S. W. Hawking
    • 1
  • Don N. Page
    • 2
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeEngland
  2. 2.Department of PhysicsThe Pennsylvania State UniversityUniversity ParkUSA

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