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Thermodynamics of black holes in anti-de Sitter space

Abstract

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.

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References

  1. Hawking, S.W.: Gravitational radiation from colliding black holes. Phys. Rev. Lett.26, 1344–1346 (1971)

    Google Scholar 

  2. Christodoulou, D.: Reversible and irreversible transformations in black-hole physics. Phys. Rev. Lett.25, 1596–1597 (1970)

    Google Scholar 

  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. Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D7, 2333–2346 (1973)

    Google Scholar 

  5. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys.43, 199–220 (1975)

    Google Scholar 

  6. Hartle, J.B., Hawking, S.W.: Path-integral derivation of black-hole radiance. Phys. Rev. D13, 2188–2203 (1976)

    Google Scholar 

  7. Gibbons, G.W., Perry, M.J.: Black holes and thermal green functions. Proc. R. Soc. London A358, 467–494 (1978)

    Google Scholar 

  8. Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D15, 2752–2756 (1977)

    Google Scholar 

  9. Gibbons, G.W., Hawking, S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D15, 2738–2751 (1977)

    Google Scholar 

  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. Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973

    Google Scholar 

  12. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys.80, 381–402 (1981)

    Google Scholar 

  13. Abbott, L.F., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys. B195, 76–96 (1982)

    Google Scholar 

  14. Gibbons, G.W., Hawking, S.W., Horowitz, G.W., Perry, M.J.: Positive mass theorems for black holes. Commun. Math. Phys. (to appear)

  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)

  16. Gibbons, G.W., Hull, C.M., Warner, N.P.: The stability of gauged supergravity. DAMTP preprint (1982)

  17. Hawking, S.W.: Black holes and thermodynamics. Phys. Rev. D13, 191–197 (1976)

    Google Scholar 

  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. 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. Gibbons, G.W., Perry, M.J.: Quantizing gravitational instantons. Nucl. Phys. B146, 90–108 (1978)

    Google Scholar 

  21. Page, D.N.: Positive-action conjecture. Phys. Rev. D18, 2733–2738 (1978)

    Google Scholar 

  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 1981

    Google Scholar 

  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 1979

    Google Scholar 

  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. Page, D.N.: Thermodynamic paradoxes. Physics Today30, 11–15 (1977)

    Google Scholar 

  26. Page, D.N.: Black hole formation in a box. Gen. Rel. Grav.13, 1117–1126 (1981)

    Google Scholar 

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Communicated by A. Jaffe

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Hawking, S.W., Page, D.N. Thermodynamics of black holes in anti-de Sitter space. Commun.Math. Phys. 87, 577–588 (1983). https://doi.org/10.1007/BF01208266

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  • DOI: https://doi.org/10.1007/BF01208266

Keywords

  • Black Hole
  • Cosmological Constant
  • Event Horizon
  • Thermal Radiation
  • Einstein Equation