Communications in Mathematical Physics

, Volume 98, Issue 3, pp 391–424 | Cite as

Asymptotically anti-de Sitter spaces

  • Marc Henneaux
  • Claudio Teitelboim
Article

Abstract

Asymptotically anti-de Sitter spaces are defined by boundary conditions on the gravitational field which obey the following criteria: (i) they are O(3, 2) invariant; (ii) they make the O(3, 2) surface integral charges finite; (iii) they include the Kerr-anti-de Sitter metric. An explicit expression of the O(3, 2) charges in terms of the canonical variables is given. These charges are shown to close in the Dirac brackets according to the anti-de Sitter algebra. The results are extended to the case ofN=1 supergravity. The coupling to gravity of a third-rank, completely antisymmetric, abelian gauge field is also considered. That coupling makes it possible to vary the cosmological constant and to compare the various anti-de Sitter spaces which are shown to have the same energy.

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References

  1. 1.
    Regge, T., Teitelboim, C.: Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. (N.Y.)88, 286 (1974)Google Scholar
  2. 2.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of spacetime. Cambridge, UK: Cambridge University Press 1973Google Scholar
  3. 3.
    Carter, B.: Hamilton-Jacobi and Schrödinger separable solutions of Einstein's equations. Commun. Math. Phys.10, 280 (1968); Demianski, M.: Acta Astron.23, 211 (1973)Google Scholar
  4. 4.
    Hawking, S.W.: The boundary conditions for gauged supergravity. Phys. Lett.126B, 175 (1983)Google Scholar
  5. 5.
    Abbott, L.F., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys. B195, 76 (1982)Google Scholar
  6. 6.
    Breitenlohner, P., Freedman, D.Z.: Stability in gauged extended supergravity. Ann. Phys. (N.Y.)144, 249 (1982)Google Scholar
  7. 7.
    Gibbons, G.W., Hull, C.M., Warner, N.P.: The stability of gauged supergravity. Nucl. Phys. B218, 173 (1983)Google Scholar
  8. 8.
    Teitelboim, C.: Surface integrals as symmetry generators in supergravity theory. Phys. Lett.69 B, 240 (1977); Deser, S., Teitelboim, C.: Supergravity has positive energy. Phys. Rev. Lett.39, 249 (1977)Google Scholar
  9. 9.
    Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys.80, 397 (1981)Google Scholar
  10. 10.
    Horowitz, G.T., Strominger, A.: Witten's expression for gravitational energy. Phys. Rev. D27, 2793 (1983)Google Scholar
  11. 11.
    Deser, S.: Positive classical gravitational energy from classical supergravity. Phys. Rev. D27, 2805 (1983)Google Scholar
  12. 12.
    Hull, C.M.: The positivity of gravitational energy and global supersymmetry. Commun. Math. Phys.90, 545 (1983)Google Scholar
  13. 13.
    Teitelboim, C.: Manifestly positive-energy expression in classical gravity: simplified derivation from supergravity. Phys. Rev. D29, 2763 (1984)Google Scholar
  14. 14.
    Aurilia, A., Nicolai, H., Townsend, P.K.: Hidden constants: Theθ parameter of QCD and the cosmological constant ofN=8 supergravity. Nucl. Phys. B176, 509 (1980)Google Scholar
  15. 15.
    Henneaux, M., Teitelboim, C.: Phys. Lett.142 B, 355 (1984)Google Scholar
  16. 16.
    Henneaux, M., Teitelboim, C.: Phys. Lett.143 B, 415 (1984)Google Scholar
  17. 17.
    Benguria, R., Cordero, P., Teitelboim, C.: Aspects of the Hamiltonian dynamics of interacting gravitational gauge and Higgs fields with applications to spherical symmetry. Nucl. Phys. B122, 61 (1977)Google Scholar
  18. 18.
    Ashtekar, A.: In: General relativity and gravitation: one hundred years after the birth of Albert Einstein. Held, A. et al.: New York: Plenum Press 1980Google Scholar
  19. 19.
    Penrose, R.: In: Relativity, groups, and topology. DeWitt, C., DeWitt, B. eds. New York: Gordon and Breach 1964Google Scholar
  20. 20.
    Avis, S.J., Isham, C.J., Storey, D.: Quantum field theory in anti-de Sitter spacetime. Phys. Rev. D18, 3565 (1978)Google Scholar
  21. 21.
    Dirac, P.A.M.: The theory of gravitation in Hamiltonian form. Proc. Roy. Soc. A246, 333 (1958)Google Scholar
  22. 22.
    Arnowitt, R., Deser, S., Misner, C.W.: In: Gravitation: an introduction to current research. Witten, L. ed. New York: Wiley 1962Google Scholar
  23. 23.
    Hanson, A., Regge, T., Teitelboim, C.: Constrained Hamiltonian systems. Acc. Naz. dei Lincei, Rome, 1976Google Scholar
  24. 24.
    Teitelboim, C.: How commutators of constraints reflect the spacetime structure. Ann. Phys. (N.Y.)79, 542 (1973). See also Teitelboim, C.: In: General relativity and gravitation: one hundred years after the birth of Albert Einstein, Vol. I. Held, A. ed. New York: Plenum Press 1980Google Scholar
  25. 25.
    Dirac, P.A.M.: Lectures on quantum mechanics. Belfer graduate school of science, Yeshiva University, New York, 1964Google Scholar
  26. 26.
    Deser, S., Zumino, B.: Broken supersymmetry and supergravity. Phys. Rev. Lett.38, 1433 (1977)Google Scholar
  27. 27.
    Townsend, P.: Cosmological constant in supergravity. Phys. Rev. D15, 2802 (1977)Google Scholar
  28. 28.
    Candelas, P., Raine, D.J.: From anti-de Sitter to Minkowski space via vacuum fluctuations. University of Teyas preprint (1983)Google Scholar
  29. 29.
    't Hooft, G.: Magnetic monopoles in unified gauge theories. Nucl. Phys. B79, 276 (1974)Google Scholar
  30. 29a.
    Polyakov, A.M.: Particle spectrum in quantum field theory. JETP Lett.20, 194 (1974)Google Scholar
  31. 30.
    Jackiw, R.: Acta Phys. Austriaca, Suppl.XXII, 383 (1980)Google Scholar
  32. 31.
    Henneaux, M.: Remarks on spacetime symmetries and non-abelian gauge fields. J. Math. Phys.23, 830 (1982)Google Scholar
  33. 32.
    Ashtekar, A., Magnon, A.: Asymptotically anti-de Sitter spacetimes. Classical and Quantum Gravity Lett.1, L 39 (1984)Google Scholar
  34. 33.
    Kosmann, Y.: Ann. Mat. Pur. Appl.XCI, 319 (1972)Google Scholar
  35. 34.
    Henneaux, M.: Gen. Rel. Grav.12, 137 (1980)Google Scholar
  36. 35.
    Cordero, P., Teitelboim, C.: Remarks on supersymmetric black holes. Phys. Lett.78B, 80 (1978)Google Scholar
  37. 36.
    Eisenhart, L.P.: Riemannian geometry. Princeton, NJ: Princeton University Press 1964Google Scholar
  38. 37.
    Brown, J.D., Henneaux, M.: In preparationGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Marc Henneaux
    • 1
  • Claudio Teitelboim
    • 1
    • 2
  1. 1.Center for Theoretical PhysicsThe University of Texas at AustinAustinUSA
  2. 2.Centro de Estudios Cientificos de Santiago, CasillaSantiago 9Chile

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