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Asymptotically anti-de Sitter spacetimes and conserved quantities in higher curvature gravitational theories

Naoya Okuyama1,* and Jun-ichirou Koga2,†

  • 1Department of Physics, Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan
  • 2Advanced Research Institute for Science and Engineering, Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan

  • *Electronic address: okuyama@gravity.phys.waseda.ac.jp
  • †Electronic address: koga@gravity.phys.waseda.ac.jp
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Phys. Rev. D 71, 084009 – Published 8 April, 2005

DOI: https://doi.org/10.1103/PhysRevD.71.084009

Abstract

We consider n-dimensional asymptotically anti-de Sitter spacetimes in higher curvature gravitational theories with n≥4, by employing the conformal completion technique. We first argue that a condition on the Ricci tensor should be supplemented to define an asymptotically anti-de Sitter spacetime in higher curvature gravitational theories and propose an alternative definition of an asymptotically anti-de Sitter spacetime. Based on that definition, we then derive a conservation law of the gravitational field and construct conserved quantities in two classes of higher curvature gravitational theories. We also show that these conserved quantities satisfy a balance equation in the same sense as in Einstein gravity and that they reproduce the results derived elsewhere. These conserved quantities are shown to be expressed as an integral of the electric part of the Weyl tensor alone and hence they vanish identically in the pure anti-de Sitter spacetime as in the case of Einstein gravity.

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References (19)

  1. J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998); E. Witten, 2, 253 (1998); S. Gubser, I. Klebanov, and A. Polyakov, Phys. Lett. B 428, 105 (1998); O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Phys. Rep. 323, 183 (2000).
  2. K. Akama, in Proceedings of the International Symposium on Gauge Theory and Gravitation, Tezukayama University, Nara, Japan, 1982 (Springer-Verlag, Berlin, 1983); Lect. Notes Phys. 176, 267 (1982); V. A. Rubakov and M. E. Shaposhinikov, Phys. Lett. 125B, 136 (1983); N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429, 263 (1998); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, 436, 257 (1998); L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); 83, 4690 (1999).
  3. P. Kraus, J. High Energy Phys. 12 (1999) 011; C. Barceló and M. Visser, Phys. Lett. B 482, 183 (2000); N. Okuyama and K. Maeda, Phys. Rev. D 70, 064030 (2004); E. Gravanis and S. Willison, Phys. Lett. B 562, 118 (2003).
  4. G. W. Gibbons, M. J. Perry, and C. N. Pope, hep-th/0408217.
  5. N. Deruelle and J. Katz, Classical Quantum Gravity 22, 421 (2005).
  6. C. G. Callan, D. Friedan, E. J. Martinec, and M. J. Perry, Nucl. Phys. B262, 593 (1985); C. G. Callan, I. R. Klebanov, and M. J. Perry, B278, 78 (1986); D. J. Gross and J. H. Sloan, B291, 41 (1987).
  7. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England, 1982).
  8. D. G. Boulware, and S. Deser, Phys. Rev. Lett. 55, 2656 (1985); J. T. Wheeler, Nucl. Phys. B268, 737 (1986); B273, 732 (1986); D. L. Wiltshire, Phys. Lett. 169B, 36 (1986); R. C. Myers and J. Z. Simon, Phys. Rev. D 38, 2434 (1988); R. G. Cai and K. S. Soh, 59, 044013 (1999); J. Crisostomo, R. Troncoso, and J. Zanelli, 62, 084013 (2000); R. G. Cai, 65, 084014 (2002).
  9. N. Deruelle, J. Katz, and S. Ogushi, Classical Quantum Gravity 21, 1971 (2004).
  10. Y. M. Cho and I. P. Neupane, Phys. Rev. D 66, 024044 (2002); A. Padilla, Classical Quantum Gravity 20, 3129 (2003).
  11. N. Deruelle and Y. Morisawa, Classical Quantum Gravity 22, 933 (2005).
  12. A. Ashtekar and A. Magnon, Classical Quantum Gravity 1, L39 (1984).
  13. A. Ashtekar and S. Das, Classical Quantum Gravity 17, L17 (2000).

  14. Although this condition is described in terms of the Weyl tensor C^abcd associated with the unphysical metric g^ab in Ref. [13], we will work with Cabcd as we described in the introduction.

  15. C. Lanczos, Ann. Math. 39, 842 (1938); D. Lovelock, J. Math. Phys. (N.Y.) 12, 498 (1971).
  16. G. W. Gibbons, H. Lü, D. N. Page, and C. N. Pope, J. Geom. Phys. 53, 49 (2005).

  17. Note, however, that it does not mean that the values of the conserved quantities in Einstein-Gauss-Bonnet gravity are equal to those in Einstein gravity multiplied by the factor Ξ′, since one needs to rescale the cosmological constant when one passes from Einstein gravity to Einstein-Gauss-Bonnet gravity.

  18. J. Lee and R. M. Wald, J. Math. Phys. (N.Y.) 31, 725 (1990); V. Iyer and R. M. Wald, Phys. Rev. D 50, 846 (1994); 52, 4430 (1995); R. M. Wald and A. Zoupas, 61, 084027 (2000).
  19. L. F. Abbott and S. Deser, Nucl. Phys. B195, 76 (1982); S. Deser and B. Tekin, Phys. Rev. Lett. 89, 101101 (2002); Phys. Rev. D 67, 084009 (2003).

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Phys. Rev. D 71, 084009– Published 8 April, 2005
  • Received 20 January 2005

© 2005 American Physical Society

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