Abstract
We prove two theorems, announced in [6], for static spacetimes that solve Einstein's equation with negative cosmological constant. The first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity, analogous to the structure theorems of Cheeger and Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with Ricci-flat conformal boundary, the convexity condition is associated with negative mass. The second theorem is a uniqueness theorem for the negative mass AdS soliton spacetime. This result lends support to the new positive mass conjecture due to Horowitz and Myers which states that the unique lowest mass solution which asymptotes to the AdS soliton is the soliton itself. This conjecture was motivated by a nonsupersymmetric version of the AdS/CFT correspondence. Our results add to the growing body of rigorous mathematical results inspired by the AdS/CFT correspondence conjecture. Our techniques exploit a special geometric feature which the universal cover of the soliton spacetime shares with familiar ``ground state'' spacetimes such as Minkowski spacetime, namely, the presence of a null line, or complete achronal null geodesic, and the totally geodesic null hypersurface that it determines. En route, we provide an analysis of the boundary data at conformal infinity for the Lorentzian signature static Einstein equations, in the spirit of the Fefferman-Graham analysis for the Riemannian signature case. This leads us to generalize to arbitrary dimension a mass definition for static asymptotically AdS spacetimes given by Chruściel and Simon. We prove equivalence of this mass definition with those of Ashtekar-Magnon and Hawking-Horowitz.
Similar content being viewed by others
References
Abbott, L., Deser, S.: Nucl. Phys. B 195, 2752 (1982)
Anderson, M.T.: Adv. Math. To appear [math.DG/0104171]
Anderson, M.T., Chruściel, P.T., Delay, E.: J. High Energy Phys. 10, 063 (2002) [gr-qc/0211006]
Ashtekar, A., Magnon, A.: Classical Quantum Gravity 1, L39 (1984)
Boucher, W., Gibbons, G.W., Horowitz, G.T.: Phys. Rev. D 30, 2447 (1984)
Cadeau, C., Woolgar, E.: Classical Quantum Gravity 8, 527 (2001) [gr-qc/0011029]
Cheeger, J., Gromoll, D.: J. Diff. Geom. 6, 119 (1971); Ann. Math. 96, 413 (1972)
Chruściel, P.T., Herzlich, M.: Preprint (2001) [math.DG/0110035]
Chruściel, P.T., Simon, W.: J. Math. Phys. 42, 1779 (2001) [gr-qc/0004032]
Constable, N.R., Myers, R.C.: J. High Energy Phys. 9910, 037 (1999) [hep-th/9908175]
de Haro, S., Skenderis, K., Solodukhin, S.N.: Commun. Math. Phys. 217, 595 (2001)~[hep-th/ 0002230]
Dijkgraaf, R., Maldacena, J., Moore, G., Verlinde, E.: Preprint (2000) [hep-th/0005003]
Fefferman, C., Graham, C.R.: Astérisque, hors série, p. 95 (1985); Graham, C.R.: Proc. 19th Winter School in Geometry and Physics, Srni, Czech Rep., Jan. 1999, [math.DG/9909042]
Galloway, G.J.: Ann. Henri Poincaré 1, 543 (2000) [math.DG/9909158]
Galloway, G.J., Schleich, K., Witt, D.M., Woolgar, E.: Phys. Rev. D 60, 104039 (1999) [gr-qc/ 9902061]
Galloway, G.J., Surya, S., Woolgar, E.: Phys. Rev. Lett. 88, 101102 (2002) [hep-th/0108170]
Galloway, G.J., Surya, S., Woolgar, E.: Class. Quant. Grav. 20, 1635 (2003) [gr-qc/0212079]
Gao, S., Wald, R.M.: Classical Quantum Gravity 17, 4999 (2000) [gr-qc/0007021]
Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Phys. Lett. B428, 105 (1998) [hep-th/9802109]
Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge: Cambridge University Press, 1973
Hawking, S.W., Horowitz, G.T.: Classical Quantum Gravity 13, 1487 (1996)
Hawking S.W., Page, D.N.: Commun. Math. Phys. 87, 577 (1983)
Henneaux, M., Teitelboim, C.: Commun. Math. Phys. 98, 391 (1985)
Henningson, M., Skenderis, K.: J. High Energy Phys. 9807, 023 (1998) [hep-th/9806087]
Horowitz, G.T., Myers, R.C.: Phys. Rev. D 59, 026005 (1999) [hep-th/9808079]
Kiem, Y., Park, D.: Phys. Rev. D 59, 044010 (1999) [hep-th/9809174]
Klebanov, I.R.: In: Quantum aspects of gauge theories, supersymmetry, and unification, Paris, 1–7 Sept 1999, available from J. High Energy Phys. Conference Archive PRHEP-tmr/99/026
Lemos, J.P.S.: Phys. Lett. B 352, 46 (1995)
Lichnerowicz, A.: C. R. Acad. Sci. 222, 432 (1946)
Maldacena, J.: Adv. Theor. Math. Phys. 2, 231 (1998) [hep-th/9711200]
Mann, R.B.: In: Internal Structure of Black Holes and Spacetime Singularities, Burko, L., and Ori, A., (eds.), Ann. Israeli Phys. Soc. 13, 311 (1998) [gr-qc/9709039]
Page, D.N.: Preprint (2001) [hep-th/0205001]
Penrose, R., Sorkin, R.D., Woolgar, E.: Preprint [gr-qc/9301015]
Petersen, P.: Riemannian geometry. Graduate Texts in Mathematics, New York: Springer-Verlag, 1998
Schoen, R., Yau, S.-T.: Commun. Math. Phys. 79, 231 (1981)
Susskind, L.: J. Math. Phys. 36, 6377 (1995) [hep-th/9409089]
Surya, S., Schleich, K., Witt, D.M.: Phys. Rev. Lett. 86, 5231 (2001) [hep-th/0101134]
't Hooft, G.: In: Salamfest, Ali, A., Ellis, J., and Randjbar-Daemi, S., (eds.), Singapore: World Scientific, 1994 [gr-qc/9310026]
Witten, E.: Commun. Math. Phys. 80, 381 (1981)
Witten, E.: Adv. Theor. Math. Phys. 2, 253 (1998) [hep-th/9802150]
Woolgar, E.: Classical Quantum Gravity 11, 1881 (1994) [gr-qc/9404019]
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Nicolai
Rights and permissions
About this article
Cite this article
Galloway, G., Surya, S. & Woolgar, E. On the Geometry and Mass of Static, Asymptotically AdS Spacetimes, and the Uniqueness of the AdS Soliton. Commun. Math. Phys. 241, 1–25 (2003). https://doi.org/10.1007/s00220-003-0912-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-003-0912-7