Communications in Mathematical Physics

, Volume 321, Issue 1, pp 85–111 | Cite as

The Massive Wave Equation in Asymptotically AdS Spacetimes



We consider the massive wave equation on asymptotically AdS spaces. We show that the timelike \({\fancyscript{F}}\) behaves like a finite timelike boundary, on which one may impose the equivalent of Dirichlet, Neumann or Robin conditions for a range of (negative) mass parameter which includes the conformally coupled case. We demonstrate well posedness for the associated initial-boundary value problems at the H 1 level of regularity. We also prove that higher regularity may be obtained, together with an asymptotic expansion for the field near \({\fancyscript{F}}\). The proofs rely on energy methods, tailored to the modified energy introduced by Breitenlohner and Freedman. We do not assume the spacetime is stationary, nor that the wave equation separates.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Maldacena, J.M.: The Large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]Google Scholar
  2. 2.
    Holzegel, G., Smulevici, J.: Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein-Gordon system. [gr-qc], 2011
  3. 3.
    Holzegel, G., Smulevici, J.: Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes. [gr-qc], 2011
  4. 4.
    Holzegel, G., Smulevici, J.: Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes. [gr-qc], 2011
  5. 5.
    Breitenlohner P., Freedman D.Z.: Stability in Gauged Extended Supergravity. Ann. Phys. 144, 249 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Ishibashi A., Wald R.M.: Dynamics in nonglobally hyperbolic static space-times. 2. General analysis of prescriptions for dynamics. Class. Quant. Grav. 20, 3815 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Ishibashi A., Wald R.M.: Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time. Class. Quant. Grav. 21, 2981 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Bachelot A.: The Dirac System on the Anti-de Sitter Universe. Commun. Math. Phys. 283, 127–167 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Holzegel, G.: Well-posedness for the massive wave equation on asymptotically anti-de Sitter spacetimes. [gr-qc], 2011
  10. 10.
    Vasy, A.: The wave equation on asymptotically Anti-de Sitter spaces. Analysis and PDE 5, no. 1 (2012)Google Scholar
  11. 11.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics 19, Providence, RI: Amer. Math. Soc., 2008Google Scholar
  12. 12.
    Ladyzhenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Applied Mathematical Sciences 49, New York: Springer-Verlag, 1985Google Scholar
  13. 13.
    Kufner, A.: Weighted Sobolev Spaces. New York: John Wiley & Sons Inc., 1985Google Scholar
  14. 14.
    Kim D.: Trace theorems for Sobolev-Slobodeckij spaces with or without weights. J. Fun. Spac. and Appl. 5(3), 243–268 (2007)MATHCrossRefGoogle Scholar
  15. 15.
    Gover, A.R., Waldron, A.: Boundary calculus for conformally compact manifolds. [math.DG], 2012
  16. 16.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics: I Functional Analysis. London: Academic Press, 1972; See also Tao, T.: “The spectral theorem and its converses for unbounded symmetric operators.” available at

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Physics, 4-183 CCISUniversity of AlbertaEdmontonCanada

Personalised recommendations