Annales Henri Poincaré

, Volume 13, Issue 4, pp 991–1038 | Cite as

Self-Gravitating Klein–Gordon Fields in Asymptotically Anti-de-Sitter Spacetimes

  • Gustav Holzegel
  • Jacques SmuleviciEmail author


We initiate the study of the spherically symmetric Einstein–Klein–Gordon system in the presence of a negative cosmological constant, a model appearing frequently in the context of high-energy physics. Due to the lack of global hyperbolicity of the solutions, the natural formulation of dynamics is that of an initial boundary value problem, with boundary conditions imposed at null infinity. We prove a local well-posedness statement for this system, with the time of existence of the solutions depending only on an invariant H 2-type norm measuring the size of the Klein–Gordon field on the initial data. The proof requires the introduction of a renormalized system of equations and relies crucially on r-weighted estimates for the wave equation on asymptotically AdS spacetimes. The results provide the basis for our companion paper establishing the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this system.


Pointwise Estimate Extension Principle Raychaudhuri Equation Penrose Diagram Pointwise Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bachelot A.: The Dirac system on the Anti-de Sitter universe. Commun. Math. Phys. 283, 127–167 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Bachelot, A.: The Klein–Gordon equation in Anti-de Sitter cosmology. arXiv:1010.1925Google Scholar
  3. 3.
    Breitenlohner P., Freedman D.Z.: Stability in gauged extended supergravity. Ann. Phys. 144, 249 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Choquet-Bruhat Y., Geroch R.P.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Christodoulou D.: The problem of a self-gravitating scalar field. Commun. Math. Phys. 105, 337–361 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Christodoulou D.: A mathematical theory of gravitational collapse. Commun. Math. Phys. 109, 613–647 (1987)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Christodoulou D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math 149, 183–217 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dafermos M.: Spherically symmetric spacetimes with a trapped surface. Class. Quant. Grav. 22, 2221–2232 (2005) gr-qc/0403032MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Dafermos M., Rodnianski I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162, 381–457 (2005) gr-qc/0309115MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Friedrich H.: Einstein equations and conformal structure: existence of anti-de Sitter-type space-times. J. Geom. Phys. 17, 125–184 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Friedrich H.: Initial boundary value problems for Einstein’s field equations and geometric uniqueness. Gen. Relat. Gravit. 41, 1947–1966 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Gubser S.S. Breaking an Abelian gauge symmetry near a black hole horizon. Phys. Rev. D 78 (2008). arXiv:065034, 0801.2977Google Scholar
  13. 13.
    Holzegel G.: On the massive wave equation on slowly rotating Kerr-AdS spacetimes. Commun. Math. Phys. 294, 169–197 (2010) arXiv:0902.0973MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Holzegel, G.: Well-posedness for the massive wave equation on asymptotically anti-de-Sitter spacetimes. arXiv:1103.0710Google Scholar
  15. 15.
    Holzegel, G., Smulevici, J.: Stability of Schwarzschild-AdS for the spherically symmetric Einstein-Klein Gordon system. arXiv:1103.3672Google Scholar
  16. 16.
    Kommemi, J.: The global structure of spherically symmetric charged scalar field spacetimes. Preprint. arXiv:1107.0949 (2011)Google Scholar
  17. 17.
    Sonner J.: A rotating holographic superconductor. Phys. Rev. D 80, 084031 (2009) arXiv:0903.0627ADSCrossRefGoogle Scholar
  18. 18.
    Vasy, A.: The wave equation on asymptotically Anti-de Sitter spaces. Anal. PDE (2009, to appear). arXiv:0911.5440Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Max-Planck-Institut für GravitationsphysikAlbert-Einstein InstitutGolmGermany

Personalised recommendations