Skip to main content
SpringerLink
Log in

Energy Multipliers for Perturbations of the Schwarzschild Metric

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the wave equations associated to metrics close to the Schwarzschild metric. We investigate spacelike energy multipliers likely to yield local decay of solutions to these wave equations, in the spirit of Morawetz. For rotationally invariant metrics, we obtain multipliers giving a control of the solutions having finitely many vanishing spherical harmonics. The structure of these multipliers is closely related to the photosphere of the metric. For Kerr metrics, in contrast, we display a region, which we call the intersphere region, where no energy inequality with the required properties can exist.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alinhac S.: On the Morawetz-Keel-Smith-Sogge inequality for the wave equation on a curved background. Publ. R. Inst. Math. Sc. Kyoto 42(3), 705–720 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chandrasekhar, C.: The Mathematical Theory of Black Holes. Int. Ser. Mon. Physics 69, Oxford: Oxford Univ. Press, 1983

  3. Christodoulou D.: Bounded variation solutions of the spherically symmetric einstein-Scalar field equations. Comm. Pure Appl. Math. XLVI, 1131–1220 (1993)

    Article  MathSciNet  Google Scholar 

  4. Dafermos, M., Rodnianski, I.: The Red-Shift Effect and Radiation Decay on Black Hole Spacetimes. http://arxiv.org/abs/gr-qc/0512119, 2005

  5. Dafermos, M., Rodnianski, I.: A Note on Energy Currents and Decay for the Wave Equation on a Schwarzschild Background. http://arxiv.org/abs/0710.071v1[math,AP], 2007

  6. Dafermos, M., Rodnianski, I.: Lectures on Black Holes and Linear Waves. http://arxiv.org/abs/0811.354v1[gr-qc], 2008

  7. Hawking, S.W., Ellis, G.F.: The Large Scale Structure of Space-Time . Cambridge Mon. Math. Physics, Cambridge: Cambridge Univ. Press, 1973

  8. Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Math. Appl. 26, Berlin-Heidelberg-New York: Springer Verlag, 1997

  9. Klainerman, S., Nicolò, F.: The Evolution Problem in General Relativity. Prog. Math. Physics 25, Besel-Boston: Birkhäuser, 2002

  10. Morawetz C.S.: Time decay for the nonlinear Klein-Gordon equation. Proc. Roy. Soc. A 306, 291–296 (1968)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Tataru, D., Tohaneanu, M.: Local Energy Estimate on Kerr Black Hole Background. http://arxiv.org/abs/0810.5766v2[math,AP], 2008

  12. Wald, R.: General Relativity. Chicago, IL: Univ. Chicago Press, 1984

Download references

Author information

Authors and Affiliations

  1. Department des Mathematiques, Université Paris-Sud, 91405, Orsay, France

    Serge Alinhac

Authors
  1. Serge Alinhac
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serge Alinhac.

Additional information

Communicated by G.W. Gibbons

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alinhac, S. Energy Multipliers for Perturbations of the Schwarzschild Metric. Commun. Math. Phys. 288, 199–224 (2009). https://doi.org/10.1007/s00220-009-0770-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0770-z

Keywords

Search

Navigation