Skip to main content
SpringerLink
Log in

A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We consider Kerr spacetimes with parameters a and M such that |a|≪M, Kerr-Newman spacetimes with parameters |Q|≪M, |a|≪M, and more generally, stationary axisymmetric black hole exterior spacetimes \((\mathcal{M},g)\) which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □ g ψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Σ τ , then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Σ τ is bounded by C times the initial energy flux through Σ0. This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ0. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Alinhac, S.: Energy multipliers for perturbations of Schwarzschild metrics. Preprint (2008)

  2. Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. arXiv:0908.2265

  3. Aretakis, S.: The wave equation on extreme Reissner-Nordström black hole spacetimes: stability and instability results. arXiv:1006.0283

  4. Bachelot, A.: Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric. Ann. Inst. H. Poincaré Phys. Théor. 16(4), 411–441 (1994)

    MathSciNet  Google Scholar 

  5. Beyer, H.: On the stability of the Kerr metric. Commun. Math. Phys. 221, 659–676 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blue, P.: Decay of the Maxwell field on the Schwarzschild manifold. J. Hyperbolic Differ. Equ. 5(4), 807–856 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blue, P., Soffer, A.: Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates. Adv. Differ. Equ. 8(5), 595–614 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Blue, P., Soffer, A.: Errata for “Global existence …Regge Wheeler equation”, 6 pp. gr-qc/0608073

  9. Blue, P., Soffer, A.: Phase space analysis on some black hole manifolds. Preprint

  10. Blue, P., Sterbenz, J.: Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Commun. Math. Phys. 268(2), 481–504 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bony, J.-F., Häfner, D.: Decay and non-decay of the local energy for the wave equation in the de Sitter-Schwarzschild metric. Commun. Math. Phys. 282(3), 697–719 (2008)

    Article  MATH  Google Scholar 

  12. Carter, B.: Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations. Commun. Math. Phys. 10, 280–310 (1968)

    MATH  Google Scholar 

  13. Carter, B.: Black hole equilibrium sates. In: Black Holes/Les Astres Occlus (École d’Été Phys. Théor., Les Houches, 1972), pp. 57–214. Gordon and Breach, New York (1973)

    Google Scholar 

  14. Christodoulou, D.: Reversible and irreversible transformations in black-hole physics. Phys. Rev. Lett. 25, 1596–1597 (1970)

    Article  Google Scholar 

  15. Christodoulou, D.: The action principle and partial differential equations. Ann. Math. Stud. 146 (1999)

  16. Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  17. Dafermos, M.: The interior of charged black holes and the problem of uniqueness in general relativity. Commun. Pure Appl. Math. 58, 0445–0504 (2005)

    Article  MathSciNet  Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dafermos, M., Rodnianski, I.: The redshift effect and radiation decay on black hole spacetimes. Comm. Pure Appl. Math. 52, 859–919 (2009). gr-qc/0512119

    Article  MathSciNet  Google Scholar 

  20. Dafermos, M., Rodnianski, I.: A note on energy currents and decay for the wave equation on a Schwarzschild background. arXiv:0710.0171

  21. Dafermos, M., Rodnianski, I.: The wave equation on Schwarzschild-de Sitter spacetimes. arXiv:0709.2766

  22. Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. arXiv:0811.0354 [gr-qc]

  23. Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: Exner, P. (ed.) XVIth International Congress on Mathematical Physics, pp. 421–433. World Scientific, Singapore (2009). arXiv:0910.4957v1

    Google Scholar 

  24. Dafermos, M., Rodnianski, I.: The black hole stability problem for linear scalar perturbations. arXiv:1010.5137

  25. Dyatlov, S.: Exponential energy decay for Kerr-de Sitter black holes beyond event horizons. arXiv:1010.5201v1

  26. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry. Adv. Theor. Math. Phys. 7, 25–52 (2003)

    MathSciNet  Google Scholar 

  27. Finster, F., Kamran, N., Smoller, J., Yau, S.T.: Decay of solutions of the wave equation in Kerr geometry. Commun. Math. Phys. 264, 465–503 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Erratum: Decay of solutions of the wave equation in Kerr geometry. Commun. Math. Phys., online first

  29. Frolov, V., Kubizñák, D.: Higher-dimensional black holes: hidden symmetries and separation of variables. Class. Quantum Gravity 25, 154005 (2008)

    Article  Google Scholar 

  30. Häfner, D.: Sur la théorie de la diffusion pour l’équation de Klein-Gordon dans la métrique de Kerr. Diss. Math. 421 (2003)

  31. Häfner, D., Nicolas, J.-P.: Scattering of massless Dirac fields by a Kerr black hole. Rev. Math. Phys. 16(1), 29–123 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics, vol. 1. Cambridge University Press, Cambridge (1973)

    Book  MATH  Google Scholar 

  33. Holzegel, G.: On the massive wave equation on slowly rotating Kerr-AdS spacetimes. Commun. Math. Phys. 294, 169–197 (2009). arXiv:0902.0973

    Article  MathSciNet  Google Scholar 

  34. Ionescu, A., Klainerman, S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175(1), 35–102 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kay, B., Wald, R.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quantum Gravity 4(4), 893–898 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kerr, R.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237–238 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kokkotas, K., Schmidt, B.: Quasi-normal modes of stars and black holes. Living Rev. Relativ. 2 (1999)

  38. Laba, I., Soffer, A.: Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds. Helv. Phys. Acta 72(4), 272–294 (1999)

    MathSciNet  Google Scholar 

  39. Laul, P., Metcalfe, J.: Localized energy estimates for wave equations on high dimensional Schwarzschild space-times. arXiv:1008.4626v2

  40. Luk, J.: Improved decay for solutions to the linear wave equation on a Schwarzschild black hole. Ann. Henri Poincaré, online first (2010). arXiv:0906.5588

  41. Marzuola, J., Metcalfe, J., Tataru, D., Tohaneanu, M.: Strichartz estimates on Schwarzschild black hole backgrounds. arXiv:0802.3942

  42. Melrose, R., Barreto, A.S., Vasy, A.: Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space. arXiv:0811.2229

  43. Morawetz, C.S.: Time decay for the nonlinear Klein-Gordon equations. Proc. R. Soc. Ser. A 206, 291–296 (1968)

    MathSciNet  Google Scholar 

  44. Nicolas, J.-P.: A non linear Klein-Gordon equation on Kerr metrics. J. Math. Pures Appl. 81, 885–914 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ralston, J.: Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22, 807–823 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  46. Schlue, V.: Linear waves on higher dimensional Schwarzschild black holes. Smith-Rayleigh-Knight essay (2010)

  47. Tataru, D.: Local decay of waves on asymptotically flat stationary space-times. arXiv:0910.5290

  48. Tataru, D., Tohaneanu, M.: Local energy estimate on Kerr black hole backgrounds. arXiv:0810.5766

  49. Tohaneanu, M.: Strichartz estimates on Kerr black hole backgrounds. arXiv:0910.1545

  50. Twainy, F.: The time decay of solutions to the scalar wave equation in Schwarzschild background. Thesis, University of California, San Diego (1989)

  51. Vasy, A.: The wave equation on asymptotically Anti-de Sitter spaces. arXiv:0911.5440

  52. Wald, R.M.: Note on the stability of the Schwarzschild metric. J. Math. Phys. 20, 1056–1058 (1979)

    Article  MathSciNet  Google Scholar 

  53. Walker, M., Penrose, R.: On quadratic first integrals of the geodesic equations for type 22 spacetimes. Commun. Math. Phys. 18, 265–274 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  54. Whiting, B.: Mode stability of the Kerr black hole. J. Math. Phys. 30, 1301 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wunsch, J., Zworski, M.: Resolvent estimates for normally hyperbolic trapped sets (2010). arXiv:1003.4640

  56. Yang, S.: Global solutions to nonlinear wave equations in time dependent inhomogeneous media. arXiv:1010.4341

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Mihalis Dafermos

  2. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544, USA

    Igor Rodnianski

Authors
  1. Mihalis Dafermos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Rodnianski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Rodnianski.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dafermos, M., Rodnianski, I. A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds. Invent. math. 185, 467–559 (2011). https://doi.org/10.1007/s00222-010-0309-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-010-0309-0

Keywords

Search

Navigation