Annales Henri Poincaré

, Volume 18, Issue 5, pp 1593–1633 | Cite as

The Trapping Effect on Degenerate Horizons

  • Yannis Angelopoulos
  • Stefanos AretakisEmail author
  • Dejan Gajic


We show that degenerate horizons exhibit a new trapping effect. Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner–Nordström up to and including the future event horizon. We show that such an estimate requires (1) a higher degree of regularity for the initial data, reminiscent of the regularity loss in the high-frequency trapping estimates on the photon sphere, and (2) the vanishing of an explicit quantity that depends on the restriction of the initial data on the horizon. The latter condition demonstrates that degenerate horizons exhibit a new \(L^{2}\) concentration phenomenon (namely, a global trapping effect, in the sense that this effect is not due to individual underlying null geodesics as in the case of the photon sphere). We moreover uncover a new stable higher-order trapping effect; we show that higher-order estimates do not hold regardless of the degree of regularity and the support of the initial data. We connect our findings to the spectrum of the stability operator in the theory of marginally outer trapped surfaces. Our methods and results play a crucial role in our upcoming works on linear and nonlinear wave equations on extremal black hole backgrounds.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. arXiv:0908.2265 (2009)
  2. 2.
    Angelopoulos, Y.: Nonlinear wave equations with null condition on extremal Reissner-Nordström spacetimes I: spherical symmetry. To appear in IMRN. arXiv:1408.4478 (2014)
  3. 3.
    Aretakis, S.: The wave equation on extreme Reissner–Nordström black hole spacetimes: stability and instability results. arXiv:1006.0283 (2010)
  4. 4.
    Aretakis, S.: Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations I. Commun. Math. Phys. 307, 17–63 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aretakis, S.: Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations II. Ann. Henri Poincaré 12, 1491–1538 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aretakis, S.: Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds. J. Funct. Anal. 263, 2770–2831 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Aretakis, S.: The characteristic gluing problem and conservation laws for the wave equation on null hypersurfaces. arXiv:1310.1365 (2013)
  8. 8.
    Aretakis, S.: A note on instabilities of extremal black holes from afar. Class. Quantum Gravity 30, 095010 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Aretakis, S.: On a foliation-covariant elliptic operator on null hypersurfaces. To appear in IMRN. arXiv:1310.1348 (2013)
  10. 10.
    Aretakis, S.: On a non-linear instability of extremal black holes. Phys. Rev. D 87, 084052 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    Aretakis, S.: Horizon instability of extremal black holes. Adv. Theor. Math. Phys. 19, 507–530 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dafermos, M., Rodnianski, I.: The redshift effect and radiation decay on black hole spacetimes. Commun. Pure Appl. Math. 62, 859–919 (2009). arXiv:0512.119
  13. 13.
    Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. In: Evolution Equations, Clay Mathematics Proceedings, Vol. 17, Am. Math. Soc. Providence, RI, pp. 97–205 (2013). arXiv:0811.0354
  14. 14.
    Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case \(|a| < m\). arXiv:1402.7034
  15. 15.
    Dafermos, M., Holzegel, G., Rodnianski, I.: The linear stability of the Schwarzschild solution to gravitational perturbations. arXiv:1601.06467 (2016)
  16. 16.
    Dain, S., Dotti, G.: The wave equation on the extreme Reissner–Nordström black hole. arXiv:1209.0213 (2012)
  17. 17.
    Dyatlov, S.: Exponential energy decay for Kerr-de Sitter black holes beyond event horizons. Math. Res. Lett. 18, 1023–1035 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gajic, D.: Linear waves in the interior of extremal black holes I. arXiv:1509.06568 (2015)
  19. 19.
    Holzegel, G., Smulevici, J.: Decay properties of Klein–Gordon fields on Kerr-AdS spacetimes. Commun. Pure Appl. Math. 66, 1751–1802 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kay, B., Wald, R.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quantum Gravity 4, 893–898 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Keir, J.: Slowly decaying waves on spherically symmetric spacetimes and an instability of ultracompact neutron stars. arXiv:1404.7036 (2014)
  22. 22.
    Klainerman, S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38, 321–332 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lucietti, J., Murata, K., Reall, H.S., Tanahashi, N.: On the horizon instability of an extreme Reissner-Nordström black hole. JHEP 1303, 035 (2013). arXiv:1212.2557
  24. 24.
    Lucietti, J., Reall, H.: Gravitational instability of an extreme Kerr black hole. Phys. Rev. D 86, 104030 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    Mars, M.: Stability of MOTS in totally geodesic null horizons. Class. Quantum Gravity 29, 145019 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Moschidis, G.: Logarithmic local energy decay for scalar waves on a general class of asymptotically flat spacetimes. arXiv:1509.08495 (2015)
  27. 27.
    Murata, K.: Instability of higher dimensional extreme black holes. Class. Quantum Gravity 30, 075002 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Murata, K., Reall, H.S., Tanahashi, N.: What happens at the horizon(s) of an extreme black hole? arXiv:1307.6800 (2013)
  29. 29.
    Ori, A.: Late-time tails in extremal Reissner–Nordström spacetime. arXiv:1305.1564 (2013)
  30. 30.
    Ralston, J.: Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22, 807–823 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Regge, T., Wheeler, J.: Stability of a Schwarzschild singularity. Phys. Rev. 108, 1063–1069 (1957)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sbierski, J.: Characterisation of the energy of Gaussian beams on Lorentzian manifolds with applications to black hole spacetimes. arXiv:1311.2477 (2013)
  33. 33.
    Sela, O.: Late-time decay of perturbations outside extremal charged black hole. arXiv:1510.06169 (2015)
  34. 34.
    Tataru, D., Tohaneanu, M.: A local energy estimate on Kerr black hole backgrounds. Int. Math. Res. Not. 2011, 248–292 (2008)zbMATHGoogle Scholar
  35. 35.
    Tsukamoto, N., Kimura, M., Harada, T.: High energy collision of particles in the vicinity of extremal black holes in higher dimensions: Banados-Silk-West process as linear instability of extremal black holes. arXiv:1310.5716 (2013)
  36. 36.
    Wunsch, J., Zworski, M.: Resolvent estimates for normally hyperbolic trapped sets. Ann. Henri Poincaré 12, 1349–1385 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Yannis Angelopoulos
    • 1
  • Stefanos Aretakis
    • 2
    • 3
    • 4
    Email author
  • Dejan Gajic
    • 5
    • 6
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA
  3. 3.Department of MathematicsUniversity of Toronto ScarboroughTorontoCanada
  4. 4.Department of MathematicsUniversity of TorontoTorontoCanada
  5. 5.Department of MathematicsImperial College LondonLondonUK
  6. 6.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

Personalised recommendations