Annales Henri Poincaré

, Volume 18, Issue 12, pp 4005–4081 | Cite as

Linear Waves in the Interior of Extremal Black Holes II

  • Dejan GajicEmail author
Open Access


We consider solutions to the linear wave equation in the interior region of extremal Kerr black holes. We show that axisymmetric solutions can be extended continuously beyond the Cauchy horizon and, moreover, that if we assume suitably fast polynomial decay in time along the event horizon, their local energy is finite. We also extend these results to non-axisymmetric solutions on slowly rotating extremal Kerr–Newman black holes. These results are the analogues of results obtained in Gajic (Commun Math Phys 353(2), 717–770, 2017) for extremal Reissner–Nordström and stand in stark contrast to previously established results for the subextremal case, where the local energy was shown to generically blow up at the Cauchy horizon.


  1. 1.
    Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. Ann. Math. 182(3), 787–853 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andersson, N., Glampedakis, K.: Late-time dynamics of rapidly rotating black holes. Phys. Rev. D 64(10), 104021 (2001)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Angelopoulos, Y., Aretakis, S., Gajic, D.: Improved decay for solutions to the wave equation on extremal Reissner–Nordström and applications. In preparationGoogle Scholar
  4. 4.
    Aretakis, S.: Horizon instability of extremal black holes. Adv. Theor. Math. Phys. 19(3), 507–530 (2015)Google Scholar
  5. 5.
    Aretakis, S.: The wave equation on extreme Reissner–Nordström black hole spacetimes: stability and instability results. arXiv:1006.0283 (2010)
  6. 6.
    Aretakis, S.: Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations i. Commun. Math. Phys. 307(1), 17–63 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Aretakis, S.: Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations ii. Ann. Henri Poincaré 12(8), 1491–1538 (2011)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Aretakis, S.: Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds. J. Funct. Anal. 263(9), 2770–2831 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Aretakis, S.: A note on instabilities of extremal black holes under scalar perturbations from afar. Class. Quantum Gravity 30(9), 095010 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Carter, B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571 (1968)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Carter, B.: Killing tensor quantum numbers and conserved currents in curved space. Phys. Rev. D 16(12), 3395–3414 (1977)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    Casals, M., Gralla, S.E., Zimmerman, P.: Horizon instability of extremal Kerr black holes: nonaxisymmetric modes and enhanced growth rate. Phys. Rev. D 94(6), 064003 (2016)ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Christodoulou, D.: Mathematical Problems of General Relativity Theory I. European Mathematical Society (EMS), Zurich (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Christodoulou, D.: The formation of black holes in general relativity. In: EMS Monographs in Mathematics. European Mathematical Society (EMS), Zurich (2009)Google Scholar
  15. 15.
    Civin, D.: Stability of Charged Rotating Black Holes for Linear Scalar Perturbations. Ph.D. Thesis (2014),
  16. 16.
    Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. Clay Math. Proc. 17, 97–205 (2013)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case \(|a|< M\). Ann. Math. 183(3), 787–913 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Dafermos, M., Holzegel, G., Rodnianski, I.: A scattering theory construction of dynamical vacuum black holes. arXiv:1306.5364 (2013)
  19. 19.
    Dafermos, M., Shlapentokh-Rothman, Y.: Time-translation invariance of scattering maps and blue-shift instabilities on Kerr black hole spacetimes. Commun. Math. Phys. 350(3), 985–1016 (2017)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Franzen, A.: Boundedness of massless scalar waves on Kerr interior backgrounds. In preparationGoogle Scholar
  21. 21.
    Franzen, A.: Boundedness of massless scalar waves on Reissner–Nordström interior backgrounds. Commun. Math. Phys. 343(2), 601–650 (2016)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Gajic, D.: Double-null foliations of Kerr–Newman. In preparation (2015)Google Scholar
  23. 23.
    Gajic, D.: Linear waves in the interior of extremal black holes I. Commun. Math. Phys. 353(2), 717–770 (2017)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Hawking, S., Ellis, G.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)CrossRefzbMATHGoogle Scholar
  25. 25.
    Hintz, P.: Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime. arXiv:1512.08003 (2015)
  26. 26.
    Klainerman, S.: Brief history of the vector-field method. Special Lecture in Honour of F. John’s 100th Anniversary (November 2010).
  27. 27.
    Lucietti, J., Murata, K., Reall, H., Tanahashi, N.: On the horizon instability of an extreme Reissner–Nordström black hole. J. High Energy Phys. 2013(3), 035 (2013)CrossRefzbMATHGoogle Scholar
  28. 28.
    Lucietti, J., Reall, H.S.: Gravitational instability of an extreme Kerr black hole. Phys. Rev. D 86(10), 104030 (2012)ADSCrossRefGoogle Scholar
  29. 29.
    Luk, J., Oh, S.-J.: Proof of linear instability of the Reissner–Nordström Cauchy horizon under scalar perturbations. Duke Math. J. 166(3), 437–493 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Luk, J., Sbierski, J.: Instability results for the wave equation in the interior of Kerr black holes. J. Funct. Anal. 271(7), 1948–1995 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Newman, E.T., Couch, E., Chinnapared, K., Exton, A., Prakash, A., Torrence, R.: Metric of a rotating, charged mass. J. Math. Phys. 6(6), 918–919 (1965)ADSCrossRefMathSciNetGoogle Scholar
  32. 32.
    Pretorius, F., Israel, W.: Quasi-spherical light cones of the Kerr geometry. Class. Quantum Gravity 15(8), 2289–2301 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Rendall, A.D.: Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 427(1872), 221–239 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  2. 2.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations