Annales Henri Poincaré

, Volume 20, Issue 5, pp 1583–1650 | Cite as

A Scattering Theory for Linear Waves on the Interior of Reissner–Nordström Black Holes

  • Christoph KehleEmail author
  • Yakov Shlapentokh-Rothman
Open Access


We develop a scattering theory for the linear wave equation \(\Box _g \psi = 0 \) on the interior of Reissner–Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution \(\psi \) on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resulting radial o.d.e. over all frequencies \(\omega \) and \(\ell \). This is non-trivial because the natural T conservation law is sign-indefinite in the black hole interior. In the physical space picture, our results imply that the Cauchy evolution from the event horizon to the Cauchy horizon is a Hilbert space isomorphism, where the past (resp. future) Hilbert space is defined by the finiteness of the degenerate T energy fluxes on both components of the event (resp. Cauchy) horizon. Finally, we prove that, in contrast to the above, for a generic set of cosmological constants \(\Lambda \), there is no analogous finite T energy scattering theory for either the linear wave equation or the Klein–Gordon equation with conformal mass on the (anti-) de Sitter–Reissner–Nordström interior.



  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. Dover Publications, Inc., New York (1992)Google Scholar
  2. 2.
    Bachelot, A.: Gravitational scattering of electromagnetic field by Schwarzschild black-hole. Ann. Inst. H. Poincaré Phys. Théor. 54(3), 261–320 (1991)ADSzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bachelot, A.: Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric. Ann. Inst. H. Poincaré Phys. Théor. 61(4), 411–441 (1994)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Borwein, P., Erdélyi, T.: Polynomials and polynomial inequalities. In: Graduate Texts in Mathematics, vol. 161. Springer, New York (1995)Google Scholar
  5. 5.
    Chandrasekhar, S., Hartle, J.B.: On crossing the Cauchy horizon of a Reissner–Nordström black-hole. Proc. Roy. Soc. Lond. Ser. A 384(1787), 301–315 (1982)ADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Civin, D.: Stability of charged rotating black holes for linear scalar perturbations. Ph.D. Thesis, University of Cambridge, Cambridge (2015)Google Scholar
  7. 7.
    Dafermos, M., Luk, J.: The interior of dynamical vacuum black holes i: The \(C^0\)-Stability of the Kerr Cauchy horizon. arXiv preprint arXiv:1710.01722 (2017)
  8. 8.
    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. of Math. (2) 183(3), 787–913 (2016)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dafermos, M., Rodnianski, I., Shlapentokh-Rothman, Y.: A scattering theory for the wave equation on Kerr black hole exteriors. Ann. Sci. Éc. Norm. Supér. (4) 51(2), 371–486 (2018)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    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)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dimock, J.: Scattering for the wave equation on the Schwarzschild metric. Gen. Relativ. Gravit. 17(4), 353–369 (1985)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dimock, J., Kay, B.S.: Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric II. J. Math. Phys. 27(10), 2520–2525 (1986)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Dimock, J., Kay, B.S.: Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric I. Ann. Phys. 175(2), 366–426 (1987)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Drouot, A.: A quantitative version of Hawking radiation. Ann. Henri Poincaré 18(3), 757–806 (2017)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Fournodavlos, G., Sbierski, J.: Generic Blow-Up Results for the Wave Equation in the Interior of a Schwarzschild Black Hole. arXiv preprint arXiv:1804.01941 (2018)
  16. 16.
    Franzen, A.T.: Boundedness of massless scalar waves on Kerr interior backgrounds. preprint (2017)Google Scholar
  17. 17.
    Franzen, A.T.: Boundedness of massless scalar waves on Reissner–Nordström interior backgrounds. Commun. Math. Phys. 343(2), 601–650 (2016)ADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Futterman, J.A.H., Handler, F.A., Matzner, R.A.: Scattering from black holes. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009)Google Scholar
  19. 19.
    Georgescu, V., Gérard, C., Häfner, D.: Asymptotic completeness for superradiant Klein–Gordon equations and applications to the de Sitter-Kerr metric. J. Eur. Math. Soc. (JEMS) 19(8), 2371–2444 (2017)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Graves, J.C., Brill, D.R.: Oscillatory character of Reissner–Nordström metric for an ideal charged wormhole. Phys. Rev. 2(120), 1507–1513 (1960)ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Gürsel, Y., Sandberg, V.D., Novikov, I.D., Starobinsky, A.A.: Evolution of scalar perturbations near the cauchy horizon of a charged black hole. Phys. Rev. D 19(2), 413–420 (1979)ADSCrossRefGoogle Scholar
  22. 22.
    Gürsel, Y., Novikov, I.D., Sandberg, V.D., Starobinsky, A.A.: Final state of the evolution of the interior of a charged black hole. Phys. Rev. D 20(6), 1260–1270 (1979)ADSCrossRefGoogle Scholar
  23. 23.
    Häfner, D.: Some mathematical aspects of the Hawking effect for rotating black holes. In: Quantum field theory and gravity, pp. 121–136. Birkhäuser/Springer Basel AG, Basel (2012)Google Scholar
  24. 24.
    Hintz, P.: Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime. Comment. Math. Helv. 92(4), 801–837 (2017)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Hintz, P., Vasy, A.: Analysis of linear waves near the Cauchy horizon of cosmological black holes. J. Math. Phys. 58(8), 081509, 45 (2017)Google Scholar
  26. 26.
    Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237–238 (1963)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Luk, J., Oh, S.J.: Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data I. the interior of the black hole region. arXiv preprint arXiv:1702.05715 (2017)
  29. 29.
    Luk, J., Oh, S.J.: Strong cosmic censorship in spherical symmetry for two-ended asymptotically flat initial data II. the exterior of the black hole region. arXiv preprint arXiv:1702.05716 (2017)
  30. 30.
    Luk, J., Oh, S.J., Shlapentokh-Rothman, Y.: A scattering approach to cauchy horizon instability and applications to mass inflation. In preparation (2018)Google Scholar
  31. 31.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Mason, L.J., Nicolas, J.P.: Conformal scattering and the Goursat problem. J. Hyperb. Differ. Equ. 1(2), 197–233 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Matzner, R.A., Zamorano, N., Sandberg, V.D.: Instability of the Cauchy horizon of Reissner–Nordström black holes. Phys. Rev. D (3) 19(10), 2821–2826 (1979)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    McNamara, J.M.: Behaviour of scalar perturbations of a Reissner–Nordström black hole inside the event horizon. Proc. R. Soc. Lond. Ser. A 364(1716), 121–134 (1978)ADSCrossRefGoogle Scholar
  35. 35.
    McNamara, J.M.: Instability of black hole inner horizons. Proc. R. Soc. Lond. Ser. A 358(1695), 499–517 (1978)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Melnyk, F.: Scattering on Reissner-Nordstrøm metric for massive charged spin 1/2 fields. Ann. Henri Poincaré 4(5), 813–846 (2003)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Mokdad, M.: Conformal scattering of maxwell fields on Reissner–Nordström–de sitter black hole spacetimes. arXiv preprint arXiv:1706.06993 (2017)
  38. 38.
    Müller zum Hagen, H., Seifert, H.J.: On characteristic initial-value and mixed problems. Gen. Relativ. Gravit. 8(4), 259–301 (1977)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    NIST Digital Library of Mathematical Functions. In: Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders B.V. (eds.), Release 1.0.16. Accessed 18 Sept 2017
  40. 40.
    Nicolas, J.P.: Conformal scattering on the Schwarzschild metric. Ann. Inst. Fourier (Grenoble) 66(3), 1175–1216 (2016)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Nordström, G.: On the energy of the gravitation field in Einstein’s theory. Verhandl. Koninkl. Ned. Akad. Wetenschap. Afdel. Natuurk 20, 1238–1245 (1918)ADSGoogle Scholar
  42. 42.
    O’Neill, B.: The Geometry of Kerr Black Holes. A K Peters Ltd., Wellesley (1995)zbMATHGoogle Scholar
  43. 43.
    Olver, F.W.J.: Error bounds for the Liouville–Green (or WKB) approximation. Proc. Camb. Philos. Soc. 57, 790–810 (1961)ADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Olver, F.W.J.: Asymptotics and special functions. AKP Classics. A K Peters Ltd., Wellesley (1997)Google Scholar
  45. 45.
    Reissner, H.: Über die eigengravitation des elektrischen feldes nach der Einsteinschen theorie. Annalen der Physik 355(9), 106–120 (1916)ADSCrossRefGoogle Scholar
  46. 46.
    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. Ser. A 427(1872), 221–239 (1990)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Sbierski, J.: On the initial value problem in general relativity and wave propagation in black-hole spacetimes. Ph.D. Thesis, University of Cambridge, Cambridge (2014)Google Scholar
  48. 48.
    Schlag, W., Soffer, A., Staubach, W.: Decay for the wave and Schrödinger evolutions on manifolds with conical ends I. Trans. Am. Math. Soc. 362(1), 19–52 (2010)zbMATHCrossRefGoogle Scholar
  49. 49.
    Taujanskas, G.: Conformal scattering of the maxwell-scalar field system on de sitter space. arXiv preprint arXiv:1809.01559 (2018)
  50. 50.
    Van de Moortel, M.: Stability and instability of the sub-extremal Reissner–Nordström black hole interior for the Einstein–Maxwell–Klein–Gordon equations in spherical symmetry. Commun. Math. Phys. 360(1), 103–168 (2018)ADSzbMATHCrossRefGoogle Scholar
  51. 51.
    Wald, R.M.: Quantum field theory in curved spacetime and black hole thermodynamics. Chicago Lectures in Physics. University of Chicago Press, Chicago, IL (1994)Google Scholar
  52. 52.
    Zamorano, N.: Interior Reissner–Nordström metric and the scalar wave equation. Phys. Rev. D (3) 26(10), 2564–2574 (1982)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

OpenAccessThis 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 MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations