Abstract
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.
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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)
Bachelot, A.: Gravitational scattering of electromagnetic field by Schwarzschild black-hole. Ann. Inst. H. Poincaré Phys. Théor. 54(3), 261–320 (1991)
Bachelot, A.: Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric. Ann. Inst. H. Poincaré Phys. Théor. 61(4), 411–441 (1994)
Borwein, P., Erdélyi, T.: Polynomials and polynomial inequalities. In: Graduate Texts in Mathematics, vol. 161. Springer, New York (1995)
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)
Civin, D.: Stability of charged rotating black holes for linear scalar perturbations. Ph.D. Thesis, University of Cambridge, Cambridge (2015)
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)
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)
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)
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)
Dimock, J.: Scattering for the wave equation on the Schwarzschild metric. Gen. Relativ. Gravit. 17(4), 353–369 (1985)
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)
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)
Drouot, A.: A quantitative version of Hawking radiation. Ann. Henri Poincaré 18(3), 757–806 (2017)
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)
Franzen, A.T.: Boundedness of massless scalar waves on Kerr interior backgrounds. preprint (2017)
Franzen, A.T.: Boundedness of massless scalar waves on Reissner–Nordström interior backgrounds. Commun. Math. Phys. 343(2), 601–650 (2016)
Futterman, J.A.H., Handler, F.A., Matzner, R.A.: Scattering from black holes. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009)
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)
Graves, J.C., Brill, D.R.: Oscillatory character of Reissner–Nordström metric for an ideal charged wormhole. Phys. Rev. 2(120), 1507–1513 (1960)
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)
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)
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)
Hintz, P.: Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime. Comment. Math. Helv. 92(4), 801–837 (2017)
Hintz, P., Vasy, A.: Analysis of linear waves near the Cauchy horizon of cosmological black holes. J. Math. Phys. 58(8), 081509, 45 (2017)
Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237–238 (1963)
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)
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)
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)
Luk, J., Oh, S.J., Shlapentokh-Rothman, Y.: A scattering approach to cauchy horizon instability and applications to mass inflation. In preparation (2018)
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)
Mason, L.J., Nicolas, J.P.: Conformal scattering and the Goursat problem. J. Hyperb. Differ. Equ. 1(2), 197–233 (2004)
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)
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)
McNamara, J.M.: Instability of black hole inner horizons. Proc. R. Soc. Lond. Ser. A 358(1695), 499–517 (1978)
Melnyk, F.: Scattering on Reissner-Nordstrøm metric for massive charged spin 1/2 fields. Ann. Henri Poincaré 4(5), 813–846 (2003)
Mokdad, M.: Conformal scattering of maxwell fields on Reissner–Nordström–de sitter black hole spacetimes. arXiv preprint arXiv:1706.06993 (2017)
Müller zum Hagen, H., Seifert, H.J.: On characteristic initial-value and mixed problems. Gen. Relativ. Gravit. 8(4), 259–301 (1977)
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.) http://dlmf.nist.gov/, Release 1.0.16. Accessed 18 Sept 2017
Nicolas, J.P.: Conformal scattering on the Schwarzschild metric. Ann. Inst. Fourier (Grenoble) 66(3), 1175–1216 (2016)
Nordström, G.: On the energy of the gravitation field in Einstein’s theory. Verhandl. Koninkl. Ned. Akad. Wetenschap. Afdel. Natuurk 20, 1238–1245 (1918)
O’Neill, B.: The Geometry of Kerr Black Holes. A K Peters Ltd., Wellesley (1995)
Olver, F.W.J.: Error bounds for the Liouville–Green (or WKB) approximation. Proc. Camb. Philos. Soc. 57, 790–810 (1961)
Olver, F.W.J.: Asymptotics and special functions. AKP Classics. A K Peters Ltd., Wellesley (1997)
Reissner, H.: Über die eigengravitation des elektrischen feldes nach der Einsteinschen theorie. Annalen der Physik 355(9), 106–120 (1916)
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)
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)
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)
Taujanskas, G.: Conformal scattering of the maxwell-scalar field system on de sitter space. arXiv preprint arXiv:1809.01559 (2018)
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)
Wald, R.M.: Quantum field theory in curved spacetime and black hole thermodynamics. Chicago Lectures in Physics. University of Chicago Press, Chicago, IL (1994)
Zamorano, N.: Interior Reissner–Nordström metric and the scalar wave equation. Phys. Rev. D (3) 26(10), 2564–2574 (1982)
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Communicated by Mihalis Dafermos.
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Kehle, C., Shlapentokh-Rothman, Y. A Scattering Theory for Linear Waves on the Interior of Reissner–Nordström Black Holes. Ann. Henri Poincaré 20, 1583–1650 (2019). https://doi.org/10.1007/s00023-019-00760-z
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DOI: https://doi.org/10.1007/s00023-019-00760-z