A scattering theory for the wave equation on Kerr black hole exteriors

Dafermos, Mihalis; Rodnianski, Igor; Shlapentokh-Rothman, Yakov

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 2, pp. 371-486.

Nous développons une théorie de la diffusion définitive en espace physique pour l'équation scalaire d'onde dans la région extérieure de la métrique de Kerr dans le cas sous-extrémal général $\left|a\right|<M$. En particulier, nous prouvons des résultats qui correspondent à « l'existence et l'unicité des états de diffusion » et la « complétude asymptotique » et nous montrons de plus que la matrice de diffusion qui envoie les champs de radiation sur l'horizon passé et l'infini nul passé aux champs sur l'horizon futur et l'infini nul futur est un opérateur borné. Ce dernier point nous permet de donner une théorie de réflexion superradiante dans le domaine temporel. Le fait que la matrice de diffusion est bornée montre en particulier que l'amplification maximale de solutions associées aux paquets d'ondes entrants d'énergie finie sur l'infini nul passé est bornée. En fréquence, cela correspond à l'affirmation nouvelle que les coefficients de réflexion et de transmission, convenablement normalisés, sont bornés uniformément, indépendamment des paramètres de fréquence. Nous complétons ceci de plus avec une démonstration que la réflexion superradiante amplifie effectivement l'énergie rayonnée à l'infini nul futur, pour les paquets d'ondes appropriés comme ci-dessus. Les résultats font usage essentiel d'un raffinement de notre démonstration récente [30] de la bornitude et de la décroissance des solutions du problème de Cauchy de façon à s'appliquer à la classe de solutions où seulement une énergie dégénérée est supposée finie. Nous montrons en contraste que l'application de diffusion analogue ne peut pas être définie pour la classe de solutions d'énergie finie non dégénérée. C'est dû au fait que le célèbre effet de décalage vers le rouge agit comme une instabilité de décalage vers le bleu quand on résout l'équation d'onde rétrograde.

We develop a definitive physical-space scattering theory for the scalar wave equation ${\square }_g\psi =0$ on Kerr exterior backgrounds in the general subextremal case $\left|a\right|<M$. In particular, we prove results corresponding to “existence and uniqueness of scattering states” and “asymptotic completeness” and we show moreover that the resulting “scattering matrix” mapping radiation fields on the past horizon ${\mathscr{H}}^{-}$ and past null infinity ${ℐ}^{-}$ to radiation fields on ${\mathscr{H}}^{+}$ and ${ℐ}^{+}$ is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity ${ℐ}^{-}$ is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalized reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity ${ℐ}^{+}$ of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [30] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.

Classification : 83C57, 35P25, 35L05, 81Uxx.
Keywords: Scattering theory, wave equation, Kerr solution, black holes.
Mot clés : Théorie de la diffusion, équation d'onde, solution de Kerr, trous noirs.

Andersson, L.; Blue, P. Hidden symmetries and decay for the wave equation on the Kerr spacetime, Ann. of Math., Volume 182 (2015), pp. 787-853 (ISSN: 0003-486X) | DOI | MR

Andersson, L.; Blue, P. Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, J. Hyperbolic Differ. Equ., Volume 12 (2015), pp. 689-743 (ISSN: 0219-8916) | DOI | MR

Alinhac, S., London Mathematical Society Lecture Note Series, 374, Cambridge Univ. Press, Cambridge, 2010, 118 pages (ISBN: 978-0-521-12822-3) | MR | Zbl

Andersson, N.; Laguna, P.; Papadopoulos, P. Dynamics of scalar fields in the background of rotating black holes II: A note on superradiance, Phys. Rev. D, Volume 58 (1998) (087503) | DOI

Aretakis, S. Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds, J. Funct. Anal., Volume 263 (2012), pp. 2770-2831 (ISSN: 0022-1236) | DOI | MR | Zbl

Alexakis, S.; Shao, A. Global uniqueness theorems for linear and nonlinear waves, J. Funct. Anal., Volume 269 (2015), pp. 3458-3499 (ISSN: 0022-1236) | DOI | MR

Alexakis, S.; Schlue, V.; Shao, A. Unique continuation from infinity for linear waves, Adv. Math., Volume 286 (2016), pp. 481-544 (ISSN: 0001-8708) | DOI | MR

Bachelot, A. Klein paradox and superradiance for the charged Klein-Gordon field, Sémin. Équ. Dériv. Partielles (2004) (exp. 23) | Numdam | MR | Zbl

Bachelot, A. Superradiance and scattering of the charged Klein-Gordon field by a step-like electrostatic potential, J. Math. Pures Appl., Volume 83 (2004), pp. 1179-1239 (ISSN: 0021-7824) | DOI | MR | Zbl

Bachelot, A. Gravitational scattering of electromagnetic field by Schwarzschild black-hole, Ann. Inst. H. Poincaré Phys. Théor., Volume 54 (1991), pp. 261-320 (ISSN: 0246-0211) | Numdam | MR | Zbl

Bachelot, A. Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric, Ann. Inst. H. Poincaré Phys. Théor., Volume 61 (1994), pp. 411-441 (ISSN: 0246-0211) | Numdam | MR | Zbl

Bachelot, A. Scattering of scalar fields by spherical gravitational collapse, J. Math. Pures Appl., Volume 76 (1997), pp. 155-210 (ISSN: 0021-7824) | DOI | MR | Zbl

Bachelot, A. The Hawking effect, Ann. Inst. H. Poincaré Phys. Théor., Volume 70 (1999), pp. 41-99 (ISSN: 0246-0211) | Numdam | MR | Zbl

Baskin, D.; Wang, F. Radiation fields on Schwarzschild spacetime, Comm. Math. Phys., Volume 331 (2014), pp. 477-506 (ISSN: 0010-3616) | DOI | MR | Zbl

Carter, B. Hamilton-Jacobi and Schrödinger separable solutions of Einstein's equations, Comm. Math. Phys., Volume 10 (1968), pp. 280-310 http://projecteuclid.org/euclid.cmp/1103841118 (ISSN: 0010-3616) | DOI | MR | Zbl

Chandrasekhar, S., International Series of Monographs on Physics, 69, The Clarendon Press, Oxford Univ. Press, New York, 1983, 646 pages (ISBN: 0-19-851291-0) | MR | Zbl

Christodoulou, D., Annals of Math. Studies, 146, Princeton Univ. Press, Princeton, NJ, 2000, 319 pages (ISBN: 0-691-04956-4; 0-691-04957-2) | MR | Zbl

Dafermos, M. The interior of charged black holes and the problem of uniqueness in general relativity, Comm. Pure Appl. Math., Volume 58 (2005), pp. 445-504 (ISSN: 0010-3640) | DOI | MR | Zbl

Dafermos, M.; Holzegel, G.; Rodnianski, I. A scattering theory construction of dynamical black hole spacetimes (preprint arXiv:1306.5534, to appear in J. Diff. Geom ) | MR

Dafermos, M.; Holzegel, G.; Rodnianski, I. The linear stability of the Schwarzschild solution to gravitational perturbations (preprint arXiv:1601.06467 ) | MR

Dimock, J. Scattering for the wave equation on the Schwarzschild metric, Gen. Relativity Gravitation, Volume 17 (1985), pp. 353-369 (ISSN: 0001-7701) | DOI | MR | Zbl

Dimock, J.; Kay, B. S. Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric II, J. Math. Phys., Volume 27 (1986), pp. 2520-2525 (ISSN: 0022-2488) | DOI | MR | Zbl

Dimock, J.; Kay, B. S. Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric I, Ann. Physics, Volume 175 (1987), pp. 366-426 (ISSN: 0003-4916) | DOI | MR | Zbl

Dafermos, M.; Luk, J. The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon (preprint arXiv:1710.01722 )

Daudé, T.; Nicoleau, F. Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-de Sitter black holes, Mem. Amer. Math. Soc., Volume 247 (2017), 113 pages (ISBN: 978-1-4704-2376-6; 978-1-4704-3701-5, ISSN: 0065-9266) | MR

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

Dafermos, M.; Rodnianski, I. Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: The cases $\left|a\right|\ll M$ or axisymmetry (preprint arXiv:1010.5132 )

Dafermos, M.; Rodnianski, I. A proof of Price's law for the collapse of a self-gravitating scalar field, Invent. math., Volume 162 (2005), pp. 381-457 (ISSN: 0020-9910) | DOI | MR | Zbl

Dafermos, M.; Rodnianski, I. The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math., Volume 62 (2009), pp. 859-919 (ISSN: 0010-3640) | DOI | MR | Zbl

Dafermos, M.; Rodnianski, I., XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, pp. 421-432 | DOI | MR | Zbl

Dafermos, M.; Rodnianski, I. A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds, Invent. math., Volume 185 (2011), pp. 467-559 (ISSN: 0020-9910) | DOI | MR | Zbl

Dafermos, M.; Rodnianski, I. The black hole stability problem for linear scalar perturbations, Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity (Damour, T. et al., eds.), World Scientific (2011), pp. 132-189

Dafermos, M.; Rodnianski, I., Evolution equations (Clay Math. Proc.), Volume 17, Amer. Math. Soc., Providence, RI, 2013, pp. 97-205 | MR | Zbl

Dafermos, M.; Rodnianski, I.; Shlapentokh-Rothman, Y. Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $\left|a\right|<M$ , Ann. of Math., Volume 183 (2016), pp. 787-913 (ISSN: 0003-486X) | DOI | MR

Dafermos, M.; Shlapentokh-Rothman, Y. Time-translation invariance of scattering maps and blue-shift instabilities on Kerr black hole spacetimes, Comm. Math. Phys., Volume 350 (2017), pp. 985-1016 (ISSN: 0010-3616) | DOI | MR

Dyatlov, S. Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole, Comm. Math. Phys., Volume 306 (2011), pp. 119-163 (ISSN: 0010-3616) | DOI | MR | Zbl

Futterman, J. A. H.; Handler, F. A.; Matzner, R. A., Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, Cambridge, 2009, 192 pages (ISBN: 978-0-521-11210-9) | MR | Zbl

Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T. A rigorous treatment of energy extraction from a rotating black hole, Comm. Math. Phys., Volume 287 (2009), pp. 829-847 (ISSN: 0010-3616) | DOI | MR | Zbl

Friedrichs, K. Über die Spektralzerlegung eines Integraloperators, Math. Ann., Volume 115 (1938), pp. 249-272 (ISSN: 0025-5831) | DOI | JFM | MR | Zbl

Friedlander, F. G. Radiation fields and hyperbolic scattering theory, Math. Proc. Cambridge Philos. Soc., Volume 88 (1980), pp. 483-515 (ISSN: 0305-0041) | DOI | MR | Zbl

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), Volume 19 (2017), pp. 2371-2444 (ISSN: 1435-9855) | DOI | MR

Hadamard, J., Oxford Univ. Press, London, 1923, 316 pages | JFM | MR

Häfner, D. Sur la théorie de la diffusion pour l'équation de Klein-Gordon dans la métrique de Kerr, Dissertationes Math. (Rozprawy Mat.), Volume 421 (2003), 102 pages (ISSN: 0012-3862) | MR | Zbl

Häfner, D. Creation of fermions by rotating charged black holes, Mém. Soc. Math. Fr., Volume 117 (2009), 158 pages (ISBN: 978-2-85629-284-6, ISSN: 0249-633X) | Numdam | MR | Zbl

Häfner, D., Quantum field theory and gravity, Birkhäuser, 2012, pp. 121-136 | MR | Zbl

Hawking, S. W. Particle creation by black holes, Comm. Math. Phys., Volume 43 (1975), pp. 199-220 http://projecteuclid.org/euclid.cmp/1103899181 (ISSN: 0010-3616) | DOI | MR

Hawking, S. W.; Ellis, G. F. R., Cambridge Monographs on Mathematical Physics, 1, Cambridge Univ. Press, 1973, 391 pages | MR | Zbl

Häfner, D.; Nicolas, J.-P. Scattering of massless Dirac fields by a Kerr black hole, Rev. Math. Phys., Volume 16 (2004), pp. 29-123 (ISSN: 0129-055X) | DOI | MR | Zbl

Kato, T. Scattering theory and perturbation of continuous spectra, Actes du Congrès International des Mathématiciens (Nice, 1970), t. 1, Gauthier-Villars, Paris (1971), pp. 135-140 | MR | Zbl

Kay, B. S. Instability of enclosed horizons, Gen. Relativity Gravitation, Volume 47 (2015) (ISSN: 0001-7701) | MR | Zbl

Klainerman, S. Remark on the asymptotic behavior of the Klein-Gordon equation in ${ℝ}^{n+1}$ , Comm. Pure Appl. Math., Volume 46 (1993), pp. 137-144 (ISSN: 0010-3640) | DOI | MR | Zbl

Lax, P. D.; Phillips, R. S., Pure and Applied Mathematics, 26, Academic Press, New York-London, 1967 | MR | Zbl

László, A.; Rácz, I. Superradiance or total reflection, Relativity and Gravitation (Springer Proceedings in Physics), Volume 157 (2014), pp. 119-127 | DOI | Zbl

Luk, J. Weak null singularities in general relativity, J. Amer. Math. Soc., Volume 31 (2018), pp. 1-63 (ISSN: 0894-0347) | MR | Zbl

Moschidis, G. Logarithmic local energy decay for scalar waves on a general class of asymptotically flat spacetimes, Ann. PDE, Volume 2 (2016), pp. Art. 5 (ISSN: 2199-2576) | DOI | MR | Zbl

Moschidis, G.; Shlapentokh-Rothman, Y. The degenerate redshift effect (in preparation)

Møller, C. General properties of the characteristic matrix in the theory of elementary particles. I, Kgl. Danske Vid. Selsk. Mat.-Fys. Medd., Volume 23 (1945), pp. 2-48 | MR

Nicolas, J.-P. Conformal scattering on the Schwarzschild metric, Ann. Inst. Fourier (Grenoble), Volume 66 (2016), pp. 1175-1216 http://aif.cedram.org/item?id=AIF_2016__66_3_1175_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Nicolas, J.-P. Scattering of linear Dirac fields by a spherically symmetric black hole, Ann. Inst. H. Poincaré Phys. Théor., Volume 62 (1995), pp. 145-179 (ISSN: 0246-0211) | Numdam | MR | Zbl

O'Neill, B., A K Peters, Ltd., Wellesley, MA, 1995, 381 pages (ISBN: 1-56881-019-9) | MR | Zbl

Olver, F. W. J., A K Peters, 1997, 572 pages (ISBN: 1-56881-069-5) | MR | Zbl

Rendall, A. D. Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations, Proc. Roy. Soc. London Ser. A, Volume 427 (1990), pp. 221-239 (ISSN: 0962-8444) | DOI | MR | Zbl

Reed, M.; Simon, B., Academic Press, 1979, 463 pages (ISBN: 0-12-585003-4) | MR | Zbl

Sbierski, J. Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes, Anal. PDE, Volume 8 (2015), pp. 1379-1420 (ISSN: 2157-5045) | DOI | MR | Zbl

Schlue, V. Decay of linear waves on higher-dimensional Schwarzschild black holes, Anal. PDE, Volume 6 (2013), pp. 515-600 (ISSN: 2157-5045) | DOI | MR | Zbl

Shlapentokh-Rothman, Y. Exponentially growing finite energy solutions for the Klein-Gordon equation on sub-extremal Kerr spacetimes, Comm. Math. Phys., Volume 329 (2014), pp. 859-891 (ISSN: 0010-3616) | DOI | MR | Zbl

Shlapentokh-Rothman, Y. Quantitative mode stability for the wave equation on the Kerr spacetime, Ann. Henri Poincaré, Volume 16 (2015), pp. 289-345 (ISSN: 1424-0637) | DOI | MR | Zbl

Starobinskii, A. A. Amplification of waves during reflection from rotating black hole, Zh. Eksp. Teor. Fiz., Volume 64 (1973), pp. 48-57

Teukolsky, S. A.; Press, W. H. Perturbations of a rotating lack hole III. Interaction of the hole with gravitational and electromagnetic radiation, Astrophys. J., Volume 193 (1974), pp. 443-451 | DOI

Tataru, D.; Tohaneanu, M. A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not., Volume 2011 (2011), pp. 248-292 (ISSN: 1073-7928) | MR | Zbl

Wald, R. M., Chicago Lectures in Physics, University of Chicago Press, Chicago, IL, 1994, 205 pages (ISBN: 0-226-87025-1; 0-226-87027-8) | MR | Zbl

Wheeler, J. A. On the mathematical description of light nuclei by the method of resonating group structure, Phys. Rev., Volume 52 (1937), pp. 1107-1122 | DOI | JFM

Whiting, B. F. Mode stability of the Kerr black hole, J. Math. Phys., Volume 30 (1989), pp. 1301-1305 (ISSN: 0022-2488) | DOI | MR | Zbl

Zeldovich, Y. Generating of Waves by a Rotating Body, Zh. Eksp. Teor. Fiz., Volume 14 (1971), pp. 180-181