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

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

doi:10.24033/asens.2358

A scattering theory for the wave equation on Kerr black hole exteriors
[Une théorie de la diffusion pour l'équation d'onde sur les extérieurs du trou noir de Kerr]

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

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

Résumé
Abstract

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 $| a | < M$ . 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 $□_{g} ψ = 0$ on Kerr exterior backgrounds in the general subextremal case $| a | < M$ . 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 $ℋ^{-}$ and past null infinity $ℐ^{-}$ to radiation fields on $ℋ^{+}$ 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.

Publié le : 2018-11-12

DOI : 10.24033/asens.2358

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.

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     title = {A scattering theory for the wave equation  on {Kerr} black hole exteriors},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {371--486},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {2},
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Dafermos, Mihalis; Rodnianski, Igor; Shlapentokh-Rothman, Yakov. A scattering theory for the wave equation  on Kerr black hole exteriors. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 2, pp. 371-486. doi : 10.24033/asens.2358. http://www.numdam.org/articles/10.24033/asens.2358/

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