Conformal scattering on the Schwarzschild metric  [ Scattering conforme en métrique de Schwarzschild ]
Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1175-1216.

Nous montrons que les résultats de décroissance connus en métrique de Schwarzschild sont suffisants pour obtenir une théorie conforme du scattering, que nous ré-interprêtons ensuite comme une théorie analytique définie en termes d’opérateurs d’ondes, avec une dynamique de comparaison explicite associée aux congruences de géodésiques isotropes principales. Le cas de la métrique de Kerr est également discuté.

We show that existing decay results for scalar fields on the Schwarzschild metric are sufficient to obtain a conformal scattering theory. Then we re-interpret this as an analytic scattering theory defined in terms of wave operators, with an explicit comparison dynamics associated with the principal null geodesic congruences. The case of the Kerr metric is also discussed.

Reçu le : 2014-08-20
Révisé le : 2014-07-12
Accepté le : 2015-10-07
Publié le : 2016-12-13
DOI : https://doi.org/10.5802/aif.3034
Classification : 35L05,  35P25,  35Q75,  83C57
Mots clés : Scattering conforme, trous noirs, équation des ondes, métrique de Schwarzschild, problème de Goursat
@article{AIF_2016__66_3_1175_0,
     author = {Nicolas, Jean-Philippe},
     title = {Conformal scattering on the Schwarzschild metric},
     journal = {Annales de l'Institut Fourier},
     pages = {1175--1216},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {3},
     year = {2016},
     doi = {10.5802/aif.3034},
     language = {en},
     url = {www.numdam.org/item/AIF_2016__66_3_1175_0/}
}
Nicolas, Jean-Philippe. Conformal scattering on the Schwarzschild metric. Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1175-1216. doi : 10.5802/aif.3034. http://www.numdam.org/item/AIF_2016__66_3_1175_0/

[1] Andersson, Lars; Blue, Pieter Hidden symmetries and decay for the wave equation on the Kerr spacetime (http://arxiv.org/abs/0908.2265)

[2] Andersson, Lars; Blue, Pieter Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior (http://arxiv.org/abs/1310.2664)

[3] Bachelot, A. Gravitational scattering of electromagnetic field by Schwarzschild black-hole, Ann. Inst. H. Poincaré Phys. Théor., Volume 54 (1991) no. 3, pp. 261-320

[4] Baez, John C. Scattering and the geometry of the solution manifold of f+λf 3 =0, J. Funct. Anal., Volume 83 (1989) no. 2, pp. 317-332 | Article

[5] Baez, John C. Scattering for the Yang-Mills equations, Trans. Amer. Math. Soc., Volume 315 (1989) no. 2, pp. 823-832 | Article

[6] Baez, John C. Conserved quantities for the Yang-Mills equations, Adv. Math., Volume 82 (1990) no. 1, pp. 126-131 | Article

[7] Baez, John C.; Segal, Irving E.; Zhou, Zheng-Fang The global Goursat problem and scattering for nonlinear wave equations, J. Funct. Anal., Volume 93 (1990) no. 2, pp. 239-269 | Article

[8] Baez, John C.; Zhou, Zheng-Fang The global Goursat problem on R×S 1 , J. Funct. Anal., Volume 83 (1989) no. 2, pp. 364-382 | Article

[9] Dafermos, Mihalis; Rodnianski, Igor The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math., Volume 62 (2009) no. 7, pp. 859-919 | Article

[10] Dafermos, Mihalis; Rodnianski, Igor Lectures on black holes and linear waves, Evolution equations (Clay Math. Proc.) Volume 17, Amer. Math. Soc., Providence, RI, 2013, pp. 97-205 (http://arxiv.org/abs/0811.0354)

[11] Dafermos, Mihalis; Rodnianski, Igor; Shlapentokh-Rothman, Yakov A scattering theory for the wave equation on Kerr black hole exteriors (http://arxiv.org/abs/1412.8379)

[12] Dimock, J. Scattering for the wave equation on the Schwarzschild metric, Gen. Relativity Gravitation, Volume 17 (1985) no. 4, pp. 353-369 | Article

[13] Dimock, J.; Kay, Bernard S. Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. II, J. Math. Phys., Volume 27 (1986) no. 10, pp. 2520-2525 | Article

[14] Dimock, J.; Kay, Bernard S. Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. I, Ann. Physics, Volume 175 (1987) no. 2, pp. 366-426 | Article

[15] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T. Decay of solutions of the wave equation in the Kerr geometry, Comm. Math. Phys., Volume 264 (2006) no. 2, pp. 465-503 | Article

[16] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T. Erratum: “Decay of solutions of the wave equation in the Kerr geometry” [Comm. Math. Phys. 264 (2006), no. 2, 465–503], Comm. Math. Phys., Volume 280 (2008) no. 2, pp. 563-573 | Article

[17] Finster, Felix; Smoller, Joel A time-independent energy estimate for outgoing scalar waves in the Kerr geometry, J. Hyperbolic Differ. Equ., Volume 5 (2008) no. 1, pp. 221-255 | Article

[18] Friedlander, F. G. On the radiation field of pulse solutions of the wave equation, Proc. Roy. Soc. Ser. A, Volume 269 (1962), pp. 53-65

[19] Friedlander, F. G. On the radiation field of pulse solutions of the wave equation. II, Proc. Roy. Soc. Ser. A, Volume 279 (1964), pp. 386-394

[20] Friedlander, F. G. On the radiation field of pulse solutions of the wave equation. III, Proc. Roy. Soc. Ser. A, Volume 299 (1967), pp. 264-278

[21] Friedlander, F. G. Radiation fields and hyperbolic scattering theory, Math. Proc. Cambridge Philos. Soc., Volume 88 (1980) no. 3, pp. 483-515 | Article

[22] Friedlander, F. G. Notes on the wave equation on asymptotically Euclidean manifolds, J. Funct. Anal., Volume 184 (2001) no. 1, pp. 1-18 | Article

[23] Häfner, Dietrich; Nicolas, Jean-Philippe Scattering of massless Dirac fields by a Kerr black hole, Rev. Math. Phys., Volume 16 (2004) no. 1, pp. 29-123 | Article

[24] Hörmander, Lars A remark on the characteristic Cauchy problem, J. Funct. Anal., Volume 93 (1990) no. 2, pp. 270-277 | Article

[25] Joudioux, Jérémie Conformal scattering for a nonlinear wave equation, J. Hyperbolic Differ. Equ., Volume 9 (2012) no. 1, pp. 1-65 | Article

[26] Lax, Peter D.; Phillips, Ralph S. Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967, xii+276 pp. (1 plate) pages

[27] Leray, Jean Hyperbolic differential equations, The Institute for Advanced Study, Princeton, N. J., 1953, 240 pages

[28] Mason, Lionel J. On Ward’s integral formula for the wave equation in plane-wave spacetimes, Twistor Newsletter, Volume 28 (1989), pp. 17-19 (http://people.maths.ox.ac.uk/lmason/Tn/28/TN28-05.pdf)

[29] Mason, Lionel J.; Nicolas, Jean-Philippe Conformal scattering and the Goursat problem, J. Hyperbolic Differ. Equ., Volume 1 (2004) no. 2, pp. 197-233 | Article

[30] Mason, Lionel J.; Nicolas, Jean-Philippe Regularity at space-like and null infinity, J. Inst. Math. Jussieu, Volume 8 (2009) no. 1, pp. 179-208 | Article

[31] Metcalfe, Jason; Tataru, Daniel; Tohaneanu, Mihai Price’s law on nonstationary space-times, Adv. Math., Volume 230 (2012) no. 3, pp. 995-1028 | Article

[32] Nicolas, Jean-Philippe Nonlinear Klein-Gordon equation on Schwarzschild-like metrics, J. Math. Pures Appl. (9), Volume 74 (1995) no. 1, pp. 35-58

[33] Penrose, R. Zero rest-mass fields including gravitation: Asymptotic behaviour, Proc. Roy. Soc. Ser. A, Volume 284 (1965), pp. 159-203

[34] Penrose, Roger Asymptotic properties of fields and space-times, Phys. Rev. Lett., Volume 10 (1963), pp. 66-68

[35] Penrose, Roger Conformal treatment of infinity, Relativité, Groupes et Topologie (Lectures, Les Houches, 1963 Summer School of Theoret. Phys., Univ. Grenoble), Gordon and Breach, New York, 1964, pp. 563-584

[36] Penrose, Roger; Rindler, Wolfgang Spinors and space-time. Vol. 1 & 2, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1984 & 1986, x+458 pages (Two-spinor calculus and relativistic fields) | Article

[37] Price, Richard H. Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations, Phys. Rev. D (3), Volume 5 (1972), pp. 2419-2438

[38] Price, Richard H. Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields, Phys. Rev. D (3), Volume 5 (1972), pp. 2439-2454

[39] Soga, Hideo Singularities of the scattering kernel for convex obstacles, J. Math. Kyoto Univ., Volume 22 (1982/83) no. 4, pp. 729-765

[40] Tataru, Daniel; Tohaneanu, Mihai A local energy estimate on Kerr black hole backgrounds, Int. Math. Res. Not. IMRN (2011) no. 2, pp. 248-292 | Article

[41] Ward, R. S. Progressing waves in flat spacetime and in plane-wave spacetimes, Classical Quantum Gravity, Volume 4 (1987) no. 3, pp. 775-778 http://stacks.iop.org/0264-9381/4/775

[42] Whittaker, E. T. On the partial differential equations of mathematical physics, Math. Ann., Volume 57 (1903) no. 3, pp. 333-355 | Article