no. 3
Scattering theory for Dirac fields near an extreme Kerr–de Sitter black hole
[Théorie de la diffusion pour des champs de Dirac au voisinage d’un trou noir extrême de type Kerr–de Sitter]
Borthwick, Jack A.1
1 Univ. Brest UMR CNRS 6205 Laboratoire de Mathématiques de Bretagne Atlantique 6. av Victor Le Gorgeu CS 93837 29238 BREST cedex 3 (France)
Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 919-997.

Dans cet article, nous développons une théorie de la diffusion pour des champs de Dirac massifs en métrique Kerr–de Sitter extrême, dans la région située entre l’horizon (double) du trou noir et l’horizon cosmologique. L’outil principal de la construction est l’existence d’un opérateur conjugué au sens de la théorie de Mourre. Par ailleurs, bien que les effets de la rotation soient amplifiées au voisinage de l’horizon double, nous montrons qu’il est néanmoins possible de se ramener à un problème de diffusion unidimensionnelle moyennant une décomposition ad-hoc de l’espace de Hilbert.

In this paper, we construct a scattering theory for classical massive Dirac fields near the “double” horizon of an extreme Kerr–de Sitter blackhole. Our main tool is the existence of a conjugate operator in the sense of Mourre theory. Additionally, despite the fact that effects of the rotation are “amplified” near the double horizon, we show that one can still reduce our study to a 1-dimensional problem through an appropriate decomposition of the Hilbert space.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3553
Classification : 35P25, 35Q75, 83C57, 35Q41
Keywords: Scattering, extremal black hole, Kerr–de Sitter blackhole, Dirac equation, Mourre theory.
Mot clés : Scattering, trou noir extrême, trou noir de de Sitter–Kerr, équation de Dirac, théorie de Mourre.

Borthwick, Jack A. 1

1 Univ. Brest UMR CNRS 6205 Laboratoire de Mathématiques de Bretagne Atlantique 6. av Victor Le Gorgeu CS 93837 29238 BREST cedex 3 (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Borthwick, Jack A. Scattering theory for Dirac fields near an extreme Kerr–de Sitter black hole. Annales de l'Institut Fourier, Tome 73 (2023) no. 3, pp. 919-997. doi : 10.5802/aif.3553. https://aif.centre-mersenne.org/articles/10.5802/aif.3553/

[1] Abrikosov, A. A. Jr. Dirac operator on the Riemann Sphere (2002) (https://arxiv.org/abs/hep-th/0212134)

[2] Amrein, Werner O.; Boutet de Monvel, Anne; Georgescu, Vladimir C 0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Modern Birkhäuser Classics, XIV, Birkhäuser Basel, 1996, xvi+460 pages | DOI | Zbl

[3] Aretakis, Stefanos Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations I, Comm. Math. Phys., Volume 307 (2011) no. 1, pp. 17-63 | DOI | MR | Zbl

[4] Aretakis, Stefanos Stability and instability of extreme Reissner–Nordström black hole spacetimes for linear scalar perturbations II, Ann. Henri Poincaré, Volume 12 (2011) no. 8, pp. 1491-1538 | DOI | MR | Zbl

[5] Batic, D.; Schmid, H. The Dirac propagator in the extreme Kerr metric, J. Phys. A, Volume 40 (2007) no. 44, pp. 13443-13451 | DOI | MR | Zbl

[6] Belgiorno, Francesco; Cacciatori, Sergio L. The absence of normalizable time-periodic solutions for the Dirac equation in the Kerr–Newman-dS black hole background, J. Phys. A, Math. Theor., Volume 42 (2009) no. 13, 135207, 15 pages | DOI | Zbl

[7] Belgiorno, Francesco; Cacciatori, Sergio L. The Dirac equation in Kerr–Newman-AdS black hole background, J. Math. Phys., Volume 51 (2010) no. 3, 033517, 32 pages | DOI | MR | Zbl

[8] Bizoń, Piotr; Friedrich, Helmut A remark about wave equations on the extreme Reissner-Nordström black hole exterior, Classical Quantum Gravity, Volume 30 (2013) no. 6, 065001, 6 pages | DOI | MR | Zbl

[9] Borthwick, Jack Maximal Kerr–de Sitter spacetimes, Classical Quantum Gravity, Volume 35 (2018) no. 21, 215006, 38 pages | DOI | MR | Zbl

[10] Camporesi, Roberto; Higuchi, Atsushi On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys., Volume 20 (1996) no. 1, pp. 1-18 | DOI | MR | Zbl

[11] Dafermos, Mihalis; Rodnianski, Igor; Shlapentokh-Rothman, Yakov A scattering theory for the wave equation on Kerr black hole exteriors, Ann. Sci. Éc. Norm. Supér. (4), Volume 51 (2018) no. 2, pp. 371-486 | DOI | MR | Zbl

[12] Dappiaggi, Claudio; Moretti, Valter; Pinamonti, Nicola Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime, Adv. Theor. Math. Phys., Volume 15 (2011) no. 2, pp. 355-447 http://projecteuclid.org/euclid.atmp/1337951927 | DOI | MR | Zbl

[13] Daudé, Thierry Scattering theory for Dirac fields in various spacetimes of the General Relativity, Thèse, Université Sciences et Technologies – Bordeaux I (2004)

[14] Daudé, Thierry Time-dependent scattering theory for charged Dirac fields on a Reissner–Nordström black hole, J. Math. Phys., Volume 51 (2010) no. 10, 102504, 57 pages | DOI | MR | Zbl

[15] Daudé, Thierry; Nicoleau, François Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr–Newman–de Sitter black holes, Memoirs of the American Mathematical Society, 1170, American Mathematical Society, Providence, RI, 2017, iv+113 pages | DOI | MR | Zbl

[16] Dereziński, Jan; Gérard, Christian Scattering theory of classical and quantum N-particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997, xii+444 pages | DOI | MR | Zbl

[17] Dollard, J.; G., Velo Asymptotic Behaviour of a Dirac Particle in a Coulomb Field, Il Nuovo Cimento A (1965-1970), Volume 45 (1966) no. 4, pp. 801-812 | DOI

[18] Evans, Lawrence C. Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010, xxii+749 pages | DOI | MR | Zbl

[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), Volume 19 (2017) no. 8, pp. 2371-2444 | DOI | MR | Zbl

[20] Georgescu, V.; Măntoiu, M. On the spectral theory of singular Dirac type Hamiltonians, J. Operator Theory, Volume 46 (2001) no. 2, pp. 289-321 | MR | Zbl

[21] Gérard, Christian; Häfner, Dietrich; Wrochna, Michal The Unruh state for massless fermions on Kerr spacetime and its Hadamard property (2020) (to appear in Ann. Sci. Éc. Norm. Supér.) | HAL

[22] Gérard, Christian; Łaba, Izabella Multiparticle quantum scattering in constant magnetic fields, Mathematical Surveys and Monographs, 90, American Mathematical Society, Providence, RI, 2002, xiv+242 pages | DOI | MR | Zbl

[23] Geroch, Robert Spinor structure of space-times in general relativity. I, J. Mathematical Phys., Volume 9 (1968), pp. 1739-1744 | DOI | MR | Zbl

[24] Geroch, Robert Spinor structure of space-times in general relativity. II, J. Mathematical Phys., Volume 11 (1970), pp. 343-348 | DOI | MR | Zbl

[25] Häfner, Dietrich Sur la théorie de la diffusion pour l’équation de Klein–Gordon dans la métrique de Kerr, Dissertationes Math., Volume 421 (2003), p. 102 | DOI | MR | Zbl

[26] 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 | DOI | MR | Zbl

[27] Helffer, Bernard; Sjöstrand, Johannes Équation de Schrödinger avec champ magnétique et équation de Harper, Journées équations aux dérivées partielles (1987), pp. 1-9 (Exp. No. VI) | DOI | Numdam | Zbl

[28] Idelon-Riton, Guillaume On scattering theory for the massive Dirac equation in Schwarzschild-Anti- de Sitter space-time and applications, Ph. D. Thesis, Université Grenoble Alpes (2016)

[29] Ince, E. L. Ordinary Differential Equations, Dover Publications, 1956

[30] Kato, Tosio Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin Heidelberg, 1980 | Zbl

[31] Lax, Peter D. Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002, xx+580 pages | MR | Zbl

[32] Mourre, E. Absence of singular continuous spectrum for certain self-adjoint operators, Commun. Math. Phys., Volume 78 (1981), pp. 391-408 | DOI | MR | Zbl

[33] Nicolas, J.-P. Dirac fields on asymptotically flat space-times, Dissertationes Math., Volume 408 (2002), p. 85 | DOI | MR | Zbl

[34] Penrose, Roger; Rindler, Wolfgang Spinors and space-time. Two-spinor calculus and relativistic fields, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1984, x+458 pages | DOI | MR | Zbl

[35] Reed, Michael; Simon, Barry Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press Inc., 1975, xv+361 pages | MR | Zbl

[36] Reed, Michael; Simon, Barry Methods of modern mathematical physics. III. Scattering theory, Academic Press Inc., 1979, xv+463 pages | MR | Zbl

[37] Sigal, I.M.; Soffer, A. Local Decay and Velocity bounds for Quantum Propagation (1988) (Princeton University)

[38] Trautman, Andrzej Spin structures on hypersurfaces and the spectrum of the Dirac operator on spheres, Spinors, twistors, Clifford algebras and quantum deformations (Sobótka Castle, 1992) (Fundamental Theories of Physics), Volume 52, Kluwer Academic Publishers, 1993, pp. 25-29 | DOI | MR | Zbl

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