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Scattering of linear Dirac fields by a spherically symmetric black-hole. (English) Zbl 0826.53072

This paper presents a time-dependent scattering theory for the linear massless Dirac system on a general Schwarzschild-type metric which covers all the usual cases of spherical black-holes. After presenting the covariant generalization of the linear Dirac system on Schwarzschild-type metrics, the existence of classical wave operators is demonstrated using Cook’s method. There is also demonstrated the asymptotic completeness of classical wave operators at the horizon of the black-hole and at infinity based on the results of A. Bachelot and A. Motet-Bachelot [Ann. Inst. Henri Poincaré, Phys. Théor. 59, No. 1, 3-68 (1993; Zbl 0793.53094)]. The methods in this paper seem to be suitable for demonstrating the existence and asymptotic completeness of Dollard- modified wave operators at infinity.
Reviewer: I.Gottlieb (Iaşi)

MSC:

53Z05 Applications of differential geometry to physics
83C57 Black holes

Citations:

Zbl 0793.53094
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References:

[1] A. Bachelot , Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric , Internal publication, U.R.A. 226 , 1993 , to appear in Ann. Inst. Henri-Poincaré, Physique Théorique . Numdam | MR 1311537 | Zbl 0809.35141 · Zbl 0809.35141
[2] A. Bachelot , Gravitational Scattering of Electromagnetic Field by Schwarzschild Black-Hole , Ann. Inst. Henri-Poincaré-Physique théorique , Vol. 54 , No. 3 , 1991 , pp. 261 - 320 . Numdam | MR 1122656 | Zbl 0743.53037 · Zbl 0743.53037
[3] A. Bachelot and A. Motet-Bachelot , Les résonances d’un Trou Noir de Schwarzschild , Ann. Inst. Henri-Poincaré, Physique théorique , Vol. 59 , No. 1 , 1993 , pp. 3 - 68 . Numdam | MR 1244181 | Zbl 0793.53094 · Zbl 0793.53094
[4] D.R. Brill and J.A. Wheeler , Interaction of Neutrinos and Gravitational Fields , Revs. Modern Phys. Vol. 29 , 3 , 1957 , pp. 465 - 479 . MR 91828 | Zbl 0078.43503 · Zbl 0078.43503
[5] Y. Choquet-Bruhat and C. Dewitt , Analysis, manifolds and physics, Part I: basics, Revised edition , 1982 , Part II: 92 applications, 1989 , North Holland . MR 678940 | Zbl 0682.58002 · Zbl 0682.58002
[6] Th. Damour , Black-Hole eddy currents , Phys. Rev. D18 , Vol. 10 , 1978 , pp. 3598 - 3604 .
[7] B.S. Dewitt , The space-time approach to quantum field theory , in Relativité, groupes et topologie, les Houches , 1983 , North Holland , 1984 . · Zbl 0596.58014
[8] J. Dimock , Scattering for the wave equation on the Schwarzschild metric , Gen. Relativ. Gravitation , Vol. 17 , No. 4 , 1985 , pp. 353 - 369 . MR 788801 | Zbl 0618.35088 · Zbl 0618.35088
[9] J. Dmock and B.S. Kay , Classical and Quantum Scattering theory for linear scalar fields on the Schwarzschild metric I , Ann. Phys. Vol. 175 , 1987 , pp. 366 - 426 . MR 887979 | Zbl 0628.53080 · Zbl 0628.53080
[10] J. Dollard and G. Velo , Asymptotic behavior of a Dirac particle in a Coulomb field , Il Nuovo Cimento , Vol. 45 , 1966 , pp. 801 - 812 .
[11] V. Enss and B. Thaller , Asymptotic observables and Coulomb scattering for the Dirac equation , Ann. Inst. Henri-Poincaré , Vol. 45 , 2 , 1986 , pp. 147 - 171 . Numdam | MR 866913 | Zbl 0615.47008 · Zbl 0615.47008
[12] I.M. Gel’fand and Z. Ya . Sapiro , Representations of the group of rotations of 3- dimensional space and their applications , Amer. Math. Soc. Transl. , Vol. 2 , 2 , 1956 , pp. 207 - 316 . MR 76290 | Zbl 0070.25902 · Zbl 0070.25902
[13] J.-P. Nicolas , Non linear Klein-Gordon equation on Schwarzschild-like metrics , to appear in J. Math. Pures et appliquées. [14] J.-P. Nicolas , Opérateur de diffusion pour le système de Dirac en métrique de Schwarzschild , to appear in C. R. Acad. Sci. Paris , Vol. 318 , 1994 . MR 1272337 | Zbl 0810.35137 · Zbl 0810.35137
[14] R. Penrose and W. Rindler , Spinors and space-time, Cambridge monographs on mathematical physics , Vol. 1 , Two-spinor calculus in relativistic fields , Cambridge University Press , 1984 . MR 776784 | Zbl 0538.53024 · Zbl 0538.53024
[15] M. Reed and B. Simon , Methods of modern mathematical physics , Vol. III , 1979 , Academic Press . Zbl 0405.47007 · Zbl 0405.47007
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