Starting with the Dirac equation outside the event horizon of a nonextreme Kerr black hole, we develop a time-dependent scattering theory for massive Dirac particles. The explicit computation of the modified wave operators at infinity is done by implementing a time-dependent logarithmic phase shift from the free dynamics to offset the long range term in the full Hamiltonian due to the presence of the gravitational force. Analytical expressions for the wave operators are also given.
REFERENCES
1.
Abramowitz
, M.
and Stegun
, I. A.
, Handbook of Mathematical Functions
(Dover
, New York
, 1965
).2.
Bachelot
, A.
, “Gravitational scattering of electromagnetic field by a Schwarzschild black hole
,” Ann. Inst. Henri Poincare, Sect. A
54
, 261
(1991
).3.
Bachelot
, A.
, “Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric
,” Ann. Inst. Henri Poincare, Sect. A
61
, 411
(1994
).4.
Bachelot
, A.
, “Quantum vacuum polarization at the black hole horizon
,” Ann. Inst. Henri Poincare, Sect. A
67
, 182
(1997
).5.
6.
Bachelot
, A.
, “Creation of fermions at the charged black hole horizon
,” Ann. Inst. Henri Poincare, Sect. A
1
, 1043
(2000
).7.
Bachelot
, A.
and Motet-Bachelot
, A.
, “Les résonances d’un trou noir de Schwarzschild
,” Ann. Inst. Henri Poincare, Sect. A
59
, 3
(1993
).8.
Batic
, D.
, Schmid
, H.
, and Winklmeier
, M.
, “On the eigenvalues of the Chandrasekhar-Page equation
,” J. Math. Phys.
46
, 012504
(2005
).9.
Batic
, D.
and Schmid
, H.
, “The Dirac propagator in the Kerr-Newman metric
,” Prog. Theor. Phys.
116
, 517
(2006
).10.
Boyer
, R. H.
and Lindquist
, R. W.
, “Maximal analytic extension of the Kerr metric
,” J. Math. Phys.
8
, 265
(1967
).11.
Chandrasekhar
, S.
, “The solution of Dirac’s equation in the Kerr geometry
,” Proc. R. Soc. London, Ser. A
349
, 571
(1976
).12.
Daudé
, T.
, Ph.D. thesis, University of Bordoux I
, 2004
.13.
Dimock
, J.
, “Scattering for the wave equation on the Schwarzschild metric
,” Gen. Relativ. Gravit.
17
, 353
(1985
).14.
Dimock
, J.
and Kay
, B. S.
, “Scattering for massive scalar fields on Coulomb potentials and Schwarzschild metrics
,” Class. Quantum Grav.
3
, 71
(1986
).15.
Dimock
, J.
and Kay
, B. S.
, “Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric II
,” J. Math. Phys.
27
, 2520
(1986
).16.
Dimock
, J.
and Kay
, B. S.
, “Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric I
,” Ann. Phys. (N.Y.)
175
, 366
(1987
).17.
Dollard
, J. D.
, “Asymptotic convergence and the Coulomb interaction
,” J. Math. Phys.
175, 729
(1964
).18.
Dollard
, J. D.
and Velo
, G.
, “Asymptotic behavior of a Dirac particle in a Coulomb field
,” Nuovo Cimento A
45
, 801
(1966
).19.
Enss
, V.
and Thaller
, B.
, “Asymptotic observables and Coulomb scattering theory for the Dirac equation
,” Ann. Inst. Henri Poincare, Sect. A
45
, 147
(1986
).20.
Erdély
,A.
, Magnus
,W.
, Oberhettinger
,F.
, and Tricomi
,F. G.
, Higher Transcendental Functions
(McGraw-Hill
, New York
, 1953
), Vol. I
;ibid. (
McGraw-Hill
, New York
, 1953
), Vol. II
.21.
Finster
, F.
, Kamran
, N.
, Smoller
, J.
, and Yau
, S. T.
, “Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry
,” Commun. Pure Appl. Math.
53
, 902
(2000
).22.
23.
Hawking
, S. W.
, “Particle creation by black holes
,” Commun. Math. Phys.
43
, 199
(1975
).24.
Häfner
, D.
, “Complétude asymptotique pour l’équation des ondes dans une classe d’espacestemps stationnaires et asymptotiquement plats
,” Ann. Inst. Fourier
51
, 779
(2001
).25.
Häfner
, D.
and Nicolas
, J. P.
, “Scattering of massless Dirac fields by a Kerr black hole
,” Rev. Math. Phys.
16
, 29
(2004
).26.
Jin
, W. M.
, “Scattering of massive Dirac fields on the Schwarzschild black hole space-time
,” Class. Quantum Grav.
15
, 3163
(1998
).27.
Kerr
, R. P.
, “Gravitational field of a spinning mass as an example of algebraically special metrics
,” Phys. Rev. Lett.
11
, 237
(1963
).28.
Melnik
, F.
, “Scattering on Reissner-Nordstrøm metric for massive charged spin fields
,” Ann. Inst. Henri Poincare, Sect. A
4
, 813
(2003
).29.
Melnik
, F.
, “The Hawking effect for spin fields
,” Commun. Math. Phys.
244
, 483
(2004
).30.
Misner
, C. W.
, Thorne
, K. S.
, and Wheeler
, S. A.
, Gravitation
(Freeman
, New York
, 1999
).31.
Mourre
, E.
, “Absence of singular continuous spectrum for certain self-adjoint operators
,” Commun. Math. Phys.
78
, 391
(1981
).32.
Newman
, E. T.
, Couch
, E.
, Chinnapared
, K.
, Exton
, A.
, Prakash
, A.
, and Torrence
, R.
, “Metric of a rotating charged mass
,” J. Math. Phys.
6
, 918
(1965
).33.
Nicolas
, J. P.
, “Scattering of linear Dirac fields by a spherically symmetric black hole
,” Ann. Inst. Henri Poincare, Sect. A
62
, 145
(1995
).34.
Nicolas
, J. P.
, “Nonlinear Klein-Gordon equation on Schwarzschild like metrics
,” J. Math. Pures Appl.
74
, 35
(1995
).35.
Sà Barreto
, A.
and Zworski
, M.
, “Distribution of resonances for spherical black holes
,” Math. Res. Lett.
4
, 103
(1998
).36.
Wald
, R. M.
, General Relativity
(The University of Chicago Press
, Chicago
, 1984
).© 2007 American Institute of Physics.
2007
American Institute of Physics
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