Propagation estimates for Dirac operators and application to scattering theory

Daudé, Thierry

doi:10.5802/aif.2074

Cited by 11 publications

(11 citation statements)

References 7 publications

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“…Time-dependent scattering theories on Kerr black holes were obtained by D. Häfner in 2003 [33] and in 2004 by D. Häfner and J.-P. Nicolas for massless Dirac fields using a Mourre estimate [34]. In 2005 T. Daudé produced scattering theories for Dirac fields in various spacetimes [19,20,21], and in 2010 he published results on time-dependent scattering for charged Dirac [22], before moving to several works on inverse scattering in general relativity. In 2014 M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman developed scattering theory for the scalar wave equation on Kerr exterior backgrounds in the subextremal case [18].…”

Section: Analytic Scattering: Brief Literature Overviewmentioning

confidence: 99%

Conformal Scattering of Maxwell fields on Reissner–Nordström–de Sitter Black Hole Spacetimes

Mokdad¹

2019

Annales De l'Institut Fourier

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We construct a complete conformal scattering theory for Maxwell fields in the static exterior region of a Reissner-Nordstrøm-de Sitter black bole spacetime. This is done using uniform energy decay results that we obtain in a separate paper [47], to show that the trace operators are injective and have closed ranges. We then solve the Goursat problem (characteristic Cauchy problem) for Maxwell fields on the null boundaries showing that the trace operators are also surjective.1 arXiv:1706.06993v1 [gr-qc] 21 Jun 2017 profile to gain information about the medium and the obstacles it contains. This reconstruction is the aim of inverse scattering. Analytic Scattering: Brief literature OverviewScattering theory proved to be a useful tool in the framework of general relativity to study the asymptotic influence of the geometry of spacetime on fields. Although in this current work we do not use an analytic approach to scattering, we very briefly touch on the history of the subject because this is part of the origin of conformal scattering and it helps to understand what new features the conformal approach bring to the domain. Scattering theory in black holes spacetimes played an essential role in the rigorous description of phenomena like superradiance, the Hawking effect, and quasi-normal modes (resonances of black holes which are related to gravitational waves). In 1980 S. Chandrasekhar [12] used the stationary approach, resorting to a Fourier transformation in time, to study quasi-linear modes of black hole spacetimes such as Schwarzschild, Reissner-Nordstrøm, and Kerr. Chandrasekhar's work systematically used the Newman-Penrose formalism to develop stationary scattering theories described in terms of the scattering matrix of transmission and reflection coefficients. And around the same time, M. Reed and B. Simon published "Scattering Theory" the third volume of their classic series [57]. Then time-dependent scattering (based on the comparison of dynamics) of classical and quantum fields on the exterior of a Schwarzschild black hole were first studied by J. Dimock in 1985 [24] and by J. Dimock and 1987 [25, 26, 27]. And in the 1990's, A. Bachelot produced an important series of papers starting with scattering theories for classical fields, Maxwell in 1990 and 1991 [1, 3], Klein-Gordon in 1994 [4] and on the Hawking effect for a spherical gravitational collapse in 1997 [2], 1999 [5] and 2000 [6]. J.-P. Nicolas in 1995 developed a scattering theory for classical massless Dirac fields [49], and a work on a non linear Klein-Gordon equation on the Schwarzschild metric (and other similar geometries) with partial scattering results obtained by conformal methods in 1995 [50]. W.M. Jin in 1998 contributed to the subject with a construction of wave operators in the massive case [36], and F. Melnyk in 2003 obtained a complete scattering for massive charged Dirac fields [44] and the Hawking effect for charged, massive spin-1/2 fields [45]. In 1999 I. Laba and A. Soffer [39] obtained complete scattering for the nonlinear Schrödinger...

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Section: Analytic Scattering: Brief Literature Overviewmentioning

confidence: 99%

Conformal Scattering of Maxwell fields on Reissner–Nordström–de Sitter Black Hole Spacetimes

Mokdad¹

2019

Annales De l'Institut Fourier

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“…We finally mention a few other rigorous results on the dynamics of quantum mechanical waves in the Kerr geometry which we we cannot describe in this article. First, the scattering theory for Dirac particle has been developed in [36,21,2].…”

Section: Dynamics Of Dirac Wavesmentioning

confidence: 99%

Linear waves in the Kerr geometry: A mathematical voyage to black hole physics

Finster

Kamran

Smoller

et al. 2009

Bull. Amer. Math. Soc.

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Abstract. This paper gives a survey of wave dynamics in the Kerr spacetime geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin s = 0, 1 2 , 1, 2, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.

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“…However, it does not give rise to a description of the full time-dependent dynamics of Dirac particles in the Kerr geometry. This was first analyzed in the framework of scattering theory in [1,5,10,21,23,24,29]. More recently, a somewhat different approach to define the dynamics of Dirac particles using an integral spectral representation of the Dirac propagator in the Hamiltonian framework was developed [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics

Röken

2017

Gen Relativ Gravit

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The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr geometry is described in the Newman-Penrose formalism by a regular Carter tetrad and the Dirac spinors and matrices are defined in a chiral Newman-Penrose dyad representation. Applying Chandrasekhar's mode ansatz, the Dirac equation is separated into radial and angular systems of ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are derived, and the decay of the associated errors is analyzed. Moreover, specific aspects of the angular eigenfunctions and eigenvalues are discussed. Finally, as an application, the scattering of massive Dirac particles by the gravitational field of a rotating Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating coordinates and for the construction of an integral spectral representation of the Dirac propagator that yields the dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon. Contents

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Propagation estimates for Dirac operators and application to scattering theory

Cited by 11 publications

References 7 publications

Conformal Scattering of Maxwell fields on Reissner–Nordström–de Sitter Black Hole Spacetimes

Conformal Scattering of Maxwell fields on Reissner–Nordström–de Sitter Black Hole Spacetimes

Linear waves in the Kerr geometry: A mathematical voyage to black hole physics

The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics

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