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

Abstract

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...

A Scattering Theory for Linear Waves on the Interior of Reissner–Nordström Black Holes

“…Comparison to scattering on the exterior of black holes. On the exterior of black holes, the scattering problem has been studied more extensively; see the pioneering works [11,13,12,2,3], the book [19] and related results on conformal scattering in [33,40,38,49]. Note that for the exterior of a Schwarzschild or Reissner-Nordström black hole, the uniform boundedness of the scattering coefficients or equivalently, the boundedness of the scattering map, can be viewed a posteriori 1 as a consequence of the global T energy identity…”

Hörmander’s method for the characteristic Cauchy problem and conformal scattering for a nonlinear wave equation

“…As mentioned before, the conformal scattering construction was extended to a non-linear wave equation [Jou12]. Similar constructions based on local energy estimates [Mok17b] were obtained on Reissner-Nordström black holes [Mok17a] for the Maxwell equations. More recently, the existence of a conformal scattering operator has been proven for Yang-Mills fields on the de Sitter background [Tau18].…”

Conformal scattering of the Maxwell-scalar field system on de Sitter space

“…. = M ∪ I , although weaker, partial compactifications leaving singularities at a finite number of points in the boundary are widely used to study, for example, black hole spacetimes [25,33,34,37,39,40]. We equipM with the rescaled (also called unphysical) metricĝ ab and call the spacetime (M,ĝ) the rescaled spacetime.…”

“…The ideas of such conformal scattering were taken up by Baez, Segal and Zhou [6][7][8][9] to study a nonlinear wave equation and to some extent Yang-Mills equations on flat space, and later by Mason and Nicolas [33,34] to study linear equations on a large class of asymptotically simple spacetimes constructed by Corvino, Schoen, Chruściel, Delay, Klainerman, Nicolò, Friedrich and others [12,13,15,16,28,29]. This spurred a programme of constructing conformal scattering theories for various fields on a variety of backgrounds and since then a number of works have appeared, many focussing on conformal scattering on black hole spacetimes 1 [23,25,37,39,40]. It should be mentioned that there have been plenty of works studying relativistic scattering theory without employing the conformal method, notably by Dimock and Kay in the 1980s [17,18] and later by Bachelot [3,4] and collaborators Nicolas, Häfner, Daudé, and Melnyk, among many others, a programme which eventually led to rigorous proofs of the Hawking effect [5,35].…”