Conformal scattering and the Goursat problem for Dirac fields in the interior of charged spherically symmetric black holes

Abstract

In this paper, we construct a conformal scattering theory for Dirac fields in the interior of a Reissner–Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution.

Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

“…Remark 6. The Goursat problems for wave equations on spinor fields were also established in some other works [53,54,64,65].…”

“…In particular, the conformal scattering theory (i.e. the geometric scattering theory) has been studied extensively from the early works by Friedlander [25][26][27][28][29], Baez et al [8], Hörmander [37] to recent ones by Mason and Nicolas [49], Joudioux [41,42], Nicolas [59], Mokdad [53,54], Taujanskas [72] and Pham [63,65].…”

“…Continuing this program, Mokdad [53,54] constructed explicitly the conformal scattering theories for the Maxwell and Dirac equations on the static exterior and dynamic interior of Reissner-Nordström de Sitter spacetimes, respectively. On the other hand, Pham [63] constructed conformal scattering theories for the scalar Reeger-Wheeler and Zerelli equations arising from the linearized gravity fields and the spin ±2 Teukolsky equations.…”

Analytic and conformal scattering in general relativity

“…Such spacetimes have an almost complete conformal compactification, the only singularity in the conformal boundary being at space-like infinity, and they can be generically time-dependent. This construction has been extended to a nonlinear Klein–Gordon equation by Jérémie Joudioux [42,43], to Maxwell fields on Schwarzschild–de Sitter-type spacetimes and to Dirac fields inside a Reissner–Nordström spacetime by Mokdad Mokdad [44] and to Maxwell potentials on asymptotically flat spacetimes by Grigalius Taujanskas and the author [45].…”

“…Recently, Nicolas Besset studied the scattering of charged scalar fields on charged backgrounds using resonance methods [13,14,15]. Moreover, the time dependent scattering of Dirac fields in the interior dynamical regions of black hole spacetimes such as Reissner-Nordström and Kerr has been investigated in the last few years by Dietrich Häfner, Mokdad Mokdad and the author [35] as well as Mokdad Mokdad with Rajai Nasser and Milos Provci [50,51]. The main principle of time dependent scattering theory is to compare a physical evolution with a simplified one in order to show that a field obeying the physical evolution approaches asymptotically a solution to the simplified comparison equation.…”