Direct and inverse scattering at fixed energy for massless charged Dirac fields by Kerr-Newman-de Sitter black holes

TL;DR

This paper develops a theory for scattering of massless charged Dirac fields around Kerr-Newman-de Sitter black holes, establishing existence, completeness, and using these results to uniquely determine the black hole's metric from fixed-energy scattering data.

Contribution

It introduces a novel approach linking time-dependent and stationary scattering operators for Dirac fields in black hole spacetimes, enabling unique inverse problem solutions.

Findings

Proved existence and asymptotic completeness of wave operators.

Derived an explicit expression for the scattering matrix at fixed energy.

Established uniqueness of black hole metric from fixed-energy scattering data.

Abstract

In this paper, we study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, we establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. We also use the miraculous property (quoting Chandrasekhar) - that the Dirac equation can be separated into radial and angular ordinary differential equations - to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, we use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitely by Frobenius method) and the Complex Angular Momentum technique on the radial equation, we are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy.