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Scattering of Dirac Fields in the Interior of Kerr-Newman(-Anti)-de Sitter Black Holes via Comparison and Symmetry Operators

Mokdad Mokdad, Milos Provci

TL;DR

This work develops a complete scattering theory for massive, charged Dirac fields in the interior of sub-extremal Kerr-Newman-(A)dS black holes, connecting the event horizon to the Cauchy horizon. The interior’s time-dependent Hamiltonian is handled via two complementary approaches: (i) Cook-style wave operators built from a carefully chosen comparison operator, and (ii) a novel symmetry operator $\mathcal Q$ that commutes with the Hamiltonian and yields a streamlined proof. The authors establish the existence, uniqueness and asymptotic completeness of the direct and inverse wave operators, and hence a unitary scattering map $S$. The analysis combines a Regge-Wheeler-type coordinate, a Newman-Penrose tetrad, and a detailed study of the Dirac operator on $\mathcal S^2$, offering tools for perturbations, geometric interpretation, and potential extensions to broader KN(A)dS spacetimes.

Abstract

In this paper we construct a scattering theory for the massive and charged Dirac fields in the interiors of sub-extremal Kerr-Newman(-anti)-de Sitter black holes. More precisely, we show existence, uniqueness and asymptotic completeness of scattering data for such Dirac fields from the event horizon of the black hole to the Cauchy horizon. Our approach relies on constructing the wave operators where the Hamiltonian of the full dynamics is time-dependent. To prove asymptotic completeness, we use two methods. The first involves a comparison operator, while for the second we introduce and employ a symmetry operator of the Dirac equation.

Scattering of Dirac Fields in the Interior of Kerr-Newman(-Anti)-de Sitter Black Holes via Comparison and Symmetry Operators

TL;DR

This work develops a complete scattering theory for massive, charged Dirac fields in the interior of sub-extremal Kerr-Newman-(A)dS black holes, connecting the event horizon to the Cauchy horizon. The interior’s time-dependent Hamiltonian is handled via two complementary approaches: (i) Cook-style wave operators built from a carefully chosen comparison operator, and (ii) a novel symmetry operator that commutes with the Hamiltonian and yields a streamlined proof. The authors establish the existence, uniqueness and asymptotic completeness of the direct and inverse wave operators, and hence a unitary scattering map . The analysis combines a Regge-Wheeler-type coordinate, a Newman-Penrose tetrad, and a detailed study of the Dirac operator on , offering tools for perturbations, geometric interpretation, and potential extensions to broader KN(A)dS spacetimes.

Abstract

In this paper we construct a scattering theory for the massive and charged Dirac fields in the interiors of sub-extremal Kerr-Newman(-anti)-de Sitter black holes. More precisely, we show existence, uniqueness and asymptotic completeness of scattering data for such Dirac fields from the event horizon of the black hole to the Cauchy horizon. Our approach relies on constructing the wave operators where the Hamiltonian of the full dynamics is time-dependent. To prove asymptotic completeness, we use two methods. The first involves a comparison operator, while for the second we introduce and employ a symmetry operator of the Dirac equation.
Paper Structure (24 sections, 19 theorems, 197 equations)

This paper contains 24 sections, 19 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Geometric setting
  3. The Kerr-Newman(-anti)-de Sitter metrics
  4. The interior of a KN(A)dS black hole
  5. Configurations in subfamilies
  6. Hypotheses on the metric
  7. The Regge-Wheeler coordinate
  8. Adapted null tetrad
  9. The Dirac field
  10. Dirac bispinor bundle
  11. The Dirac equation and the conserved current
  12. Newman-Penrose formalism
  13. Hamiltonian formulation
  14. The Dirac operator on $\mathcal{S}^2$
  15. Scattering theory
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $r_\pm \in (0, +\infty)$ with $r_-<r_+$, and let $f(r)$ be a real-valued function of one variable ($r$) satisfyingThat is, in place of $\Delta_r$.H1--H3. Assume moreover that eq:reggewheeler holds, and let $r(\tau)$ be the inverse function of $\tau(r)$. Finally, set Then

Theorems & Definitions (51)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Remark 2.3
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 41 more