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An Integral Representation for the Dirac Propagator in the Reissner-Nordström Geometry in Eddington-Finkelstein Coordinates

Felix Finster, Christoph Krpoun

TL;DR

This work derives a rigorous integral representation for the Dirac propagator in Reissner-Nordström spacetime using horizon-penetrating Eddington-Finkelstein coordinates. By separating variables (Chandrasekhar method) into radial and angular parts, the authors construct radial Jost solutions and Green’s matrices, enabling a resolvent-based expression of the spectral measure and a propagator formula that remains valid across the event and Cauchy horizons. The key contribution is the explicit linkage between the spectral data and the radial ODE solutions, culminating in a compact propagator formula that isolates horizon-crossing dynamics and depends on transmission coefficients. This framework lays the groundwork for spectral analysis of the fermionic signature operator and paves the way for understanding fermion propagation in charged black-hole spacetimes with precise analytic control.

Abstract

The Cauchy problem for the massive Dirac equation is studied in the Reissner-Nordström geometry in horizon-penetrating Eddington-Finkelstein-type coordinates. We derive an integral representation for the Dirac propagator involving the solutions of the ordinary differential equations which arise in the separation of variables. Our integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon.

An Integral Representation for the Dirac Propagator in the Reissner-Nordström Geometry in Eddington-Finkelstein Coordinates

TL;DR

This work derives a rigorous integral representation for the Dirac propagator in Reissner-Nordström spacetime using horizon-penetrating Eddington-Finkelstein coordinates. By separating variables (Chandrasekhar method) into radial and angular parts, the authors construct radial Jost solutions and Green’s matrices, enabling a resolvent-based expression of the spectral measure and a propagator formula that remains valid across the event and Cauchy horizons. The key contribution is the explicit linkage between the spectral data and the radial ODE solutions, culminating in a compact propagator formula that isolates horizon-crossing dynamics and depends on transmission coefficients. This framework lays the groundwork for spectral analysis of the fermionic signature operator and paves the way for understanding fermion propagation in charged black-hole spacetimes with precise analytic control.

Abstract

The Cauchy problem for the massive Dirac equation is studied in the Reissner-Nordström geometry in horizon-penetrating Eddington-Finkelstein-type coordinates. We derive an integral representation for the Dirac propagator involving the solutions of the ordinary differential equations which arise in the separation of variables. Our integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon.
Paper Structure (13 sections, 19 theorems, 93 equations)

This paper contains 13 sections, 19 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Dirac Equation in a Globally Hyperbolic Spacetime
  4. The Dirac Equation in the Reissner-Nordström Geometry
  5. Functional Analytic Preparations
  6. Computation of the Resolvent
  7. The Green's Matrix
  8. The Radial Jost Solutions
  9. Construction of the Green's Matrix
  10. Computation of the Green's Matrix
  11. Integral Representation of the Dirac Propagator
  12. Abstract Representation
  13. Main Theorem

Key Result

Lemma 2.1

In a stationary spacetime, the Hamiltonian $H$ with domain ${\mathscr{D}}(H)$ is a symmetric operator on $\mathscr{H}$.

Theorems & Definitions (35)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 25 more