An Integral Representation for the Dirac Propagator in the Reissner-Nordstr\"om Geometry in Eddington-Finkelstein Coordinates

TL;DR

This paper derives an integral representation for the Dirac propagator in Reissner-Nordström spacetime using Eddington-Finkelstein coordinates, enabling analysis of Dirac particles across the event horizon.

Contribution

It introduces a novel integral representation of the Dirac propagator in Reissner-Nordström geometry, accounting for horizon-crossing dynamics.

Findings

Provides an explicit integral formula involving separated solutions.

Describes Dirac particle behavior across the event horizon.

Applicable up to the Cauchy horizon.

Abstract

The Cauchy problem for the massive Dirac equation is studied in the Reissner-Nordstr\"om 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.

Equations

$$\psi (\tau )=e^{-i\tau H}{\psi }_0={\int }_{\sigma (H)}{e}^{-i\omega \tau }\text{d}{E}_{\omega }{\psi }_0,$$

$$⟨\cdot |\cdot ⟩:$$

$$⟨\psi |\varphi ⟩:={\int }_{\text{mycal}M}\prec \psi |\varphi {\succ }_x\text{d}{\mu }_{\text{mycal}M}.$$

$$(\mathcal{D}-m)\psi =0,$$

$$\mathcal{D}=i{G}^k{\partial }_k+\mathcal{B}:C^{\infty }(\text{mycal}M,S\text{mycal}M)⟶C^{\infty }(\text{mycal}M,S\text{mycal}M),$$

$$\{G^j,G^k\}=2g^{jk}\text{mathbbm}{1}_{S_x\text{mycal}M}.$$

$$(\psi |\varphi )={\int }_{\text{mycal}N}\prec \psi |{\nu }^j{G}_j\varphi {\succ }_x\text{d}{\mu }_{\text{mycal}N}(x),$$

$$i\partial_{\tau}\psi=H\psi\qquad\text{with}\qquad H:=-(G^{\tau})^{-1}\bigg{(}i\sum_{\alpha=1}^{3}G^{\alpha}\partial_{\alpha}+{\mathcal{B}}-m\bigg{)}\>,$$

$$\mathcal{D}(H)=C_0^{\infty }(\text{mycal}N,S\text{mycal}M).$$

$$0={\partial }_{\tau }(\psi |\varphi ).$$

$$0=(\partial_{\tau}\psi|\phi)+(\psi|\partial_{\tau}\phi)=-i\Big{(}(H\psi|\phi)-(\psi|H\phi)\Big{)}\>.$$

$$g=$$

$$-r^2\text{d}\vartheta \otimes \text{d}\vartheta -r^2\sin (\vartheta {)}^2\text{d}\phi \otimes \text{d}\phi$$

$$\frac{\text{d}u}{\text{d}r}=\frac{r^2}{\Delta }.$$

$$r_{\pm }=M\pm \sqrt{M^2-Q^2}$$

$$\Psi$$

$${\Gamma }_{\text{trafo}}D(\mathcal{D}-m)D^{-1}\Psi$$

$$D:=\frac{\sqrt{r}}{r_{+}}\left[\begin{array}{cccc}|\Delta {|}^{1/2}& 0& 0& 0\\ 0& r_{+}& 0& 0\\ 0& 0& r_{+}& 0\\ 0& 0& 0& |\Delta {|}^{1/2}\end{array}\right],{\Gamma }_{\text{trafo}}:=r\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right].$$

$$\mathcal{R}$$

$$\mathcal{A}$$

$$\text{centering}{\mathcal{D}}_0:=\text{@add@centering}$$

$${\mathcal{D}}_1:=$$

$${\mathcal{L}}_{\pm }:=$$

$$\prec \psi |\varphi {\succ }_x=-⟨\Psi ,\left(\begin{array}{cc}0& \text{mathbbm}{1}_{{\mathbb{C}}^2}\\ \text{mathbbm}{1}_{{\mathbb{C}}^2}& 0\end{array}\right)\Phi {⟩}_{{\mathbb{C}}^4}.$$

$$\Psi =e^{-i(k+\frac{1}{2})\phi }\frac{1}{r_{+}}\left[\begin{array}{c}{X}_{+}(\tau ,r)Y_l(\vartheta {)}_{+}\\ r_{+}{X}_{-}(\tau ,r)Y_l(\vartheta {)}_{-}\\ r_{+}{X}_{-}(\tau ,r)Y_l(\vartheta {)}_{+}\\ X_{+}(\tau ,r)Y_l(\vartheta {)}_{-}\end{array}\right]\text{with}\omega \in \mathbb{R}\text{ and }k,l\in \mathbb{Z},$$

$$\mathcal{R}\Psi =\xi \Psi \text{and}\mathcal{A}\Psi =-\xi \Psi$$

$$\left[\begin{array}{cc}(2r^2-\Delta ){\partial }_{\tau }+\Delta {\partial }_r& |\Delta {|}^{1/2}(imr-\xi )\\ -ϵ(\Delta )|\Delta {|}^{1/2}(imr+\xi )& -\Delta ({\partial }_{\tau }-{\partial }_r)\end{array}\right]\left(\begin{array}{c}{X}_{+}(\tau ,r)\\ r_{+}{X}_{-}(\tau ,r)\end{array}\right)$$

$$\displaystyle\bigg{(}\begin{bmatrix}0&{\mathcal{L}}_{-}\\ -{\mathcal{L}}_{+}&0\end{bmatrix}-\xi\,\mathbbm{1}_{\mathbb{C}^{4}}\bigg{)}\begin{pmatrix}Y_{l}(\vartheta)_{+}\\[1.99997pt] Y_{l}(\vartheta)_{-}\end{pmatrix}$$

$$\displaystyle\big{\langle}e^{-i(k+\frac{1}{2})\varphi}\,Y_{kl}(\vartheta),e^{-i(k^{\prime}+\frac{1}{2})\varphi}\,Y_{k^{\prime}l^{\prime}}(\vartheta)\big{\rangle}_{L^{2}(S^{2})}=\delta_{k,k^{\prime}}\>\delta_{l,l^{\prime}}\>.$$

$$i{\partial }_{\tau }X(\tau ,r)=H_{\xi }X(\tau ,r)$$

$$H_{\xi }:=\underset{=:C^{-1}(r)\in M(2,\mathbb{C})}{\underbrace{\left[\begin{array}{cc}-\frac{i}{(2r^2-\Delta )}& 0\\ 0& \frac{i}{\Delta }\end{array}\right]}}\left[\begin{array}{cc}\Delta {\partial }_r& |\Delta {|}^{1/2}(imr-\xi )\\ -ϵ(\Delta )|\Delta {|}^{1/2}(imr+\xi )& \Delta {\partial }_r\end{array}\right],$$

$$\left[\begin{array}{cc}-(2r^2-\Delta )i\omega +\Delta {\partial }_r& |\Delta {|}^{1/2}(imr-\xi )\\ -ϵ(\Delta )|\Delta {|}^{1/2}(imr+\xi )& \Delta (i\omega +{\partial }_r)\end{array}\right]\left(\begin{array}{c}{X}_{+}^{(\omega )}(r)\\ r_{+}{X}_{-}^{(\omega )}(r)\end{array}\right)$$

$$X^{(\omega )}(u)=\left[\begin{array}{cc}\cosh (\Theta )& \sinh (\Theta )\\ \sinh (\Theta )& \cosh (\Theta )\end{array}\right]\left(\begin{array}{c}{f}_{\infty }^{(1)}{e}^{i{\Phi }_{+}(u)}\\ f_{\infty }^{(2)}{e}^{-i{\Phi }_{-}(u)}\end{array}\right)+E_{\infty }(u)$$

$$\displaystyle\centering\Phi_{\pm}(u):=\pm w_{1,2}\,u+M\bigg{(}\pm 2\omega\pm\dfrac{m^{2}}{w_{1,2}}\bigg{)}\ln(u)\@add@centering$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 The Dirac Equation in a Globally Hyperbolic Spacetime

4. 2.2 The Dirac Equation in the Reissner-Nordström Geometry

5. 3 Functional Analytic Preparations

6. 4 Computation of the Resolvent

7. 4.1 The Green’s Matrix

8. 4.2 The Radial Jost Solutions

9. 4.3 Construction of the Green’s Matrix

10. 4.4 Computation of the Green’s Matrix

11. 5 Integral Representation of the Dirac Propagator

12. 5.1 Abstract Representation

13. 5.2 Main Theorem

1. Introduction

The Dirac equation in curved spacetime describes the dynamics of quantum mechanical waves in the presence of classical gravitational fields. The propagation of Dirac waves in black hole geometries is of particular interest with respect to Hawking radiation, the stability of black holes and the fermionic signature operator. So far, this problem has been studied mainly in the exterior region of the black hole [6, 5]. Here we turn attention to the behavior of Dirac waves across and inside the event horizon. We consider the Reissner-Nordström geometry, which describes a spherical symmetric and charged black hole. Our starting point is the Dirac equation in Eddington-Finkelstein coordinates as derived in [14]. Our main result is to derive a corresponding integral representation of the propagator

[TABLE]

where $\text{\rm{d}}E_{\omega}$ is the spectral measure of the Dirac Hamiltonian $H$, and $\psi_{0}$ is smooth initial data with compact support. We express the spectral measure explicitly in terms of the fundamental solutions of the radial ordinary differential equation (ODE) arising in Chandrasekhar’s separation of variables (see Theorem 5.5). We remark that similar results have been derived previously in the Kerr geometry [11]. The novel feature of the present paper is the charge of the black hole. Moreover, we work out the integral representation in more detail and simplify the formulas considerably. Our integral representation will be used in a follow-up paper to compute the spectrum of the fermionic signature operator [8].

The paper is organized as follows. In Section 2 we give the necessary preliminaries on the Dirac equation in globally hyperbolic spacetimes and on its separation in the Reissner-Nordström geometry in Eddington-Finkelstein-type coordinates. In Section 3 we introduce Dirichlet-type boundary conditions inside the Cauchy horizon and show that the resulting Hamiltonian is essentially self-adjoint. Moreover, we express the spectral measure via Stone’s formula in terms of the resolvent. In Section 4 the resolvent is computed in terms of the fundamental solutions. To this end, we construct Jost solutions and use the conservation law for the radial flux in order to compute the Green’s matrix. In Section 5 we use the obtained formulas for the resolvent in order to express the spectral measure explicitly in terms of the fundamental solutions. This gives the simple and useful analytic expression for the Dirac propagator as stated in Theorem 5.5.

2. Preliminaries

2.1. The Dirac Equation in a Globally Hyperbolic Spacetime

We begin with preliminaries on the Dirac equation in globally hyperbolic spacetimes, following the presentation in [9]. Thus let $(\mycal M,g)$ be a four dimensional, smooth, globally hyperbolic Lorentzian spin manifold. For the signature of the metric we use the convention $(+,-,-,-)$. As proven in [1], $\mycal M$ admits a smooth foliation $(\mycal N_{\tau})_{\tau\in\mathbb{R}}$ by Cauchy hypersurfaces. We denote the corresponding spinor bundle by $S\mycal M$. Its fibers $S_{x}\mycal M$ are endowed with an inner product $\mathopen{\prec}.|.\mathclose{\succ}_{x}$ of signature $(2,2)$. The smooth sections of the spinor bundle are denoted by $C^{\infty}(\mycal M,S\mycal M)$. Likewise, $C_{0}^{\infty}(\mycal M,S\mycal M)$ are the smooth sections with compact support. We also refer to these sections as wave functions and usually denote them by $\psi$ or $\phi$. On the wave functions, we introduce the Lorentz invariant inner product

[TABLE]

We consider the Dirac equation for a given mass parameter $m\geq 0$. We write the Dirac equation as

[TABLE]

where the Dirac operator takes the form

[TABLE]

and $G^{k}:T_{x}\mycal M\longrightarrow L(S_{x}\mycal M)$ are the Dirac matrices. They fulfill the anti-commutation relations

[TABLE]

One can understand this map as a representation of the Clifford multiplication in components of the general Dirac matrices. We will use the Feynman dagger notation reading $\not{\nu}=G^{j}\nu_{j}$. The connection part of the covariant derivative is summarized in the term ${\mathcal{B}}$. We remark that the Dirac equation can be written alternatively as $\mathcal{D}=iG^{j}\nabla_{j}$, where $\nabla$ is the Levi-Civita spin connection on $S\mycal M$. For more details on the Dirac equation in curved spacetimes we refer to [9] or [15].

Given initial data on a Cauchy surface, the Dirac equation admits unique global solutions. Choosing compactly supported initial data, due to finite propagation speed, the resulting solution also has compact support on any other Cauchy surface. Such solutions are referred to as being spatially compact. The smooth, spatially compact solutions are denoted by $C^{\infty}_{\text{sc}}(\mycal M,S\mycal M)$. On such solutions, one has the scalar product

[TABLE]

where $\nu$ is the future directed-normal on $\mycal N$ (due to current conservation, the scalar product is in fact independent of the choice of $\mycal N$; for details see [9, Section 2]). Forming the completion gives the Hilbert space $(\mathscr{H},(\,\cdot\,|\,\cdot\,))$.

In this paper, we restrict attention to stationary spacetimes, meaning that there is a Killing field $K$ which is asymptotically timelike (for the general definition see [16]). We always choose the foliation $(\mycal N_{\tau})_{\tau\in\mathbb{R}}$ such that the Killing field is given by $K=\partial_{\tau}$. In this case, it is useful to write spacetime as a product $\mycal M=\mathbb{R}\times\mycal N$. Moreover, we can write the Dirac equation in the Hamiltonian form

[TABLE]

where the Hamiltonian $H$ is an operator acting on $\mathscr{H}$ with dense domain

[TABLE]

Lemma 2.1**.**

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

Proof.

Since the scalar product in (2.1) is independent of the choice of the Cauchy surface, we know that for all $\psi,\phi\in{\mathscr{D}}(H)$,

[TABLE]

Since the Dirac matrices $G^{k}$, as well as the normal vector field $\nu$ and the volume form do not depend on $\tau$, we only need to differentiate the wave functions. We thus obtain

[TABLE]

This concludes the proof. ∎

2.2. The Dirac Equation in the Reissner-Nordström Geometry

We work in Eddington Finkelstein coordinates $(\tau,r,\vartheta,\varphi)$ in the range $\mathbb{R}\times(0,\infty)\times(0,\pi)\times[0,2\pi)$ as defined in [2, 14]. In these coordinates, the metric takes the form

[TABLE]

with $\Delta\equiv\Delta(r)=r^{2}-2Mr+Q^{2}$ and $\tau=t+u-r$. Here $u$ is the Regge-Wheeler coordinate (“tortoise coordinate”) which is defined in terms of $r$ by

[TABLE]

The zeros of the function $\Delta$ denoted by

[TABLE]

describe the event horizon and Cauchy horizon, respectively. The region $\{r>r_{-}\}$ outside the Cauchy horizon is a globally hyperbolic spacetime. The interior of the Cauchy horizon $\{r<r_{-}\}$, however, is not globally hyperbolic because of the spacetime singularity at $r=0$. Our spacetime has the topology $\mycal M\cong\mathbb{R}^{2}\times S^{2}$.

In [14] the Dirac equation was computed and separated in a specific gauge in which the Dirac matrices are in the Weyl representation. Starting from this representation, it is most convenient to transform the Dirac wave function and the Dirac operator as

[TABLE]

with the transformation matrices

[TABLE]

Here $\mathcal{R}$ and $\mathcal{A}$ are the radial and angular operators given by

[TABLE]

where the linear operators ${\mathcal{D}}_{0/1},\,{\mathcal{L}}_{\pm}:C^{\infty}(\mycal M,S\mycal M)\longrightarrow C^{\infty}(\mycal M,S\mycal M)$ have the from

[TABLE]

In this formulation, the spin inner product takes the form

[TABLE]

Employing the separation ansatz

[TABLE]

we obtain the eigenvalue problems

[TABLE]

with a separation constant $\xi$. In this way, the Dirac equation decouples into a radial and angular ODE of the form

[TABLE]

In view of (2.11), the angular operator in (2.15) does not involve $\tau$-derivatives (note that this is no longer the case in the Kerr geometry). This angular operator is an essentially selfadjoint operator on $L^{2}(S^{2},\mathbb{C}^{2})$ with dense domain $C^{\infty}(S^{2},\mathbb{C}^{2})$, having a purely discrete spectrum (for details see [6, Section 3 and Appendix A]). More specifically, the angular operator is the spin-weighted spherical operator for $s=\frac{1}{2}$ as analyzed in [13]. We denote the corresponding orthonormal eigenvector basis by ${}_{\frac{1}{2}}Y_{kl}=e^{-i(k+\frac{1}{2})\varphi}\,Y_{kl}(\vartheta)$ with $l,k\in\mathbb{Z}$, i.e.

[TABLE]

Restricting attention to one angular momentum mode, it suffices to solve the PDE (2.14) in $\tau$ and $r$ for $\xi=\lambda_{kl}$, where $\lambda_{kl}$ denotes the corresponding angular eigenvalue. Since the angular eigenfunctions are orthonormal, the general Cauchy problem can be solved by decomposing the initial data into angular momentum modes, solving the PDE (2.14) for each mode and taking the superposition.

Similar to (2.2), the PDE (2.14) can again be written in the Hamiltonian form:

Lemma 2.2**.**

The radial equation (2.14) can be written in the Schrödinger-type form

[TABLE]

with the Hamiltonian $H_{\xi}$ given by

[TABLE]

where $X(\tau,r)$ is the radial part of the wave function from (2.13).

Proof.

Follows by direct computation. ∎

Since $\tau$ is a Killing-field we can write $X(\tau,r)=e^{-i\omega\tau}X^{(\omega)}(r)$ and get a first order radial ODE given by

[TABLE]

More details on the previous steps and the asymptotics of the solutions to the radial ODE (2.18) are worked out in [14]. We here state the main results, which can be obtained from [14, Theorem 1.1] by setting the angular momentum $a$ equal to zero.

Lemma 2.3**.**

For the case $|\omega|<m$ one solution of the radial ODE (2.18) has exponential decay and the other one exponential decrease for $u\rightarrow\infty$. In the case $|\omega|>m$, on the other hand, the solutions have following asymptotics,

1. (i)

(Asymptotics at infinity)* Let $w_{1}\in\mathbb{C}$ be the root of $\omega^{2}-m^{2}$ contained in the convex hull of $\mathbb{R}_{+}$ and $\mathbb{R}_{+}\cdot i$ and $w_{2}=-w_{1}$ the other root, and let $\Theta:=\frac{1}{4}\ln(\frac{\omega-m}{\omega+m})$, then there is $f_{\infty}:=(f_{\infty}^{(1)},f_{\infty}^{(2)})^{T}\in\mathbb{R}^{2}\setminus\{0\}$ with*

[TABLE]

for the asymptotic phases

[TABLE]

and for an error function $E_{\infty}(u)$ with polynomial decay. More precisely, there is $c\in\mathbb{R}_{+}$ with

[TABLE] 2. (ii)

(Asymptotics at the Cauchy horizon)* For every non-trivial solution $X$,*

[TABLE]

with $h_{r_{-}}:=(h_{r_{-}}^{(1)},h_{r_{-}}^{(2)})^{T}\in\mathbb{R}^{2}\setminus\{0\}$, with $E_{r_{-}}$ such that for $r$ sufficiently close to $r_{-}$ and suitable constants $a,b\in\mathbb{R}_{+}$,

[TABLE]

3. Functional Analytic Preparations

In this section we bring the scalar product (2.1) into a more explicit form. Moreover, we set up the Cauchy problem in a way where spectral methods in Hilbert space become applicable. In order to obtain a unitary time evolution, we must consider a spacetime region which includes the Cauchy horizon, and we must introduce reflecting boundary conditions on the timelike surface $r=r_{0}<r_{-}$. Moreover, we must make sure that the surfaces $\{\tau=\text{const}\}$ are spacelike. Noting that the metric coefficient $g_{rr}$ in (2.3) has a zero at

[TABLE]

we are led to choosing $r_{0}$ in the interval

[TABLE]

We thus consider the spacetime region $M\subset\mycal M$ defined by

[TABLE]

This spacetime is foliated by the space-like hypersurfaces $(N_{\tau})_{\tau\in\mathbb{R}}$ given by

[TABLE]

Each hypersurface has the boundary $\partial N_{\tau}=\partial M\cap N_{\tau}\simeq S^{2}$. These $N_{\tau}\subset\mycal N_{\tau}$ give rise to a space-like foliation of $M$. The Killing field $K=\partial_{\tau}$ is tangential to $\partial M$ and is timelike on this hypersurface (because $g(K,K)=g_{00}=\Delta(r_{0})/r^{2}>0$). We denote the corresponding spinor bundle by $SM$.

We first compute the scalar product in this spacetime region.

Lemma 3.1**.**

The scalar product (2.1) can be written as

[TABLE]

(where the capital letters always denote the transformed Dirac wave function (2.5)) with

[TABLE]

Note that, in view of the lower bound in (3.1), the matrix $\Gamma$ is strictly positive, showing that (3.2) is indeed a scalar product.

Proof.

By direct computation, we see that the volume form is

[TABLE]

It remains to compute the combination $G^{j}\nu_{j}$. The normal $\nu$ is determined by the four equations

[TABLE]

By direct computation, we find

[TABLE]

This corresponds to the co-vector

[TABLE]

Reading off the transformed Gamma-matrix $\overline{G^{\tau}}$ from (2.7), we need to transform it back with the relation

[TABLE]

Using the form of the spin inner product (2.12), we obtain

[TABLE]

concluding the proof. ∎

In order to obtain a Cauchy problem with a well-defined, unitary time evolution, we need to introduce suitable boundary conditions at $r=r_{0}$. Following the procedure in [10], we introduce the reflecting boundary conditions

[TABLE]

where $\not{n}$ is the inner normal on $\partial M$. For the Cauchy problem, we set $N=N_{\tau}|_{\tau=0}$. We choose initial data in the class

[TABLE]

We denote the Hilbert space generated by these functions (with the scalar product computed in Lemma 3.1) by $\mathscr{H}_{N}$. The following lemma was proved in [10].

Lemma 3.2**.**

For initial data $\psi_{0}$ in the class (3.4), the Cauchy problem with boundary conditions

[TABLE]

has a unique, global solution $\psi\in C^{\infty}_{\text{sc}}(M,SM)$. Evaluating this solution at subsequent times $\tau$ and $\tau^{\prime}$ gives rise to a unique unitary time evolution operator leaving the domain $C^{\infty}_{\text{\rm{init}}}(N)$ invariant, i.e.

[TABLE]

Having a dense domain which is invariant under the time evolution makes it possible to apply Chernoff’s method [3] to obtain the following result. More details can be found in [10].

Lemma 3.3**.**

The Dirac Hamiltonian $H$ in the Reissner-Nordström geometry in Eddington-Finkelstein coordinates with domain of definition

[TABLE]

is essentially self-adjoint on the Hilbert space $\mathscr{H}_{N}$.

For ease in notation, we denote the self-adjoint extension of the Hamiltonian again by $H$. By the spectral theorem for self-adjoint operators we can express the solution of the Cauchy problem for any $\psi_{0}\in\mathscr{H}_{N}$ as

[TABLE]

where $\omega\in\sigma(H)$ are the spectral values, and $\text{\rm{d}}E_{\omega}$ is the corresponding spectral measure of $H$. As explained after (2.16), from now on we restrict attention to one angular momentum mode and consider the corresponding two-component Dirac equation (2.17). For the later explicit analysis, it is helpful to rewrite the spectral measure with the help of Stone’s formula.

Lemma 3.4**.**

The solution of the Cauchy problem can be written as

[TABLE]

where $(H_{\xi}-\omega\mp i\varepsilon)^{-1}$ are the resolvents of the Dirac Hamiltonian $H_{\xi}$ in the upper and lower half-planes and $X_{0}(r)\in C^{\infty}_{\text{\rm{init}}}(N)$ is the initial data for a fixed angular mode $k,l$.

Proof.

Using the properties of the spectral measure, we obtain

[TABLE]

Applying Stone’s formula (see [17, Theorem VII.13]) gives the result. ∎

4. Computation of the Resolvent

4.1. The Green’s Matrix

Our next task is to calculate the resolvent $(H_{\xi}-\omega\mp i\varepsilon)^{-1}$ appearing in Proposition 3.4 in the upper and lower half planes. It is most convenient to work with the Hamiltonian for the radial equation (2.14) after separation of variables, denoted by $H_{\xi}$.

We introduce the abbreviation $\omega\mp i\varepsilon=:\omega_{\varepsilon}$. Moreover, it is useful for the computations to write the operator $H_{\xi}-\omega_{\varepsilon}\mathbbm{1}_{\mathbb{C}^{2}}$ as

[TABLE]

with the radial differential operator

[TABLE]

This makes it possible to write the radial equation as

[TABLE]

Our strategy is to invert this differential operator with the help of the so-called Green’s matrix $G(r;r^{\prime})_{\omega_{\varepsilon}}$, being a distributional solution to the equation

[TABLE]

The Green’s matrix can be expressed in terms of the fundamental solutions of the radial equations. We postpone the detailed computations to the next section (Section 4.2). Here we explain how the resolvent can be computed from the Green’s matrix.

Lemma 4.1**.**

Let $X_{0}\in C^{\infty}_{\text{\rm{init}}}(N)$ be initial data for a fixed angular mode $k,l$. Then the resolvent acting on $X_{0}$ can be expressed in terms of the Green’s matrix $G(r;r^{\prime})_{\omega_{\varepsilon}}$ in (4.3) as

[TABLE]

where $C(r)$ is the matrix

[TABLE]

Proof.

By direct computation using (4.3), one verifies that the operator $H_{\xi}-\omega_{\varepsilon}\mathbbm{1}_{\mathbb{C}^{2}}$ in (4.1) has the inverse

[TABLE]

This gives the result. ∎

4.2. The Radial Jost Solutions

We now define the fundamental solutions which we will later use to compute the Green’s matrix. Our method is to construct Jost solutions $\mathcal{J}_{\pm}$. In preparation, we re-writing the first order radial system of ODEs stemming from the matrix Dirac equation in (2.18) into two second order scalar equations, sometimes referred to as the Jost equation. We consider the three regions

[TABLE]

separately. Choosing the Regge-Wheeler coordinate $u\in\mathbb{R}$ in one of these regions, the radial solutions $X^{(\omega)}(r(u))$ satisfy the equations

[TABLE]

where $E_{\omega,l}(u)$ and $K^{\pm}_{\omega,l}(u)_{\pm}$ are smooth functions. The explicit form of these functions is rather involved and will not given here. However, for the construction of the Jost solutions, it suffices to analyze the asymptotic form of these functions as $u\rightarrow\pm\infty$. We recall the basic idea for $X^{(\omega)}_{+}$ and the asymptotics $u\rightarrow+\infty$. We rewrite (4.5) in the form

[TABLE]

with a complex constant $\Omega_{+}$ and a potential $W^{+}$. Having the complex coefficients, the differential operator on the left side can be inverted with the help of an explicitly given Green’s kernel $S(u,v)$, i.e.

[TABLE]

This makes it possible to formulate a Lippmann-Schwinger equation of the form

[TABLE]

Using that $W$ decays as $u\rightarrow\infty$, one can perform an expansion in powers of $W$,

[TABLE]

Similar to the Picard-Lindelöf iteration, one can show that this series converges, giving rise to a unique solution with prescribed asymptotics as $u\rightarrow\infty$.

This method has been worked out for the Dirac and wave equations in the Kerr geometry in Boyer-Lindquist coordinates in [7, 12] as well in Eddington-Finkelstein coordinates in [11] (for basics and more details see also [4]).

Lemma 4.2**.**

For every angular momentum mode $k$ and $l$ and $\omega_{\varepsilon}\in\{\omega_{\varepsilon}\neq 0\,|\,\omega_{\varepsilon}\in\mathbb{C}\}$, there are unique radial Jost solution to the complexified Schrödinger-type equations (4.5) with the asymptotic boundary conditions

[TABLE]

for suitable complex numbers $\Omega_{\pm}$. Furthermore, these solutions are analytic and smooth in $u$ and $\omega_{\varepsilon}$.

Proof.

In the case $\operatorname{Im}\omega_{\varepsilon}>0$, the Jost solutions have been constructed in [7, 12]. By complex conjugation of the Schrödinger-type equation, we obtain corresponding solutions for $\operatorname{Im}\omega_{\varepsilon}<0$. ∎

We note that the asymptotics (4.9) and (4.10) for $\mathcal{J}_{\pm}$ can also be written as

[TABLE]

with a decaying error term $E(u)$. This shows that $\Omega_{+}$ encodes the asymptotic phase and amplitude of the wave.

We now collect the Jost solutions in the respective regions (4.4) and introduce a convenient notation. We can use the same notation for the Jost solutions constructed near the event horizon from inside and outside (and similarly near the Cauchy horizons). because these solutions have the same asymptotics in the Regge-Wheeler coordinates of the respective spacetime region. Note that the Regge-Wheeler coordinate $u$ tends to $-\infty$ at the event horizon and to $+\infty$ at the Cauchy horizon.

In order to find Jost solutions, we must ensure that the integral in (4.7) and the series (4.8) converge. Depending on the signs of $\operatorname{Im}{(\omega_{\varepsilon})}$ and $\operatorname{Re}{(\omega_{\varepsilon})}$, we thus obtain different solutions as compiled in the next lemma.

Lemma 4.3**.**

We introduce possible Jost solutions with the following asymptotics:

1. (i)

Near spatial infinity:

[TABLE] 2. (ii)

Near the event horizon $r_{+}$:

[TABLE]

where, depending on $\operatorname{Im}(\omega)$, the above functions are well-defined and in $L^{2}_{\text{\rm{loc}}}$ near the event horizon:

[TABLE] 3. (iii)

Near the Cauchy horizon $r_{-}$:

[TABLE]

where, depending on $\operatorname{Im}(\omega)$, the above functions are well-defined and in $L^{2}_{\text{\rm{loc}}}$ near the Cauchy horizon:

[TABLE]

Here $f_{\infty,1/2}\neq 0$ and $h_{\pm,1/2}\neq 0$ are constants and $b\in\mathbb{R}_{+}$.

We remark that, in the limit $\varepsilon\searrow 0$, the asymptotics of the solutions near infinity and near the horizons was worked out in [14, Theorem 1.1].

Proof.

We begin with the Jost solutions near spatial infinity. According to Theorem 2.3, the solutions have the plane wave asymptotics with asymptotic phases given by (2.19). It suffices to consider the linear term in (2.19), because for large $u$ it dominates the logarithm. In the case $|\omega_{\varepsilon}|>|m|$, by a Taylor expansion of the square root we obtain

[TABLE]

Since the imaginary term determines the decay behavior, it suffices to consider the second term. This shows that the combinations of real and imaginary parts in (4.11) and (4.12) ensure that the exponentials $e^{\pm i\phi_{\pm}(u)}$ decay at infinity. This property is precisely what is needed in order for the Jost solutions to be well-defined. In the remaining case $|\omega_{\varepsilon}|\leq|m|$, using that $\operatorname{Re}\sqrt{m^{2}-\omega_{\varepsilon}^{2}}>0$ we conclude that the exponentials $e^{\pm i\phi_{\pm}(u)}$ again decay at infinity.

Now, we look at the Jost functions near the event and Cauchy horizons. We only consider the Cauchy horizon, because the event horizon can be treated similarly. Near the Cauchy horizon, the fundamental solutions have the plane wave asymptotics as worked out in Theorem 2.3. Using that the potential $W$ in the Lippmann-Schwinger equation (4.7) decays exponentially, it turns out that there are two convergent Jost solutions $\widehat{\mathcal{J}}_{-}$ and $\widecheck{\mathcal{J}}_{-}(u)$ defined for $\omega_{\varepsilon}$ close to the real axis, which form a fundamental system (for details see [7, Section 3]). ∎

Finally, the boundary condition at $r=r_{0}$ as initial data for the radial ODE, we obtain a solution $\mathcal{J}_{\partial M}$ with the following boundary values.

1. (iv)

At the boundary $\partial M$:

[TABLE]

4.3. Construction of the Green’s Matrix

Our next goal is to express the Green’s matrix (4.3) in terms of Jost solutions. To this end, we make the general “gluing ansätze” depending on $r^{\prime}$ and $\operatorname{Im}(\omega_{\varepsilon})$

[TABLE]

where $\Phi_{1}$ and $\Phi_{2}$ are the Jost solutions defined in the respective regions

[TABLE]

Here and in what follows, it is more convenient to work again with the radial variable $r\in[r_{0},\infty)$. The corresponding Regge-Wheeler coordinate is obtained in the respective regions (i.e., inside the Cauchy horizon, between the horizons, and in the asymptotic end) by (2.4). We point out that when extending the solutions $\Phi_{L}$ or $\Phi_{R}$ across the event or Cauchy horizons, we need to make sure that their $L^{2}$-norms are finite, i.e.

[TABLE]

where we work with the $L^{2}$-scalar product in (3.2).

We now explain in detail how the functions $\Phi_{L}$ and $\Phi_{R}$ can be chosen. We consider the cases separately when $\omega_{\varepsilon}$ is in the upper and lower half plane.

1. (i)

$\operatorname{Im}(\omega_{\varepsilon})<0\text{ and }|\omega_{\varepsilon}|\leq m\quad\text{or}\quad\operatorname{Im}(\omega_{\varepsilon})<0\text{ and }\operatorname{Re}(\omega_{\varepsilon})<0\text{ and }|\omega_{\varepsilon}|>m$:

We begin with $r^{\prime}$ outside the event horizon. Due to the choice of of $\omega_{\varepsilon}$, $\widehat{\mathcal{J}_{\infty}}(r)$ decays as $r\rightarrow\infty$, and we can set

[TABLE]

For $r<r^{\prime}$ we have two possibilities. One is to take the solution $\widehat{\mathcal{J}_{+}}$. This solution decays exponentially at the event horizon. Extending it by zero up to $r=r_{0}$ gives a solution in $L^{2}$. This leads us to the ansatz

[TABLE]

The other possibility is to take another fundamental solution and to extend it across the event and Cauchy horizons up to the boundary at $r=r_{0}$. Because of the negative imaginary part of $\omega_{\varepsilon}$, only the constant function $\widecheck{\mathcal{J}}_{-}(r)$ is square integrable at the Cauchy horizon. However, for almost all values of $\omega$, this solution does not satisfy the boundary conditions at $r_{0}$. For this reason, we are forced to choose the fundamental solution (4.13) with two free parameters $a,b\,\in\mathbb{C}$ (here the superscript $\infty$ clarifies that $r^{\prime}$ lies in the asymptotic end). These solutions are shown in Figure 1.

We next consider the case $r_{-}<r^{\prime}\leq r_{+}$. By the same argumentation as above, there are no non-trivial solutions crossing the Cauchy horizon. This leads to the conclusion that the wave needs to vanish at $r^{\prime}$ and is continuously extended by zero to $r_{0}$.

For $r>r_{+}$, on the other hand, we have two possible functions. The constant function needs to be the same inside and outside the event horizon, but the decaying solution can have two different prefactors in the linear combination for waves. This gives us enough freedom to match the solutions. The different fundamental solutions are illustrated in Figure 2. We begin with the most general linear combination and simplify it afterward. It has the form

[TABLE]

Now we can express $\widehat{\mathcal{J}_{\infty}}(r)$ as a linear combination of $\widecheck{\mathcal{J}_{+}}(r)$ and $\widehat{\mathcal{J}_{+}}(r)$. This results in

[TABLE]

with two free parameters $a,c_{L}\in\mathbb{C}$ (the parameter $c$, on the other hand, is determined by the matching conditions on the event horizon). Thus, we have the solutions

[TABLE]

It remains to consider the case $r_{0}\leq r^{\prime}\leq r_{-}$. On the left side of $r^{\prime}$ the boundary conditions at $r_{0}$ admit, up to a multiple, a unique solution, i.e.

[TABLE]

For $r>r_{-}$, only the solution $\widecheck{\mathcal{J}}_{-}(r)$ can be extended in $L^{2}$ across the Cauchy horizon. This solution must be extended also across the event horizon and must be matched to the decaying solution (see Figure 3 for a detailed sketch). Now the matching across the Cauchy horizon determines the constants $c_{L}$ and $c$, whereas the matching across the event horizon determines $a$ and $c_{R}$. We conclude that

[TABLE]

with two free parameter $d,e\in\mathbb{C}$ (and $c_{L},c_{R},c$ and $a$ fixed by the matching conditions). 2. (ii)

$\operatorname{Im}(\omega_{\varepsilon})>0\text{ and }|\omega_{\varepsilon}|\leq m\quad\text{or}\quad\operatorname{Im}(\omega_{\varepsilon})>0\text{ and }\operatorname{Re}(\omega_{\varepsilon})<0\text{ and }|\omega_{\varepsilon}|>m$:

The procedure to find the fundamental solutions in this case is similar as in the first case. One only needs to keep in mind that the roles of the event and Cauchy horizons are interchanged, in the sense that at the Cauchy horizon, both fundamental solutions are in $L^{2}$, whereas at the event horizon, one fundamental solution is singular.

The remaining cases are similar. We summarize the results in following Lemma.

Lemma 4.4**.**

The extended Jost solutions $\Phi_{1}$ and $\Phi_{2}$ can be expressed with the free parameter $c_{1}$ and $c_{2}$ as well as the constants $a_{1},\,a_{2},\,a_{3}$ and $a_{4}$ as

1. (i)

$\operatorname{Im}(\omega_{\varepsilon})<0\text{ and }|\omega_{\varepsilon}|\leq m\quad\text{or}\quad\operatorname{Im}(\omega_{\varepsilon})<0\text{ and }\operatorname{Re}(\omega_{\varepsilon})<0\text{ and }|\omega_{\varepsilon}|>m$:

**

• •

$\Phi^{\infty}_{1}(r)=c_{1}\,\Theta(r-r_{+})\widehat{\mathcal{J}_{+}}(r)$

$\Phi^{\infty}_{2}(r)=c_{2}\,\widehat{\mathcal{J}_{\infty}}(r)$ *

• •

$\Phi^{r_{+}}_{1}(r)=c_{1}\,\Theta(r_{+}-r)\widehat{\mathcal{J}_{+}}(r)$

*$\Phi^{r_{+}}_{2}(r)=c_{2}\,\Theta(r-r_{+})\widehat{\mathcal{J}_{\infty}}(r)+a_{1}\,\Theta(r_{+}-r)\widecheck{\mathcal{J}_{+}}(r)$

• •

$\Phi^{r_{-}}_{1}(r)=c_{1}\,\mathcal{J}_{\partial M}(r)$

$\Phi^{r_{-}}_{2}(r)=c_{2}\,\Theta(r-r_{-})\widecheck{\mathcal{J}_{-}}(r)+a_{1}\,\Theta(r-r_{-})\Theta(r_{+}-r)\widehat{\mathcal{J}_{+}}\\ \text{}\qquad\quad+a_{2}\,\Theta(r-r_{+})\widehat{\mathcal{J}_{+}}(r)+a_{3}\,\Theta(r-r_{-})\Theta(r_{+}-r)\widecheck{\mathcal{J}_{+}}(r)+a_{4}\,\Theta(r-r_{+})\widehat{\mathcal{J}_{\infty}}(r)$ *

** 2. (ii)

$\operatorname{Im}(\omega_{\varepsilon})>0\text{ and }|\omega_{\varepsilon}|\leq m\quad\text{or}\quad\operatorname{Im}(\omega_{\varepsilon})>0\text{ and }\operatorname{Re}(\omega_{\varepsilon})<0\text{ and }|\omega_{\varepsilon}|>m$:

**

• •

$\Phi^{\infty}_{1}(r)=c_{1}\,\Theta(r-r_{+})\widecheck{\mathcal{J}_{+}}(r)+a_{1}\,\Theta(r-r_{-})\Theta(r_{+}-r)\widecheck{\mathcal{J}_{-}}(r)\\ \text{}\qquad\quad+a_{2}\,\Theta(r-r_{-})\Theta(r_{+}-r)\widehat{\mathcal{J}_{-}}(r)+a_{3}\,\Theta(r_{-}-r)\widehat{\mathcal{J}_{-}}(r)+a_{4}\,\Theta(r_{-}-r)\mathcal{J}_{\partial M}(r)$

*$\Phi^{\infty}_{2}(r)=c_{2}\,\widecheck{\mathcal{J}_{\infty}}(r)$

• •

$\Phi^{r_{+}}_{1}(r)=c_{1}\,\Theta(r_{-}-r)\mathcal{J}_{\partial M}+a_{1}\,\Theta(r-r_{-})\widecheck{\mathcal{J}_{-}}(r)$

$\Phi^{r_{+}}_{2}(r)=c_{2}\,\Theta(r-r_{-})\widehat{\mathcal{J}_{-}}(r)$

**

• •

$\Phi^{r_{-}}_{1}(r)=c_{1}\,\Theta(r_{-}-r)\mathcal{J}_{\partial M}(r)$

*$\Phi^{r_{-}}_{2}(r)=c_{2}\,\Theta(r_{-}-r)\widehat{\mathcal{J}_{-}}(r)$

** 3. (iii)

$\operatorname{Im}(\omega_{\varepsilon})<0\text{ and }|\omega_{\varepsilon}|\leq m\quad\text{or}\quad\operatorname{Im}(\omega_{\varepsilon})<0\text{ and }\operatorname{Re}(\omega_{\varepsilon})>0\text{ and }|\omega_{\varepsilon}|>m$:

**

• •

Similar to the first case, but with interchanged $\widehat{\mathcal{J}_{\infty}}$ and $\widecheck{\mathcal{J}_{\infty}}$ when $|\omega_{\varepsilon}|>m$.

** 4. (iv)

$\operatorname{Im}(\omega_{\varepsilon})>0\text{ and }\quad|\omega_{\varepsilon}|\leq m\quad\text{or}\quad\operatorname{Im}(\omega_{\varepsilon})>0\text{ and }\operatorname{Re}(\omega_{\varepsilon})>0\text{ and }|\omega_{\varepsilon}|>m$:

**

• •

Similar to the second case, but with interchanged $\widehat{\mathcal{J}_{\infty}}$ and $\widecheck{\mathcal{J}_{\infty}}$ when $|\omega_{\varepsilon}|>m$.

Proof.

Details of the derivation are discussed above. ∎

4.4. Computation of the Green’s Matrix

In this subsection, we will use the global Jost solutions constructed above to compute the Green’s matrix. For scalar Jost solutions, this construction is well-known and uses the conservation of the Wronskian. In our setting of two-component solutions, we make use instead of the conservation of the Dirac current $\mathopen{\prec}\Phi,\gamma^{\mu}\Phi\mathclose{\succ}$ in radial directions. More precisely, we define a Gram matrix, referred to as the radial flux matrix. We want to highlight that the corresponding conservation law holds only after taking the limit $\varepsilon\searrow 0$ (otherwise the currents on different time slices do not cancel each other).

Lemma 4.5**.**

The radial flux matrix $h_{ij}$ defined by

[TABLE]

with the matrix $A$ given by

[TABLE]

is independent of $r$.

Proof.

By acting with the partial derivative of $r$ we end up with

[TABLE]

where $V$ is the matrix on the right side of (4.2). A straightforward calculation gives the result. ∎

Following up, we make the ansatz for $G(r;r^{\prime})$ when $r^{\prime}\notin(r_{-},\,r_{+})$

[TABLE]

and for $r^{\prime}\in(r_{-},\,r_{+})$

[TABLE]

with $c_{ij}\in\,\mathbb{C}$. We want to highlight that the above tensor notation corresponds to the bra-ket notation

[TABLE]

Lemma 4.6**.**

The Green’s matrix is well-defined and bounded in the three regions. Additionally, the coefficients $c_{ij}$ for $r^{\prime}\notin(r_{-},\,r_{+})$ are defined like

[TABLE]

where $h^{ij}$ is the inverse matrix of $h_{ij}$ from Lemma 4.5. For the case $r^{\prime}\in(r_{-},\,r_{+})$ one ends up with

[TABLE]

Proof.

The proof is a straight forward computation. We want to solve the distributional equation (4.3)

[TABLE]

Inserting the ansatz for $r^{\prime}\notin(r_{-},\,r_{+})$ we end up with

[TABLE]

Additionally we have following completeness relation on our Hilbert space

[TABLE]

with

[TABLE]

Combining both parts gives the result. The steps are identical for the case $r^{\prime}\in(r_{-},\,r_{+})$.

By Lemma 4.4 we have global defined Jost solutions which are all bounded. Together with the computed coefficients one can see that all Green’s matrices are bounded in the corresponding interval for $r^{\prime}$. Since the Jost solutions are globally defined in $r$ they are well-defined. ∎

5. Integral Representation of the Dirac Propagator

5.1. Abstract Representation

We can now state a first step towards our main result.

Proposition 5.1**.**

Let $H$ be the Hamiltonian of the Dirac equation in the Reissner-Nordström geometry in Eddington-Finkelstein coordinates. Then the corresponding Dirac propagator has the integral representation

[TABLE]

Here the resolvent can be expressed in forms of the Green’s matrices $G(r;r^{\prime})_{\omega_{\varepsilon}}$ from the ansatz (4.15) and (4.16) plus the coefficients from Lemma 4.6. It has the expression

[TABLE]

with

[TABLE]

where $X_{0}\in C^{\infty}_{\text{\rm{init}}}(N)$ is the initial data for a fixed angular momentum mode $k,l$.

Proof.

Combining the results from Propositions 3.4, 4.1 and 4.6, the remaining task is to interchange the limit $\varepsilon\searrow 0$ with the integral and then to take the limit $a\rightarrow\infty$ in (3.5). Since all extended Jost functions are bounded (for details see Subsection 4.3), we can apply Lebesgue’s dominated convergence theorem to take the limit $\varepsilon\searrow 0$ inside the integral. The limit $a\rightarrow\infty$ exists by Stone’s theorem. ∎

5.2. Main Theorem

By further calculations it is possible to bring the result from Proposition 5.1 in a much more handy form. We begin by computing the Green’s matrices in the upper and lower complex plane separately and taking the limit $\varepsilon\searrow 0$ for $r^{\prime}\in(r_{+},\infty)$. Afterward, we will take the difference and find a more compact expression for the integral representation. In the end we can extends this to $r^{\prime}\in(r_{-},\infty)$.

Lemma 5.2**.**

For $r^{\prime}\in(r_{+},\infty)$ we can express the differences of the Green’s function as

[TABLE]

where $g_{11}=g_{22}=1$, $g_{12}=\frac{a}{b}$, $g_{21}=\frac{d}{c}$ with $a,\,b,\,c,\,d\in\mathbb{C}^{2}$ and $A$ the matrix from Lemma 4.5. Additionally, $\chi(r)=(\chi_{1}(r),\chi_{2}(r))^{T}$ are the limits of the Jost solutions from Lemma 4.3 defined on the real axis and $G(r;r^{\prime})_{>0}$ describes the Green’s matrix on the upper, as well as $G(r;r^{\prime})_{<0}$ on the lower complex half plane.

Proof.

By looking at Lemma 4.4 with $r^{\prime}\in(r_{+},\infty)$ for $\operatorname{Im}(\omega_{\varepsilon})>0$ we get two extend Jost solutions $\Phi_{1}(r)$ and $\Phi_{2}(r)$. Since one can express any Jost solutions as a linear combination of two others we choose for $\Phi_{2}(r)$ a different ansatz to simplify the calculations

[TABLE]

The coefficients of the linear combinations are $a,\,b\in\mathbb{C}$. First we will calculate the coefficients $c_{ij}$ with those functions. After that we plug everything in the ansatz (4.15) and compute the result. By direct computations one ends up with

[TABLE]

(note that in the considered region $r^{\prime}\in(r_{+},\infty)$, the function $\Delta(r^{\prime})$ is positive). Now we begin by computing the first term from (4.15)

[TABLE]

Repeating the steps for the second term ends up with

[TABLE]

Taking the sum of both terms leads to

[TABLE]

Since the resolvent is bounded we can take the $\varepsilon$-limit inside the integral from Proposition 5.1 and apply the limit on the Jost solutions from the upper and lower complex plane. More importantly the Jost solutions from the lower and upper plane coincide on the real axis. We define following limits

[TABLE]

Therefore we end up with the result for the upper half plane

[TABLE]

For the lower half plane we take the Jost solutions

[TABLE]

with $c,\,d\,\in\mathbb{C}$. Repeating similar steps for the lower half plane we end up with

[TABLE]

Taking the difference and setting the coefficients as the matrix entries gives the result. ∎

In a next step we want to compute the spectral projector from the difference of the Green’s matrices only outside the black hole. This is possible since the Green’s matrix is only pointwise evaluated in the resolvent making it possible to split the integral due to linearity.

Lemma 5.3**.**

The spectral projector of the Dirac propagator in Eddington-Finkelstein coordinates for $r^{\prime}\in(r_{+},\infty)$ on initial data $X_{0}\in C^{\infty}_{\text{\rm{init}}}(N)$ has the form

[TABLE]

with $t_{11}=t_{22}=1$ and $t_{12}=\frac{a}{b}=\overline{t_{21}}$. We restrict \Gamma\equiv\Gamma\big{|}_{2\times 2} as the upper left $2\times 2$-block of the matrix in (3.3).

Proof.

By linearity we can pull the difference of the resolvents and by dominated convergence the $\varepsilon$-limit from Proposition 5.1 into the integral over $r^{\prime}$. Then we use the result from Lemma 5.2 and obtain the result by direct computations.

To derive the expression for $\frac{d}{c}$ we use the radial flux from Lemma 4.5. Since we are in the limit $\varepsilon\rightarrow 0$ we can compare the linear combinations of $\Phi_{2}^{+}(r)$ and $\Phi_{2}^{-}(r)$ on the real line (The plus and minus denotes the corresponding limits from the upper and lower complex half plane). We use the fact that the radial flux is conserved $\forall\,r$ and that \Phi_{2}^{+}\big{|}_{(r_{-},r_{+})}=0, which one can see in Lemma 4.4 or in Figure 1. Then we can use continuity and evaluate the expression

[TABLE]

This gives the result. ∎

We want to highlight that one can also derive the relation between the quotients $\frac{a}{b}$ and $\frac{d}{c}$ by using that the Hamiltonian is symmetric with respect to the conserved scalar product from Lemma 3.1. By direct computation, one sees that the resolvent needs to be symmetric with respect to the adjoint operator of the underlying Hilbert space. Thus, the equation

[TABLE]

needs to be satisfied, which again determines the quotient $d/c$ (here the dagger denotes the adjoint with respect to the scalar product $L^{2}(\text{\rm{d}}r)$).

We next extend the spectral projection to the region $r^{\prime}\in(r_{-},\infty)$.

Lemma 5.4**.**

The spectral projector of the Dirac propagator in Eddington-Finkelstein coordinates on initial data $X_{0}\in C^{\infty}_{\text{\rm{init}}}(N)$ from Lemma 5.3 extends to $r^{\prime}\in(r_{-},\infty)$, i.e.

[TABLE]

Proof.

Since the spectral projector is a symmetric operator we can interchange the variables $r\leftrightarrow r^{\prime}$. Therefore, the relation in Lemma 5.3 also holds if $r\in(r_{+},\infty)$ and $r^{\prime}\in(r_{-},\infty)$. With this in mind, it remains to consider the case $r,r^{\prime}\in(r_{-},r_{+})$. To this end, one repeats the computational steps in Lemma 5.2 for the ansatz (4.16), again with the same fundamental solutions $\Phi_{1}$ and $\Phi_{2}$ in (5.1) and (5.2). A straightforward computation shows that the spectral projector is again of the form as in (5.3). ∎

We are now in the position to state the main theorem of this paper.

Theorem 5.5**.**

The Dirac propagator in Proposition 5.1 can be expressed in terms of globally defined fundamental solutions $\chi_{i}(r,\omega)$ for $i\in\{1,2\}$ and $r\,\in(r_{-},\infty)$ as

[TABLE]

with $\widehat{X}_{i}(\omega):\mathbb{C}\rightarrow\mathbb{C}$ smooth functions defined by

[TABLE]

Here $(\cdot|\cdot)$ denotes the conserved scalar product on the hypersurfaces defined in (2.1) and given more explicitly in (3.2), restricted to the upper left $2\times 2$ block. Moreover, $t_{ij}$ are the entries of the matrix

[TABLE]

were $X_{0}(r)\,\in C^{\infty}_{\text{\rm{init}}}(N)$. Here $a,\,b\,\in\mathbb{C}$ are the transmission coefficients of the radial ODE defined in (5.1).

Proof.

We combine the results from Proposition 5.1, Lemma 5.3 and Lemma 5.4 and evaluate the integral over $r^{\prime}$ only in the range $(r_{-},\infty)$. This is possible because of the boundary conditions the wave is reflected on $\partial M$ and never comes back through the Cauchy horizon. Thus all interactions behind the Cauchy horizon do not contribute. Using the expression of the scalar product on the Cauchy hypersurfaces from Lemma 3.1 gives the final expression. ∎

Acknowledgments: We would like to thank Olaf Müller for helpful discussions. C.K. gratefully acknowledges support by the Heinrich-Böll-Stiftung.