The massive Dirac equation in Kerr geometry: separability in Eddington–Finkelstein-type coordinates and asymptotics

Abstract

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington–Finkelstein-type coordinates is shown. To this end, Kerr geometry is described by a Carter tetrad and the Dirac spinors and matrices are given in a chiral Newman–Penrose dyad representation. Applying Chandrasekhar’s mode ansatz, the Dirac equation is separated into systems of radial and angular ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are explicitly derived. Their decay is analyzed by means of error estimates. Moreover, the eigenfunctions and eigenvalues of the angular system are discussed. Finally, as an application, the scattering of Dirac waves by the gravitational field of a Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in Kerr geometry in horizon-penetrating coordinates and for the construction of a functional analytic integral representation of the Dirac propagator.

Appendices

Appendix 1: The Newman–Penrose formalism

Let $(\mathfrak {M}, \varvec{g})$ be a Lorentzian 4-manifold endowed with the unique, torsion-free Levi–Civita connection $\varvec{\omega }$ and dual basis $(\varvec{e}_{\mu })$ and $(\varvec{e}^{\mu })$, $\mu \in \{0, 1, 2, 3\}$, on sections of the tangent and cotangent bundles $T\mathfrak {M}$ and $T^{\star }\mathfrak {M}$, respectively. Further, let $F\mathfrak {M}$ and $F^{\star }\mathfrak {M}$ be local (orthonormal or null) frame bundles on $\mathfrak {M}$. The dual basis on sections of $F\mathfrak {M}$ and $F^{\star }\mathfrak {M}$ each consist of four vectors denoted by $(\varvec{e}_{(a)})$ and $(\varvec{e}^{(a)})$, $a \in \{0, 1, 2, 3\}$. In terms of the original basis vectors, these can be written as $\varvec{e}_{(a)} = e^{\mu }_{(a)} \varvec{e}_{\mu }$ and $\varvec{e}^{(a)} = e_{\mu }^{(a)} \varvec{e}^{\mu }$, where $e^{\mu }_{(a)}$ is a linear map from $T\mathfrak {M}$ to $F\mathfrak {M}$, namely a ($4 \times 4$)-matrix, and $e_{\mu }^{(a)}$ is its inverse. In the Newman–Penrose formalism [25], the local basis vectors are given by two real-valued null vectors, $\varvec{l} = \varvec{e}_{(0)} = \varvec{e}^{(1)}$ and $\varvec{n} = \varvec{e}_{(1)} = \varvec{e}^{(0)}$, as well as by a conjugate-complex pair of null vectors, $\varvec{m} = \varvec{e}_{(2)} = - \varvec{e}^{(3)}$ and $\varvec{\overline{m}} = \varvec{e}_{(3)} = - \varvec{e} ^{(2)}$. These have to satisfy the null conditions

$$\begin{aligned} \varvec{l} \cdot \varvec{l} = \varvec{n} \cdot \varvec{n} = \varvec{m} \cdot \varvec{m} = \varvec{\overline{m}} \cdot \varvec{\overline{m}} = 0, \end{aligned}$$

(43)

the orthogonality conditions

$$\begin{aligned} \varvec{l} \cdot \varvec{m} = \varvec{l} \cdot \varvec{\overline{m}} = \varvec{n} \cdot \varvec{m} = \varvec{n} \cdot \varvec{\overline{m}} = 0, \end{aligned}$$

(44)

and the cross-normalization conditions (which depend on the signature convention)

$$\begin{aligned} \varvec{l} \cdot \varvec{n} = - \varvec{m} \cdot \varvec{\overline{m}} = 1. \end{aligned}$$

(45)

The local metric in the Newman–Penrose formalism is non-degenerate and constant. It reads

$$\begin{aligned} \varvec{\eta } = g_{\mu \nu } e^{\mu }_{(a)} e^{\nu }_{(b)} \varvec{e}^{(a)} \otimes \varvec{e}^{(b)} = \varvec{l} \otimes \varvec{n} + \varvec{n} \otimes \varvec{l} - \varvec{m} \otimes \varvec{\overline{m}} - \varvec{\overline{m}} \otimes \varvec{m}. \end{aligned}$$

Since the Levi–Civita connection is torsion-free, the first Maurer–Cartan equation of structure simplifies to

$$\begin{aligned} \text{ d }\varvec{e}^{(a)} = \gamma ^{(a)}_{(b) (c)} \varvec{e}^{(b)} \wedge \varvec{e}^{(c)}, \end{aligned}$$

(46)

where the Ricci rotation coefficients $\gamma ^{(a)}_{(b) (c)}$ are related to the connection via

$$\begin{aligned} \gamma ^{(a)}_{(b) (c)} \varvec{e}^{(c)} = e_{\mu }^{(a)} \text{ d }e^{\mu }_{(b)} + e_{\mu }^{(a)} e^{\nu }_{(b)} \, \omega ^{\mu }_{\nu }. \end{aligned}$$

In the Newman–Penrose formalism, (46) becomes

$$\begin{aligned} \text{ d }\varvec{l}&= 2 \mathfrak {R}{(\epsilon )} \, \varvec{n} \wedge \varvec{l} - 2 \varvec{n} \wedge \mathfrak {R}{(\kappa \, \varvec{\overline{m}})} - 2 \varvec{l} \wedge \mathfrak {R}{\bigl ([\tau -\overline{\alpha } - \beta ] \, \varvec{\overline{m}}\bigr )} + 2 \text{ i } \, \mathfrak {I}{(\varrho )} \, \varvec{m} \wedge \varvec{\overline{m}} \nonumber \\ \text{ d }\varvec{n}&= 2 \mathfrak {R}{(\gamma )} \, \varvec{n} \wedge \varvec{l} - 2 \varvec{n} \wedge \mathfrak {R}{\bigl ([\overline{\alpha } + \beta - \overline{\pi }] \, \varvec{\overline{m}}\bigr )} + 2 \varvec{l} \wedge \mathfrak {R}{(\overline{\nu } \, \overline{\varvec{m}})} + 2 \text{ i } \, \mathfrak {I}{(\mu )} \, \varvec{m} \wedge \varvec{\overline{m}} \nonumber \\ \text{ d }\varvec{m}&= \overline{\text{ d }\varvec{\overline{m}}} = (\overline{\pi } + \tau ) \, \varvec{n} \wedge \varvec{l} + \bigl (2 \text{ i } \, \mathfrak {I}{(\epsilon )} - \varrho \bigr ) \, \varvec{n} \wedge \varvec{m} - \sigma \, \varvec{n} \wedge \varvec{\overline{m}} + \bigl (\overline{\mu } + 2 \text{ i } \, \mathfrak {I}{(\gamma )}\bigr ) \, \varvec{l} \wedge \varvec{m} \nonumber \\&\qquad \qquad + \overline{\lambda } \, \varvec{l} \wedge \varvec{\overline{m}} - (\overline{\alpha } - \beta ) \, \varvec{m} \wedge \varvec{\overline{m}} \end{aligned}$$

(47)

with the so-called spin coefficients

$$\begin{aligned} \kappa = \gamma _{(2) (0) (0)}&\quad \varrho = \gamma _{(2) (0) (3)}&\quad \epsilon = \tfrac{1}{2} \bigl (\gamma _{(1) (0) (0)} + \gamma _{(2) (3) (0)}\bigr ) \nonumber \\ \sigma = \gamma _{(2) (0) (2)}&\quad \mu = \gamma _{(1) (3) (2)}&\quad \gamma = \tfrac{1}{2} \bigl (\gamma _{(1) (0) (1)} + \gamma _{(2) (3) (1)}\bigr ) \nonumber \\ \lambda = \gamma _{(1) (3) (3)}&\quad \tau = \gamma _{(2) (0) (1)}&\quad \alpha = \tfrac{1}{2} \bigl (\gamma _{(1) (0) (3)} + \gamma _{(2) (3) (3)}\bigr ) \\ \nu = \gamma _{(1) (3) (1)}&\quad \pi = \gamma _{(1) (3) (0)}&\quad \beta = \tfrac{1}{2} \bigl (\gamma _{(1) (0) (2)} + \gamma _{(2) (3) (2)}\bigr ) \nonumber \end{aligned}$$

(48)

and the real and imaginary parts $\mathfrak {R}{(\cdot )}$ and $\mathfrak {I}{(\cdot )}$, respectively. The transformations employed in this study are elements of the 2-parameter subgroup of local Lorentz transformations known as class III or spin-boost Lorentz transformations. These renormalize the real-valued Newman–Penrose vectors $\varvec{l}$ and $\varvec{n}$, but leave their directions unchanged, and rotate the conjugate-complex pair $\varvec{m}$ and $\varvec{\overline{m}}$ by an angle $\psi$ in the $(\varvec{m}, \varvec{\overline{m}})$-plane [8]

$$\begin{aligned} \varvec{l} \mapsto \varvec{l}' = \varsigma \, \varvec{l}, \quad \varvec{n} \mapsto \varvec{n}' = \varsigma ^{-1} \, \varvec{n}, \quad \varvec{m} \mapsto \varvec{m}' = e^{\text{ i }\psi } \varvec{m}, \quad \varvec{\overline{m}} \mapsto \varvec{\overline{m}}' = e^{- \text{ i } \psi } \, \varvec{\overline{m}}, \end{aligned}$$

(49)

where $\varsigma \in \mathbb {R} \backslash \{0\}$ and $\psi \in \mathbb {R}$ are functions depending on the spacetime coordinates $(x^{\mu })$. There are various aspects of the Newman–Penrose formalism that are not discussed in this appendix such as the different classes of local Lorentz transformations or the Weyl scalars and their algebraic classification. They can be found elsewhere in the literature. The interested reader may be referred to [26, 28].

Appendix 2: The general relativistic Dirac equation

The general relativistic, massive Dirac equation without an external potential is given by [19, 32]

$$\begin{aligned} \bigl (\gamma ^{\mu } \nabla _{\mu } + \text{ i } m\bigr ) \Psi (x^{\mu }) = 0, \end{aligned}$$

where the $\gamma ^{\mu }$ are the general relativistic Dirac matrices, which satisfy the anticommutator relations $\{\gamma ^{\mu }, \gamma ^{\nu }\} = 2 g^{\mu \nu } \text{ id }_{\mathbb {C}^4}$, $\Psi (x^{\mu })$ is a Dirac 4-spinor on sections $S_x\mathfrak {M} \simeq \mathbb {C}^4$ of the spin bundle $S\mathfrak {M} = \mathfrak {M} \times \mathbb {C}^4$ on $\mathfrak {M}$, $\varvec{\nabla }$ denotes the metric connection on $S\mathfrak {M}$, and m is the fermion rest mass. Using the chiral 2-spinor representation of the Dirac 4-spinors and matrices [20]

$$\begin{aligned} \Psi = \left( \begin{array}{c} P^A \\ \overline{Q}_{\dot{B}} \\ \end{array}\right) \quad \text{ and } \quad \gamma ^{\mu } = \sqrt{2} \left( \begin{array}{ll} 0 &{} \quad \sigma ^{\mu A \dot{B}}\\ \sigma ^{\mu }_{A \dot{B}} &{} \quad 0 \\ \end{array}\right) , \quad A \in \{1, 2\}, \quad \dot{B} \in \{\dot{1}, \dot{2}\}, \end{aligned}$$

with the 2-component spinors $P^A$ and $\overline{Q}_{\dot{B}}$ and the Hermitian $(2 \times 2)$-Infeld–van der Waerden symbols $\sigma ^{\mu }_{A \dot{B}}$, we obtain the following 2-spinor form of the Dirac equation

$$\begin{aligned} \begin{aligned} \nabla _{A \dot{B} } P^A + \text{ i } \mu _{\star } \overline{Q}_{\dot{B}}&= 0 \\ \nabla _{A \dot{B}} Q^A + \text{ i } \mu _{\star } \overline{P}_{\dot{B}}&= 0, \end{aligned} \end{aligned}$$

(50)

where $\mu _{\star } := m/\sqrt{2}$ and $\nabla _{A \dot{B}} = \sigma ^{\mu }_{A \dot{B}} \nabla _{\mu }$. Note that dotted indices are subjected to conjugated complex transformations. Next, let $\zeta _{(k)}$ and $\zeta ^{(k)}$, $k \in \{1, 2\}$, be (dual) local Newman–Penrose bases for the Dirac 2-spinors. The associated local spinor components can be expressed by $\mathscr {Y}^{(k)} = \zeta ^{(k)}_{A} \mathscr {Y}^{A}$ and $\mathscr {Y}_{(k)} = \zeta _{(k)}^{A} \mathscr {Y}_{A}$ with the original 2-spinor components $\mathscr {Y}^{A}, \mathscr {Y}_{A}\in \mathbb {C}^2$ as well as the $(2 \times 2)$-matrix $\zeta ^{(k)}_{A}$ and its inverse $\zeta _{(k)}^{A}$. In this representation, the metric connection reads

$$\begin{aligned} \nabla _{(k) (\dot{l})} \mathscr {Y}^{(m)} = \zeta ^{A}_{(k)} \overline{\zeta }^{\dot{B}}_{(\dot{l})} \, \zeta ^{(m)}_{C} \nabla _{A \dot{B}} \, \mathscr {Y}^C = \partial _{(k) (\dot{l})} \mathscr {Y}^{(m)} + \Gamma ^{(m)}_{(n) (k) (\dot{l})} \mathscr {Y}^{(n)}, \end{aligned}$$

(51)

where $\partial _{(k) (\dot{l})} = \sigma ^{\mu }_{(k) (\dot{l})} \partial _{\mu }$ and

$$\begin{aligned} \Gamma ^{(m)}_{(n) (k) (\dot{l})}= & {} \Gamma ^{(m) (\dot{o})}_{(n) (\dot{o}) (k) (\dot{l})} \,\nonumber \\ {}= & {} \sqrt{2} \, \epsilon ^{(m) (q)} \, \epsilon ^{(\dot{o}) (\dot{p})} \, \sigma ^{\mu }_{(q) (\dot{p})} \,\, \sigma ^{\nu }_{(n) (\dot{o})}\,\, \sigma ^{\lambda }_{(k) (\dot{l})} \, e_{\mu }^{(a)} \, e_{\nu }^{(b)} \, e_{\lambda }^{(c)} \, \gamma _{(a) (b) (c)}.\nonumber \\ \end{aligned}$$

(52)

We point out that the 2-dimensional Levi–Civita symbol $\varvec{\epsilon }$ acts as skew metric on $\mathbb {C}^2$. Furthermore, the Infeld–van der Waerden symbols yield

$$\begin{aligned} \sigma ^{\mu }_{(k) (\dot{l})} = \left( \begin{array}{cc} l^{\mu } &{} m^{\mu } \\ \overline{m}^{\, \mu } &{} n^{\mu } \\ \end{array}\right) . \end{aligned}$$

(53)

By means of (51–53), (48), and the definitions $\mathscr {F}_1 := P^{(1)}, \mathscr {F}_2 := P^{(2)}, \mathscr {G}_1 := \overline{Q}^{(\dot{2})},$ as well as $\mathscr {G}_2 := - \overline{Q}^{(\dot{1})}$, the general relativistic Dirac equation (50) in the Newman–Penrose formalism becomes

$$\begin{aligned} \begin{aligned} (l^{\mu } \partial _{\mu } + \varepsilon - \varrho ) \mathscr {F}_1 + (\overline{m}^{\mu } \partial _{\mu } + \pi - \alpha ) \mathscr {F}_2&= \text{ i } \mu _{\star } \mathscr {G}_1 \\ (n^{\mu } \partial _{\mu } + \mu - \gamma ) \mathscr {F}_2 + (m^{\mu } \partial _{\mu } + \beta - \tau ) \mathscr {F}_1&= \text{ i } \mu _{\star } \mathscr {G}_2 \\ (l^{\mu } \partial _{\mu } + \overline{\varepsilon } - \overline{\varrho }) \mathscr {G}_2 - (m^{\mu } \partial _{\mu } + \overline{\pi } - \overline{\alpha }) \mathscr {G}_1&= \text{ i } \mu _{\star } \mathscr {F}_2 \\ (n^{\mu } \partial _{\mu } + \overline{\mu } - \overline{\gamma }) \mathscr {G}_1 - (\overline{m}^{\mu } \partial _{\mu } + \overline{\beta } - \overline{\tau }) \mathscr {G}_2&= \text{ i } \mu _{\star } \mathscr {F}_1. \end{aligned} \end{aligned}$$

(54)