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An Integral Spectral Representation of the Massive Dirac Propagator in the Kerr Geometry in Eddington-Finkelstein-type Coordinates

We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive a functional analytic integral representation of the associated propagator using the spectral theorem for unbounded self-adjoint operators, Stone's formula, and quantities arising in the analysis of Chandrasekhar's separation of variables. This integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon. In the derivation, we first write the Dirac equation in Hamiltonian form and show the essential self-adjointness of the Hamiltonian. For the latter purpose, as the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon, we cannot use standard elliptic methods of proof. Instead, we employ a new, general method for mixed initial-boundary value problems that combines results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. In this regard and since the time evolution may not be unitary because of Dirac particles impinging on the ring singularity, we also impose a suitable Dirichlet-type boundary condition on a time-like inner hypersurface placed inside the Cauchy horizon, which has no effect on the dynamics outside the Cauchy horizon. We then compute the resolvent of the Dirac Hamiltonian via the projector onto a finite-dimensional, invariant spectral eigenspace of the angular operator and the radial Green's matrix stemming from Chandrasekhar's separation of variables. Applying Stone's formula to the spectral measure of the Hamiltonian in the spectral decomposition of the Dirac propagator, that is, by expressing the spectral measure in terms of this resolvent, we obtain an explicit integral representation of the propagator.


I. INTRODUCTION
In [9], a functional analytic integral representation of the propagator of the massive Dirac equation in the non-extreme Kerr geometry outside the event horizon is derived in Boyer-Lindquist coordinates (t, r, θ, ϕ) with t ∈ R, r ∈ R >0 , θ ∈ [0, π], and ϕ ∈ [0, 2π). It has been used to study the long-time behavior (including decay rates) and the escape probability of Dirac particles [10]. The shortcoming of this integral representation is, however, that it yields a solution of the Cauchy problem only outside the event horizon. In the present paper, we construct a generalized integral representation that describes the complete dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon. The methods used in the derivation of our integral representation are quite different from those employed in [9], as is now outlined. We work with horizon-penetrating advanced Eddington-Finkelstein-type coordinates τ (t, r), r, θ, φ(ϕ, r) [17], i.e., with an analytic extension of Boyer-Lindquist coordinates that covers both the exterior and interior black hole regions without exhibiting singularities at the horizons and features a proper coordinate time τ required, e.g., for the initial value formulation. Furthermore, we employ a regular Carter tetrad, that is, a symmetric Newman-Penrose null frame, which makes use of, on the one hand, the discrete time and angle reversal isometries of Kerr geometry and, on the other hand, its Petrov type. After computing the corresponding spin coefficients, we explicitly determine the massive Dirac equation in Hamiltonian form i∂ τ ψ(τ, r, θ, φ) = Hψ(τ, r, θ, φ) , where H denotes the Hamiltonian and ψ the Dirac 4-spinor. Moreover, we introduce a scalar product on the solution space and show that the Dirac Hamiltonian is symmetric with respect to this scalar product (on smooth and compactly supported Dirac 4-spinors). We also establish that it coincides with the canonical scalar product obtained by integrating the normal component (defined with respect to space-like hypersurfaces) of the Dirac current. We point out that in the present setting, the Dirac equation as well as the scalar product are smooth at the horizons. Since we apply the spectral theorem in the derivation of the propagator, we need to construct a unique self-adjoint extension of the Hamiltonian.
To this end, in order to have a unitary time evolution, we first prevent that Dirac particles may impinge on the curvature singularity by shielding it imposing a Dirichlet-type boundary condition on a time-like hypersurface inside the Cauchy horizon. Clearly, this boundary condition changes the dynamics of the Dirac particles because they are now reflected on the inner boundary surface. However, the dynamics outside the Cauchy horizon is not affected, as the reflected particles cannot reenter this particular region (see FIG. 1 on page 13). Second, since the Hamiltonian is not elliptic at the horizons, we employ the method for non-uniformly elliptic boundary value problems introduced in [13], yielding a unique self-adjoint extension. This makes it possible to write down the Dirac propagator in spectral form where dE ω is the spectral measure of H and ψ 0 := ψ(τ = 0) is the initial data. Then, according to Stone's formula, we represent the spectral measure via the resolvents (H − ω ∓ i ) −1 with > 0, and express them in terms of the solutions of the radial and angular ODE systems resulting from Chandrasekhar's separation of variables. In more detail, employing Chandrasekhar's separation ansatz in (1) and projecting onto a finite-dimensional, invariant angular eigenspace, the Dirac Hamiltonian becomes a matrix-valued first-order ordinary differential operator in the radial variable. The resolvent of this operator can be determined by means of the Green's matrix of the radial ODE system. We remark that, since this system cannot be solved analytically without making suitable approximations or by considering asymptotics, in the present work, we use the asymptotic solutions at infinity and at the event and Cauchy horizons for guidance in the implicit construction of the functions required for the computation of the Green's matrix. Subsequently, by summing over all angular modes, we obtain the full resolvent in separated form. The integral representation of the Dirac propagator is thus given by the formula in which k labels the azimuthal modes (for the explicit forms of the resolvents see Theorem IV.3).
The article is organized as follows. In Section II, we provide the mathematical framework for Kerr geometry and for the massive Dirac equation. Moreover, we recall required results from the asymptotic analysis of the radial ODE system and from the spectral analysis of the angular ODE system without giving proofs. We derive the Hamiltonian formulation and a suitable scalar product for the Hilbert space of solutions of the Cauchy problem in Section III. Furthermore, we verify the symmetry of the Hamiltonian with respect to this scalar product. In Section IV, we show that the Hamiltonian is essentially self-adjoint and construct the integral representation of the propagator.

II. PRELIMINARIES
We recall the necessary basics on the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates, the general relativistic, massive Dirac equation in the Newman-Penrose formalism, and Chandrasekhar's separation of variables (including asymptotic and spectral results for the solutions of the corresponding radial and angular ODE systems).
The non-extreme Kerr geometry is a connected, orientable and time-orientable, smooth, asymptotically flat Lorentzian 4-manifold (M, g) with topology S 2 × R 2 , for which the metric g is stationary and axisymmetric and given in horizon-penetrating advanced Eddington-Finkelstein-type coordinates (τ, r, θ, φ) with τ ∈ R, r ∈ R >0 , θ ∈ [0, π], and φ ∈ [0, 2π) by [17] where M is the mass and aM the angular momentum of the black hole with 0 ≤ a < M , and Σ = Σ(r, θ) := r 2 + a 2 cos 2 (θ). The event and Cauchy horizons are located at r ± := M ± √ M 2 − a 2 , respectively. The advanced Eddington-Finkelstein-type coordinates are an analytic extension of the common Boyer-Lindquist coordinates (t, r, θ, ϕ) with t ∈ R, r ∈ R >0 , θ ∈ [0, π], and ϕ ∈ [0, 2π) [3], covering both the exterior and interior black hole regions while being regular at the horizons. In terms of the Boyer-Lindquist coordinates, the advanced Eddington-Finkelstein-type time and azimuthal angle coordinates read This horizon-penetrating coordinate system possesses a proper coordinate time, unlike the original advanced Eddington-Finkelstein (null) coordinates [7,8]. It is advantageous to describe Kerr geometry in the Newman-Penrose formalism using a regular Carter tetrad [4,17] with ∆ = ∆(r) := (r − r + )(r − r − ) = r 2 − 2M r + a 2 being the horizon function because this frame is adapted to the two principal null directions of the Weyl tensor and to the fundamental discrete time and angle reversal isometries. Thus, since Kerr geometry is algebraically special and of Petrov type D, one has the computational advantage that the four spin coefficients κ, σ, λ, and ν as well as the four Weyl scalars Ψ 0 , Ψ 1 , Ψ 3 , and Ψ 4 vanish [15], and that specific spin coefficients are linearly dependent. Substituting the Carter tetrad (4) into -and solving -the first Maurer-Cartan equation of structure, we obtain the spin coefficients [17] Introducing a spin bundle SM = M × C 4 on M with fibers S x M C 4 , x ∈ M, we can formulate the general relativistic, massive Dirac equation (without an external potential) where ∇ is the metric connection on SM, γ µ are the Dirac matrices, ψ is the Dirac 4-spinor defined on the fibers S x M, and m is the fermion rest mass. In the Newman-Penrose formalism -by employing a local dyad spinor frame -this equation becomes the coupled first-order PDE system with ψ = (F 1 , F 2 , −G 1 , −G 2 ) T and µ := m/ √ 2 [5]. Substituting the Carter tetrad (4) and the associated spin coefficients (5) into the system (6), and applying the transformation where P := diag r − ia cos (θ), r − ia cos (θ), r + ia cos (θ), r + ia cos (θ) , we find which is the starting point for the derivation of the Hamiltonian formulation of the massive Dirac equation on a Kerr background geometry in horizon-penetrating coordinates presented in the next section. We note in passing that the system (8) corresponds to the transformed Dirac equation where γ 0 := diag(1, 1, −1, −1). This will become relevant both in the construction of the Hamiltonian formulation and the scalar product. Finally, for the later computation of the resolvent of the Dirac Hamiltonian, we require specific results arising from Chandrasekhar's separation of variables of the system (8). More precisely, we need the asymptotics of the radial ODE system at infinity, at the event horizon, and at the Cauchy horizon, as well as certain information about the eigenvalues and eigenfunctions of the angular ODE system. In the following, these results are recalled. For a detailed analysis and proofs see [17]. Substituting the separation ansatz in which ω ∈ R and k ∈ Z + 1/2, into (8) yields the first-order radial and angular ODE systems where are matrix-valued radial and angular operators, R := |∆| R + , r + R − T and T := (T + , T − ) T are radial and angular vector-valued functions, and ξ is the constant of separation. The asymptotics and decay properties of the solutions of the radial ODE system at infinity, the event horizon, and the Cauchy horizon are specified in the lemmas below.
Lemma II.1. Every nontrivial solution R of (10) is asymptotically as r → ∞ of the form with the asymptotic diagonalization matrix where the asymptotic phases ∞ T = 0, as well as an error with polynomial decay Lemma II.2. Every nontrivial solution R of (10) is asymptotically as r r ± of the form with the constants g r± = g (1) r± , g Kerr := a/(2M r ± ), as well as an error with exponential decay The spectral properties of the eigenvalues and eigenfunctions of the angular ODE system are summarized in the following proposition.
Proposition II.3. For any ω ∈ R and k ∈ Z + 1/2, the differential operator (12) has a complete set of orthonormal eigenfunctions (T l ) l∈Z in L 2 (0, π), sin (θ) dθ 2 . The corresponding eigenvalues ξ l are real-valued and non-degenerate, and can thus be ordered as ξ l < ξ l+1 . Moreover, the eigenfunctions are pointwise bounded and smooth away from the poles, Both the eigenfunctions T l and the eigenvalues ξ l depend smoothly on ω.

III. HAMILTONIAN FORMULATION OF THE MASSIVE DIRAC EQUATION IN KERR GEOMETRY AND THE CANONICAL SCALAR PRODUCT
In order to derive the Hamiltonian formulation of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates, it is advantageous to first rewrite the system (8) in the form and are matrix-valued differential operators with Separating the τ -derivative and multiplying by the inverse of the matrix (cf. Eq. (9)) as well as by the imaginary unit leads to where R (3) and A (3) contain the first-order spatial as well as the zero-order contributions of the operators (13) and (14), respectively. The Dirac Hamiltonian H may be recast in the more convenient form with the matrix-valued coefficients and the potential where the quantities B k , k ∈ {1, 2, 3, 4}, are the (2 × 2)-blocks Furthermore, we work with the scalar product defined on the space-like hypersurface N τ := {τ = const., r, θ, φ} [9], where denotes the indefinite spin scalar product of signature (2, 2), ψ := ψ † S the adjoint Dirac spinor, / ν = γ µ ν µ the Clifford contraction of the future-directed, time-like normal ν, and dµ |Nτ = |det(g |Nτ )| dφ dθ dr with the induced (Riemannian) metric g |Nτ is the invariant measure on N τ .
(Note that this scalar product is independent of the choice of the specific space-like hypersurface.) The matrix S is defined via the relation With (15) and the spinor transformation (7), we find and thus, using (25), we obtain for the matrix S the term The vector field ν is determined by means of the conditions ν|∂ r |g = ν|∂ θ |g = ν|∂ φ |g = 0 and ν|ν |g = 1 , where · | · |g := g( · , · ) is the spacetime inner product on the manifold M given by (2), yielding The corresponding dual co-vector reads Moreover, the induced metric g |Nτ on N τ is the restriction of (2) from M to the submanifold N τ is the Jacobian determinant in the volume measure dµ |Nτ . Expressing the scalar product (23), which is invariant under spinor transformations, in terms of the primed wave functions (7) employed in (16) and substituting (26) and (27), we obtain Again using (15), that is with γ τ = −P P † −1 γ 0 γ τ / √ Σ, the scalar product (28) becomes where Note that in the above derivation, we have first applied the relation √ Σ 1 1 C 4 = PP † , then the transformation law for the matrix S , namely S = P † S P, and finally we have used the fact that both S and the product PS are self-adjoint, which leads to the relation PS = S P † . Besides, the integration limits are suppressed for ease of notation if possible and given if necessary. The eigenvalues λ 1 , λ 2 of the matrix (30) are positive and with algebraic multiplicities µ A (λ 1 ) = 2 = µ A (λ 2 ) showing that (29) is indeed positive-definite.
Theorem III.1. The Dirac Hamiltonian (17) is symmetric with respect to the scalar product (29).
Proof. In order to establish the symmetry, namely that we begin by splitting the potential V given in (21) into mass-independent and mass-dependent parts with the (2 × 2)-blocks B k defined in (22), therefore obtaining anti-self-adjoint and self-adjoint matrices with Γ τ = Γ τ † defined in (30). This leads to Integration by parts of the first triple integral and substitution of the relations (31) in the remaining two triple integrals yields We remark that in the integration by parts, the angular derivatives do not give rise to boundary terms because the two-dimensional submanifold S 2 is compact without boundary. The radial boundary terms on the other hand vanish due to Dirichlet-type boundary conditions imposed on the Dirac spinors. More precisely, the radial boundary terms read .
Direct computation of the matrix Γ τ α r gives and hence the radial boundary terms become In order for this expression to vanish, we impose the radial Dirichlet-type boundary condition As in the next section only Dirac spinors with supp φ = [r 0 , ∞)×S 2 and thus radial boundary conditions at a specific time-like inner boundary r = r 0 < r − and at infinity, for which the terms in (33) vanish separately, are considered, we may bring the radial Dirichlet-type boundary condition at r = r 0 into a more suitable form as follows. (Note that at infinity, we merely require proper decay of the Dirac spinors.) By means of the spin scalar product (24) and the relation S γ r = i Γ τ α r /Σ, we may represent (33) as ≺ψ |γ r φ |{τ }×{r0}×S 2 = 0 .
Introducing n as the unit normal to the hypersurfaces {τ } × S 2 , we can write (34) in the form where the slash again denotes Clifford multiplication. With / n 2 = −1 1 C 4 , the above implication can be easily verified by We point out that the mixed terms in the last line cancel each other out. Next, the explicit calculation of the square bracket in (32) yields Moreover, all three matrix products Γ τ α j , j ∈ {r, θ, φ}, are anti-self-adjoint Therefore, we immediately find that

IV. SPECTRAL ANALYSIS OF THE DIRAC HAMILTONIAN AND INTEGRAL REPRESENTATION OF THE PROPAGATOR
In this section, we show that the Dirac Hamiltonian in the non-extreme Kerr geometry in horizonpenetrating Eddington-Finkelstein-type coordinates is essentially self-adjoint. Moreover, we construct a specific integral representation of the Dirac propagator, which yields the dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon. In more detail, we derive an explicit expression for the spectral measure dE ω arising in the formal spectral decomposition of the propagator Since this spectral decomposition only applies to self-adjoint operators, we first require a unique selfadjoint extension of the Dirac Hamiltonian. This problem involves the technical difficulty that the Dirac Hamiltonian in Kerr geometry is not elliptic at the event and Cauchy horizons, which can be easily seen from the evaluation of the determinant of the principal symbol The Hamiltonian fails to be elliptic if the determinant vanishes, that is, if g ij ξ i ξ j = 0 for ξ being non-zero. With one can verify that this holds true only at the event and Cauchy horizons. Hence, in the construction of the extension, we employ the method for non-uniformly elliptic boundary value problems presented in [13], combining results from the theory of symmetric hyperbolic systems with elliptic methods [2,6,14,18]. Then, by means of Stone's formula, we express the spectral measure in terms of the resolvent of the Dirac Hamiltonian, which we compute using the Green's matrix of the radial ODE system (10)  Figure 1). These hypersurfaces constitute a foliation of M along the time direction characterized by the parameter τ . At ∂M , we assume the radial Dirichlet-type boundary condition (35), which has the effect that Dirac particles are reflected away from the singularity, to obtain a unitary time evolution without changing the dynamics outside the Cauchy horizon. Also near ∂M , we have a locally time-like Killing vector field K, which is a linear combination of the Killing fields ∂ τ and ∂ φ describing the stationarity and axisymmetry of Kerr geometry. It is given by [15], and corresponds to the Killing field K = ∂ t of [13] that is represented by a coordinate system describing an observer who is co-moving along the flow lines of the Killing field. Using the results of the previous section, we can set up a Hilbert space H , (·|·) |Nτ with H = L 2 (N τ , SM ), where SM denotes the spin bundle of M , and the scalar product (29). In this setting, we find a unique global solution of the Cauchy problem for the massive Dirac equation in the class C ∞ sc (M , SM ). Lemma IV.1. The Cauchy problem for the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates i∂ τ ψ = Hψ with the radial Dirichlet-type boundary condition at ∂M given by where the initial data is smooth, compactly supported outside, across, and inside the event horizon, up to the Cauchy horizon, and is compatible with the boundary condition, i.e., has a unique global solution in the class of smooth wave functions with spatially compact support Evaluating this solution at subsequent times τ and τ gives rise to a unique unitary propagator This lemma is essential as a technical tool in the construction of the self-adjoint extension of the Dirac Hamiltonian. Its proof is shown in detail in [13].

A. Self-adjoint Extension of the Dirac Hamiltonian
We introduce a theorem for the existence of a unique self-adjoint extension of the Dirac Hamiltonian H, which is defined by (17) and (18) For the proof, we again refer to [13], in which the construction of a self-adjoint extension of the Dirac Hamiltonian is discussed for a more general class of non-uniformly elliptic mixed initial/boundary value problems for spacetimes with dimension d ≥ 3.

B. Resolvent of the Dirac Hamiltonian and Integral Representation of the Propagator
In order to construct an explicit expression for the spectral measure in the spectral decomposition of the Dirac propagator (36), we use Stone's formula and thus the resolvent of the essentially self-adjoint Hamiltonian H defined in Theorem IV.2. As the spectrum of the Hamiltonian σ(H) ⊆ R is on the real line, the resolvent (H − ω c ) −1 ∈ L(H ) exists for all ω c ∈ C\R with real part (ω c ) = ω ∈ σ(H), and is given uniquely.