An Integral Representation of the Massive Dirac Propagator in Kerr Geometry in Eddington–Finkelstein-type Coordinates Felix Finster and Christian R¨oken Universit¨ at Regensburg, Fakult¨ at f¨ ur Mathematik, 93040 Regensburg, Germany (Dated: November 2017)
We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating Eddington–Finkelstein-type coordinates and derive a functional analytic integral representation of the propagator using the spectral theorem and the solutions of the ODEs arising in 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. We then construct a unique self-adjoint extension of the Hamiltonian. To this end, as the Dirac Hamiltonian fails to be elliptic at the horizons, we combine results from the theory of symmetric hyperbolic systems with elliptic methods. Moreover, since the time evolution is not unitary because the particles may impinge on the singularity, we impose a suitable Dirichlet-type boundary condition inside the Cauchy horizon, having no effect on the outside dynamics. Finally, we obtain an explicit expression for the spectral decomposition of the propagator by applying Stone’s formula to the spectral measure of the Hamiltonian and expressing the resolvent in terms of the solutions of the separated radial and angular ODE systems.
arXiv:1606.01509v2 [gr-qc] 2 Nov 2017
ABSTRACT.
Contents
I. Introduction
2
II. Preliminaries
3
III. Hamiltonian Formulation of the Massive Dirac Equation in Kerr Geometry and the Canonical Scalar Product 6 IV. Spectral Analysis of the Dirac Hamiltonian and Integral Representation of the Propagator A. Self-adjoint Extension of the Dirac Hamiltonian B. Resolvent of the Dirac Hamiltonian and Integral Representation of the Propagator
12 14 14
Appendix: Functions for the Construction of the Green’s Matrix of the Radial ODE System
20
Acknowledgments
21
References
21
e-mail:
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2 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) ,
(1)
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 Z ψ = e−iτ H ψ0 = e−iωτ ψ0 dEω , R
where dEω is the spectral measure of H and ψ0 := 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 Z 1 X −ikφ ψ(τ, r, θ, φ) = e lim e−iωτ (Hk − ω − i)−1 − (Hk − ω + i)−1 (r, θ; r0 , θ0 ) ψ0,k (r0 , θ0 ) dω , &0 R 2πi k∈Z
in which k labels the azimuthal modes (for the explicit forms of the resolvents see Theorem IV.3).
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 × R2 , 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] 2M r 2M r g = 1− dτ ⊗ dτ − dr − a sin2 (θ) dφ ⊗ dτ + dτ ⊗ dr − a sin2 (θ) dφ Σ Σ (2) 2M r dr − a sin2 (θ) dφ ⊗ dr − a sin2 (θ) dφ − Σ dθ ⊗ dθ − Σ sin2 (θ) dφ ⊗ dφ , − 1+ Σ where M is the mass and aM the angular momentum of the black hole with 0 ≤ a < M√ , and Σ = Σ(r, θ) := r2 + a2 cos2 (θ). The event and Cauchy horizons are located at r± := M ± M 2 − a2 , 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 τ := t +
2 + a2 r+ r2 + a2 ln |r − r+ | − − ln |r − r− | r+ − r− r+ − r−
(3) φ := ϕ +
a r+ − r−
r − r+ . ln r − r−
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] l= √
1 [∆ + 4M r] ∂τ + ∆ ∂r + 2a ∂φ 2Σ r+
r+ n= √ (∂τ − ∂r ) 2Σ (4) 1 m= √ ia sin (θ) ∂τ + ∂θ + i csc (θ) ∂φ 2Σ
1 m = −√ ia sin (θ) ∂τ − ∂θ + i csc (θ) ∂φ 2Σ
4 with ∆ = ∆(r) := (r − r+ )(r − r− ) = r2 − 2M r + a2 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 Ψ0 , Ψ1 , Ψ3 , and Ψ4 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] κ = σ = λ = ν = 0,
γ=−
23/2
√
r+ , Σ (r − ia cos (θ))
π = −τ = √
ia sin (θ) , 2Σ (r − ia cos (θ))
α = −β = −
2 1 r + a2 cot (θ) − ira sin (θ) . 3/2 (2Σ)
µ = −√
=
r2 − a2 − 2ia cos (θ) (r − M ) √ 23/2 Σ r+ (r − ia cos (θ))
r+ , 2Σ (r − ia cos (θ))
% = −√
∆ 2Σ r+ (r − ia cos (θ))
(5)
Introducing a spin bundle SM = M × C4 on M with fibers Sx M ' C4 , x ∈ M, we can formulate the general relativistic, massive Dirac equation (without an external potential) γ µ ∇µ + im ψ(xµ ) = 0 , µ ∈ {0, 1, 2, 3} , where ∇ is the metric connection on SM, γ µ are the Dirac matrices, ψ is the Dirac 4-spinor defined on the fibers Sx 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 (nµ ∂µ + µ − γ)G1 − (m µ ∂µ + β − τ )G2 = iµ? F1 (lµ ∂µ + ε − %)G2 − (mµ ∂µ + π − α)G1 = iµ? F2 (6) µ (l ∂µ + ε − %)F1 + (m µ ∂µ + π − α)F2 = iµ? G1 (nµ ∂µ + µ − γ)F2 + (mµ ∂µ + β − τ )F1 = iµ? G2 √ with ψ = (F1 , F2 , −G1 , −G2 )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 ψ 0 = Pψ = (H1 , H2 , −J1 , −J2 )T ,
γ 0µ = Pγ µ P −1 ,
(7)
where p p p p P := diag r − ia cos (θ), r − ia cos (θ), r + ia cos (θ), r + ia cos (θ) , we find √ r+ ∂τ − ∂r J1 + ia sin (θ) ∂τ − ∂θ + i csc (θ) ∂φ − 2−1 cot (θ) J2 = 2 iµ? r + ia cos (θ) H1 −1 [∆ + 4M r] ∂τ + ∆ ∂r + 2a ∂φ + r − M J2 − ia sin (θ) ∂τ + ∂θ + i csc (θ) ∂φ + 2−1 cot (θ) J1 r+ √ = 2 iµ? r + ia cos (θ) H2 −1 r+ [∆ + 4M r] ∂τ + ∆ ∂r + 2a ∂φ + r − M H1 − ia sin (θ) ∂τ − ∂θ + i csc (θ) ∂φ − 2−1 cot (θ) H2 √ = 2 iµ? r − ia cos (θ) J1 √ r+ ∂τ − ∂r H2 + ia sin (θ) ∂τ + ∂θ + i csc (θ) ∂φ + 2−1 cot (θ) H1 = 2 iµ? r − ia cos (θ) J2 , (8)
5 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 √ − Σ γ 0 P † P −1 γ 0µ ∇µ + P ∂µ P −1 + im ψ 0 = 0 , (9) 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 H1 = e−i(ωτ +kφ) R+ (r)T+ (θ) H2 = e−i(ωτ +kφ) R− (r)T− (θ) J1 = e−i(ωτ +kφ) R− (r)T+ (θ) J2 = e−i(ωτ +kφ) R+ (r)T− (θ) , in which ω ∈ R and k ∈ Z + 1/2, into (8) yields the first-order radial and angular ODE systems p |∆| 0 1 e Or? R = 2 ξ Re r + a2 sign(∆) 0 Oθ T = ξ T ,
(10) (11)
where r? := r +
2 + a2 r+ r2 + a2 ln |r − r+ | − − ln |r − r− | r+ − r− r+ − r−
is the Regge–Wheeler coordinate, −ω(∆ + 4M r) − 2ak i Or? := 11C2 ∂r? + 2 p r + a2 2|∆| sign(∆) µ? r √
Oθ :=
2µ? a cos (θ)
−
p
2|∆| µ? r
ω∆
−∂θ − 2−1 cot (θ) + aω sin (θ) + k csc(θ)
(12) − 2µ? a cos (θ) p T T |∆| R+ , r+ R− are matrix-valued radial and angular operators, Re := and 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 Re of (10) is asymptotically as r → ∞ of the form ! (1) f exp iφ (r ) ∞ + ? e ? ) = Re∞ (r? ) + E∞ (r? ) = D∞ + E∞ (r? ) R(r (2) f∞ exp −iφ− (r? ) ∂θ + 2
−1
cot (θ) + aω sin (θ) + k csc(θ)
√
with the asymptotic diagonalization matrix ! cosh (Ω) sinh (Ω) sinh (Ω) cosh (Ω) D∞ := ! 1 cosh (Ω) + i sinh (Ω) sinh (Ω) + i cosh (Ω) √ sinh (Ω) + i cosh (Ω) cosh (Ω) + i sinh (Ω) 2
for ω 2 ≥ 2µ2?
for ω 2 < 2µ2? ,
6 where
Ω :=
1 ln 4 1 4 ln
! √ ω − 2µ? √ ω + 2µ?
for ω 2 ≥ 2µ2?
! √ 2µ? − ω √ 2µ? + ω
for ω 2 < 2µ2? ,
the asymptotic phases
φ± (r? ) := sign(ω) ×
p ω 2 − 2µ2? r? + 2M − p 2 2 2µ? − ω ir? + 2M
(1)
(2) T
the constants f∞ = f∞ , f∞
µ2? ±ω − p ω 2 − 2µ2? iµ2? ±ω − p 2µ2? − ω 2
! ln (r? )
for ω 2 ≥ 2µ2?
! ln (r? )
for ω 2 < 2µ2? ,
6= 0, as well as an error with polynomial decay
e ? ) − Re∞ (r? ) ≤ a/r? kE∞ (r? )k = R(r
for a suitable constant a ∈ R>0 . Lemma II.2. Every nontrivial solution Re of (10) is asymptotically as r & r± of the form h i ! (±) (1) gr± exp 2i ω + kΩKerr r? e e + Er± (r? ) R(r? ) = Rr± (r? ) + Er± (r? ) = (2) gr± (1)
(2) T
with the constants gr± = gr± , gr± decay
(±)
6= 0 and ΩKerr := a/(2M r± ), as well as an error with exponential
e ? ) − Rer (r? ) ≤ p± exp (±q± r? ) kEr± (r? )k = R(r ± for suitable constants p± , q± ∈ R>0 . 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 2 of orthonormal eigenfunctions (Tl )l∈Z in L2 (0, π), sin (θ) dθ . The corresponding eigenvalues ξl are real-valued and non-degenerate, and can thus be ordered as ξl < ξl+1 . Moreover, the eigenfunctions are pointwise bounded and smooth away from the poles, 2 2 Tl ∈ L∞ (0, π) ∩ C ∞ (0, π) . Both the eigenfunctions Tl and the eigenvalues ξl 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 (R + A )ψ 0 = 0 ,
7 where √ − 2 iµ? r √ 0 −D− 0 0 − 2 iµ r 0 −D ? + √ R := D+ 0 2 iµ? r √ 0 0 D− 0 2 iµ? r
(13)
and √ A :=
2µ? a cos (θ) √ 0 0 L 0 2µ? a cos (θ) √ L 0 L 2µ? a cos (θ) √ 0 0 L 0 0 2µ? a cos (θ)
(14)
are matrix-valued differential operators with −1 D+ := r+ [∆ + 4M r] ∂τ + ∆ ∂r + 2a ∂φ + r − M
D− := r+ (∂τ − ∂r ) L := ia sin (θ) ∂τ + ∂θ + i csc(θ) ∂φ + 2−1 cot (θ) . Separating the τ -derivative and multiplying by the inverse of the matrix 0 0 −r+ −ia sin (θ) −1 √ [∆ + 4M r] 0 0 ia sin (θ) −r+ γ e0τ := − Σ γ 0 P † P −1 γ 0τ = −1 r+ [∆ + 4M r] −ia sin (θ) 0 0 ia sin (θ) r+ 0 0
(15)
(cf. Eq. (9)) as well as by the imaginary unit leads to i∂τ ψ 0 = −i (e γ 0τ )−1 R (3) + A (3) ψ 0 =: Hψ 0 ,
(16)
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 H = αj ∂j + V ,
j ∈ {r, θ, φ} ,
(17)
with the matrix-valued coefficients
i∆ r+ a sin (θ) 0 0 r−1 ∆a sin (θ) −i(∆ + 4M r) 1 0 0 + αr := − −1 0 0 −i(∆ + 4M r) r+ ∆a sin (θ) Σ + 2M r 0 0 r+ a sin (θ) i∆
−a sin (θ) ir+ 0 0 ir−1 [∆ + 4M r] a sin (θ) 1 0 0 + αθ := − −1 0 0 −a sin (θ) −ir+ [∆ + 4M r] Σ + 2M r 0 0 −ir+ a sin (θ)
(18)
ia r+ csc(θ) 0 0 r−1 csc(θ)(∆ − 2Σ) 1 −ia 0 0 + αφ := − −1 0 0 −ia r csc(θ)(∆ − 2Σ) Σ + 2M r + 0 0 r+ csc(θ) ia
(19)
(20)
8 and the potential 1 V := − Σ + 2M r
B1 B2 B3 B4
,
(21)
where the quantities Bk , k ∈ {1, 2, 3, 4}, are the (2 × 2)-blocks i(r − M ) − 2−1 a cos (θ) 2−1 i r+ cot (θ) B1 := −1 −1 −1 r+ a sin (θ)(r − M ) + 2 i cot (θ)(∆ + 4M r) 2 a cos (θ) B2 := √
−
√ − 2µ? r+ r − ia cos (θ)
B4 :=
√ −1 2 iµ? a sin (θ) r − ia cos (θ) − 2 r+ µ? (∆ + 4M r) r − ia cos (θ)
√
B3 :=
√ − 2 iµ? a sin (θ) r − ia cos (θ)
−1 2 r+ µ? (∆
+ 4M r) r + ia cos (θ)
√
√ − 2 iµ? a sin (θ) r + ia cos (θ) −2−1 a cos (θ) −1
−2
(22) 2 iµ? a sin (θ) r + ia cos (θ) √ − 2µ? r+ r + ia cos (θ)
−1 r+ a sin (θ)(r − M ) − 2−1 i cot (θ)(∆ + 4M r)
i(r − M ) + 2
ir+ cot (θ)
Furthermore, we work with the scalar product Z (ψ|φ)|Nτ :=
−1
.
a cos (θ)
≺ψ|/ ν φ dµNτ
(23)
Nτ
defined on the space-like hypersurface Nτ := {τ = const., r, θ, φ} [9], where ≺ · | · : Sx M × Sx M → C , (ψ, φ) 7→ ψ ? φ
(24)
denotes the indefinite spin scalar product of signature (2, 2), ψ ? := ψ † S the adjoint Dirac spinor, ν / = γ µ νµ the Clifford contraction of the future-directed, time-like normal ν, and dµ|Nτ = q |det(g |Nτ )| dφ dθ dr with the induced (Riemannian) metric g |Nτ is the invariant measure on Nτ . (Note that this scalar product is independent of the choice of the specific space-like hypersurface.) The matrix S is defined via the relation γ µ† := S γ µ S −1 .
(25)
With (15) and the spinor transformation (7), we find −1 0 0µ 1 γµ = − √ P † γ γ e P Σ and thus, using (25), we obtain for the matrix S the term 0 0 S = 1 0
0 0 0 1
1 0 0 0
0 1 . 0 0
The vector field ν is determined by means of the conditions hν|∂r i|g = hν|∂θ i|g = hν|∂φ i|g = 0
and
hν|νi|g = 1 ,
9 where h · | · i|g := g( · , · ) is the spacetime inner product on the manifold M given by (2), yielding ν=
2M r 1+ Σ
1/2
2M r ∂τ − Σ
2M r 1+ Σ
−1/2 ∂r .
The corresponding dual co-vector reads ν=
1+
2M r Σ
−1/2 dτ .
(26)
Moreover, the induced metric g |Nτ on Nτ is the restriction of (2) from M to the submanifold Nτ g |Nτ
2M r =− 1+ dr − a sin2 (θ) dφ ⊗ dr − a sin2 (θ) dφ − Σ dθ ⊗ dθ − Σ sin2 (θ) dφ ⊗ dφ , Σ
for which q
1/2 2M r |det(g |Nτ )| = Σ sin (θ) 1 + Σ
(27)
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 ZZZ ψ 0† S 0 γ 0τ φ0 Σ sin (θ) dφ dθ dr . (28) (ψ 0 |φ0 )|Nτ = −1 0 0τ √ Again using (15), that is with γ 0τ = −P P † γ γ e / Σ, the scalar product (28) becomes ZZZ −1 0 0τ 0 √ ψ 0† S 0 P P † γ γ e φ Σ sin (θ) dφ dθ dr (ψ 0 |φ0 )|Nτ = − ZZZ
ψ 0† PP † S 0 P P †
ZZZ
ψ 0† PS P †
=−
=−
ZZZ =−
ZZZ =
−1
ψ 0† S P † P †
−1
γ0 γ e0τ φ0 sin (θ) dφ dθ dr
γ0 γ e0τ φ0 sin (θ) dφ dθ dr
−1
(29)
γ0 γ e0τ φ0 sin (θ) dφ dθ dr
ψ 0† Γτ φ0 sin (θ) dφ dθ dr ,
where −1 r+ [∆ + 4M r] −ia sin (θ) 0 0 ia sin (θ) r+ 0 0 . = 0 0 r+ ia sin (θ) −1 0 0 −ia sin (θ) r+ [∆ + 4M r]
Γτ := −S γ 0 γ e0τ
(30)
√ Note that in the above derivation, we have first applied the relation Σ 11C4 = PP † , then the transformation law for the matrix S 0 , namely S = P † S 0 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
10 integration limits are suppressed for ease of notation if possible and given if necessary. The eigenvalues λ1 , λ2 of the matrix (30) are positive and with algebraic multiplicities µA (λ1 ) = 2 = µA (λ2 ) s 2 ∆ + 4M r 1 ∆ + 4M r r+ + + 4a2 sin2 (θ) > 0 λ1 = + r+ − 2 r+ r+
s 2 1 ∆ + 4M r ∆ + 4M r λ2 = r+ + − 4 (Σ + 2M r) > 0 , − r+ + 2 r+ r+ 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 (ψ 0 |Hφ0 )|Nτ = (Hψ 0 |φ0 )|Nτ , we begin by splitting the potential V given in (21) into mass-independent and mass-dependent parts V = V0 + Vµ? , where V0 := −
1 Σ + 2M r
B1 0C2 0C2 B4
and Vµ? := −
1 Σ + 2M r
0C2 B2 B3 0C2
with the (2 × 2)-blocks Bk defined in (22), therefore obtaining anti-self-adjoint and self-adjoint matrices Γτ V0 = −V0† Γτ and Γτ Vµ? = Vµ†? Γτ τ
(31)
τ†
with Γ = Γ
defined in (30). This leads to ZZZ 0 0 ψ 0† Γτ H φ0 sin (θ) dφ dθ dr (ψ |Hφ )|Nτ = ZZZ =
ψ 0† Γτ αj ∂j (φ0 ) sin (θ) dφ dθ dr +
ZZZ +
ZZZ
ψ 0† Γτ V0 φ0 sin (θ) dφ dθ dr
ψ 0† Γτ Vµ? φ0 sin (θ) dφ dθ dr .
Integration by parts of the first triple integral and substitution of the relations (31) in the remaining two triple integrals yields ZZZ ZZZ 0 0 0 0† τ j (ψ |Hφ )|Nτ = − ∂j ψ Γ α sin (θ) φ dφ dθ dr − ψ 0† V0† Γτ φ0 sin (θ) dφ dθ dr ZZZ
ψ 0† Vµ†? Γτ φ0 sin (θ) dφ dθ dr
ZZZ
∂j (ψ 0† ) Γτ αj φ0 sin (θ) dφ dθ dr
+
=−
ZZZ
ψ 0† ∂j (Γτ ) αj + Γτ ∂j (αj ) + Γτ αθ cot (θ) φ0 sin (θ) dφ dθ dr
ZZZ
(V0 ψ 0 )† Γτ φ0 sin (θ) dφ dθ dr +
−
−
(32)
ZZZ
(Vµ? ψ 0 )† Γτ φ0 sin (θ) dφ dθ dr .
11 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 ZZ r2 ψ 0† Γτ αr φ0 sin (θ) dφ dθ . r1
S2
Direct computation of the matrix Γτ αr gives ∆ ∆ Γ α = i diag − , r+ , r+ , − r+ r+ τ
r
and hence the radial boundary terms become ZZ r2 ∆ ∆ ir+ − 2 ψ 0 1 φ01 + ψ 0 2 φ02 + ψ 0 3 φ03 − 2 ψ 0 4 φ04 sin (θ) dφ dθ . r+ r+ r1 S2 In order for this expression to vanish, we impose the radial Dirichlet-type boundary condition 2 X
∆ 0 0 ∆ 0 0 0 0 0 0 (−1) − 2 ψ 1 φ1 + ψ 2 φ2 + ψ 3 φ3 − 2 ψ 4 φ4 = 0 . r+ r+ ri i=1 i
(33)
As in the next section only Dirac spinors with supp φ0 = [r0 , ∞)×S 2 and thus radial boundary conditions at a specific time-like inner boundary r = r0 < 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 = r0 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 0 γ 0r = i Γτ αr /Σ, we may represent (33) as ≺ψ 0 |γ 0r φ0 |{τ }×{r0 }×S 2 = 0 .
(34)
Introducing n as the unit normal to the hypersurfaces {τ } × S 2 , we can write (34) in the form ≺ψ 0 |/ n φ0 |{τ }×{r0 }×S 2 = 0
⇐
0 (/ n − i)ψ|{τ }×{r0 }×S 2 = 0 ,
(35)
2 where the slash again denotes Clifford multiplication. With n / = −11C4 , the above implication can be easily verified by
≺/ n ψ 0 |/ n φ0 |{τ }×{r0 }×S 2 = ≺ψ 0 |/ n2 φ0 |{τ }×{r0 }×S 2 = − ≺ψ 0 |φ0 |{τ }×{r0 }×S 2 = −≺ − i ψ 0 | − i φ0 |{τ }×{r0 }×S 2 ⇔ ≺(/ n − i) ψ 0 |(/ n − i) φ0 |{τ }×{r0 }×S 2 = 0 . 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 ∂j (Γτ ) αj + Γτ ∂j (αj ) + Γτ αθ cot (θ) = −2V0† Γτ . Moreover, all three matrix products Γτ αj , j ∈ {r, θ, φ}, are anti-self-adjoint Γτ αj = −αj† Γτ † = −αj† Γτ .
12 Therefore, we immediately find that ZZZ ZZZ (ψ 0 |Hφ0 )|Nτ = ∂j (ψ 0† ) αj† Γτ φ0 sin (θ) dφ dθ dr + 2 (V0 ψ 0 )† Γτ φ0 sin (θ) dφ dθ dr ZZZ −
ZZZ
0
(αj ∂j ψ 0 )† Γτ φ0 sin (θ) dφ dθ dr +
ZZZ +
ZZZ
IV.
τ
ZZZ
(Vµ? ψ 0 )† Γτ φ0 sin (θ) dφ dθ dr
ZZZ
(V0 ψ 0 )† Γτ φ0 sin (θ) dφ dθ dr
(V0 ψ ) Γ φ sin (θ) dφ dθ dr +
=
=
0 †
(Vµ? ψ 0 )† Γτ φ0 sin (θ) dφ dθ dr
(Hψ 0 )† Γτ φ0 sin (θ) dφ dθ dr = (Hψ 0 |φ0 )|Nτ .
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 dEω arising in the formal spectral decomposition of the propagator Z −iτ H U (τ ) = e = e−iωτ dEω . (36) R
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 P (x, ξ) = −i (γ 0τ )−1 γ 0j ξj , where x = (τ, r, θ, φ) and ξ ∈ Tx? M. A short computation yields det P (x, ξ) =
g ij ξi ξj gτ τ
2 .
The Hamiltonian fails to be elliptic if the determinant vanishes, that is, if g ij ξi ξj = 0 for ξ being non-zero. With g |Nτ = −Σ−1 ∆ ∂r ⊗ ∂r + a [∂r ⊗ ∂φ + ∂φ ⊗ ∂r ] + ∂θ ⊗ ∂θ + csc2 (θ) ∂φ ⊗ ∂φ , 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)
13
FIG. 1: Carter–Penrose diagram of the region M of Kerr geometry with constant-τ hypersurfaces Nτ1 and Nτ2 cut-off at the boundary ∂M . Cauchy data is propagated in τ -direction along the constant-τ hypersurfaces. A radial Dirichlet-type boundary condition imposed on ∂M leads to a reflection of the Dirac particles, which ensures unitarity, without affecting their dynamics outside the Cauchy horizon.
and the angular projector corresponding to the angular ODE system (11) that arise in Chandrasekhar’s separation of variables. In the following, in order to formulate the Cauchy problem for the Dirac equation in Hamiltonian form and the domain of definition of the self-adjoint extension of the Hamiltonian, the geometrical and functional-analytic settings are stated. Let (M, g) be the non-extreme Kerr geometry with the metric (2) in horizon-penetrating Eddington– Finkelstein-type coordinates (3). We consider the subset M := {τ, r > r0 , θ, φ} ⊂ M with r0 < r− . Furthermore, we introduce the time-like inner boundary ∂M := {τ, r = r0 , θ, φ} of M and the family of space-like hypersurfaces N = (Nτ )τ ∈R with Nτ := {τ = const., r > r0 , θ, φ} and boundaries ∂Nτ := ∂M ∩ Nτ ' S 2 (see 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 K = ∂τ + b ∂φ , where b = b(r0 ) ∈ R\{0} is a constant [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 = L2 (Nτ , 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 Csc (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 0 = Hψ 0 0 0 ∞ ψ|τ =0 = ψ0 ∈ C0 Nτ =0 , SM with the radial Dirichlet-type boundary condition at ∂M given by 0 (/ n − i)ψ|∂M = 0,
14 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., (/ n − i)(H p ψ00 ) = 0 ∀ p ∈ N0 , has a unique global solution in the class of smooth wave functions with spatially compact support 0 ∞ ψ ∈ Csc (M , SM ) (/ n − i) H p ψ 0 |∂M = 0 ∀ p ∈ N0 . Evaluating this solution at subsequent times τ and τ 0 gives rise to a unique unitary propagator 0 ∞ ∞ U τ ,τ : Csc Nτ , SM → Csc Nτ 0 , SM . 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)-(22), in a specific domain of the Hilbert space H , (·|·)|Nτ . Theorem IV.2. The massive Dirac Hamiltonian H in the non-extreme Kerr geometry in horizonpenetrating advanced Eddington–Finkelstein-type coordinates with domain of definition n − i)(H p ψ 0 )|∂Nτ = 0 ∀ p ∈ N0 ⊂ H D(H) = ψ 0 ∈ C0∞ Nτ , SM (/ is essentially self-adjoint. 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 − ωc )−1 ∈ L(H ) exists for all ωc ∈ C\R with real part 0
(
r → r+ and =(ω ) > 0 r → r− and =(ω ) < 0
for
for
=(ω ) > 0 if |ω |2 ≥ 2µ2? 0 or |ω |2 < 2µ2? and 0
lim Φ2 = γχ1 + δχ2
for
=(ω ) < 0 ,
&0
&0
20 where α, β, γ, and δ are constants. The corresponding Wronskian yields ( β W (χ1 , χ2 ) for =(ω ) > 0 lim W (Φ1 , Φ2 ) = &0 −γ W (χ1 , χ2 ) for =(ω ) < 0 . Then, we obtain for the non-zero 2 × 2 blocks of lim&0 (H − ω − i)−1 − (H − ω + i)−1 in (44) χk,1 (r)χl,2 (r0 ) χk,1 (r)χl,1 (r0 ) 2 − X 1 ∆(r0 ) r+ Tk,l lim [G(r; r0 )l,k,ω+i − G(r; r0 )l,k,ω−i ] = χk,2 (r)χl,2 (r0 ) χk,2 (r)χl,1 (r0 ) &0 W (χ1 , χ2 )(r0 ) − k,l=1 ∆(r0 ) r+
with the coefficients T1,1 =
α , β
T1,2 = T2,1 = 1, and T2,2 =
δ . γ
Appendix: Functions for the Construction of the Green’s Matrix of the Radial ODE System
We specify the functions Φ1 (r, r0 ) and Φ2 (r, r0 ) that are used for the construction of the radial Green’s matrix that solves Eq. (41). Since their explicit forms are not known, we describe them in terms of asymptotic expansions. To this end, we define, on the one hand, functions with suitable decay at infinity ( " # c 1 =(ω ) < 0 if |ω |2 ≥ 2µ2? 1 1,∞ (∞) b for Φ (r) := p exp iφ+ (r? ) 11C2 + O 0 r? 0 if |ω |2 < 2µ2? |∆|
q (∞) (r) := c2,∞ Φ
" # 1 0 exp −iφ− (r? ) 11C2 + O 1 r?
for
( =(ω ) > 0 if |ω |2 ≥ 2µ2? 0
h i 0 b (−) (r) := c1,r 11C2 + O exp (−qr? ) Φ − 1
for
=(ω ) < 0
for
=(ω ) > 0 .
h i h i c2,r− (−) q (−) (r) := p exp 2i ω + kΩKerr r? 11C2 + O exp (−qr? ) Φ |∆|
1 0
The quantities c1/2,∞ and c1/2,r± are constants. To clarify our notation, we point out that the superscripts (∞), (+), and (−) designate the asymptotic expansions at infinity, the event horizon, and the Cauchy horizon, respectively. We furthermore remark that the existence of ODE solutions with these asymptotics follows from the construction of Jost solutions (for which one rewrites the radial first-order system (40) as a second-order scalar equation and proceeds as in [1] or [12]) and that our asymptotic expansions ensure that the solutions are square integrable near the horizons. For example,
21 as the Regge–Wheeler coordinate r? tends to minus infinity at the event horizon, the exponential fac (+) tor exp 2i [ω + kΩKerr ] r? tends to zero if =(ω ) < 0. However, this exponential factor would not be square integrable if =(ω ) > 0. Last, we introduce a function that satisfies the Dirichlet-type boundary condition at r = r0 p −r+ / |∆| (2) , Φ∂M (r) := c0 Φ∂M (r) 1 (2)
where c0 is a constant and Φ∂M denotes the second component. Then, in case |ω |2 ≥ 2µ2? and =(ω ) < 0 or |ω |2 < 2µ2? , =(ω ) < 0, and 0, the radial functions Φ1 and Φ2 can be expressed as b (∞) (r) Φ1 (r, r+ < r0 ) = Θ(r − r0 ) Φ b (+) (r) Φ2 (r, r+ < r0 ) = Θ(r0 − r) Θ(r − r+ ) Φ b (−) (r) + Θ(r − r+ ) Φ b (∞) (r) Φ1 (r, r− < r0 < r+ ) = Θ(r − r0 ) Θ(r+ − r) Φ b (+) (r) Φ2 (r, r− < r0 < r+ ) = Θ(r − r0 ) Θ(r+ − r) Φ
(46)
b (−) (r) + Θ(r − r+ ) Φ b (∞) (r) Φ1 (r, r0 < r0 < r− ) = Θ(r − r0 ) Θ(r+ − r) Φ Φ2 (r, r0 < r0 < r− ) = Θ(r0 − r) Φ∂M (r) , whereas in case |ω |2 ≥ 2µ2? and =(ω ) > 0 or |ω |2 < 2µ2? , =(ω ) > 0, and 0, and 0 or |ω |2 < 2µ2? , =(ω ) < 0, and
