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

Röken, Christian

doi:10.1007/s10714-017-2194-y

Cited by 17 publications

(15 citation statements)

References 36 publications

(82 reference statements)

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“…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) [22] by…”

Section: Preliminariesmentioning

confidence: 99%

“…This horizon-penetrating coordinate system possesses a proper time function, unlike the original advanced Eddington-Finkelstein (null) coordinates [8,9]. It is advantageous to describe the Kerr geometry in the Newman-Penrose formalism using a regular Carter tetrad [4,22]…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus, since the 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 [18], and that specific spin coefficients are linearly dependent. Substituting the Carter tetrad (3) into -and solving -the first Maurer-Cartan equation of structure, we obtain the spin coefficients [22]…”

Section: Preliminariesmentioning

confidence: 99%

“…In the following, these results are recalled. For a detailed analysis and proofs see [22]. Substituting the separation ansatz…”

Section: Preliminariesmentioning

confidence: 99%

“…The methods used in the derivation of our integral spectral representation are quite different from those employed in [11], as is now outlined. We work with horizon-penetrating advanced Eddington-Finkelstein-type coordinates [22], i.e., with an analytic extension of Boyer-Lindquist coordinates that simultaneously covers both the exterior and the interior black hole region without exhibiting singularities at the horizons and features a proper time function τ . Furthermore, we employ a regular Carter tetrad, i.e., a symmetric Newman-Penrose null frame, which reflects, on the one hand, the discrete time and angle reversal isometries of the Kerr geometry and, on the other hand, its Petrov type.…”

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confidence: 99%

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An integral spectral representation of the massive Dirac propagator in the Kerr geometry in Eddington–Finkelstein-type coordinates

Finster

Röken

2018

Advances in Theoretical and Mathematical Physics

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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. Contents

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…In the following, these results are recalled. For a detailed analysis and proofs see [22]. Substituting the separation ansatz…”

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

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An integral spectral representation of the massive Dirac propagator in the Kerr geometry in Eddington–Finkelstein-type coordinates

Finster

Röken

2018

Advances in Theoretical and Mathematical Physics

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Scattering‐state solutions to the Dirac spinors with negative‐energy eigenstates in a rotating spheroid

2023

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Based on the work of Fu et al. (2020), we derive the rest seven scattering-state solutions to the Dirac equation when E = −im and establish a relation between differential scattering cross-section, 𝜎 i * (p, 𝜃, 𝜑), and stellar matter density, 𝜇, using the long-wave approximation. It is found that the sensitivity of average scattering cross-sections 𝜎 i (p, 𝜃) to the change in the stellar matter density is proportional to 𝜇 2 . We find that the average scattering amplitudes f i (p, 𝜃), as well as average scattering cross-sections 𝜎 i (p, 𝜃), are independent of the masses of particles, m, for four scattering states 𝜒 (i) , i = 0, 1, 2 and 3, while f i (p, 𝜃) and 𝜎 i * (p, 𝜃) depend on m, for the rest four scattering states, 𝜙 (i) , i = 0, 1, 2 and 3. By measuring the scattering cross-section, we can determine the density of the ellipsoid and obtain its gravitational properties. This work will be useful in understanding the properties of anti-Dirac spinors and the physical effects in a rotating spheroid.

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Dirac spinor scattering states with positive‐energy in rotating spheroid models

Gao,

Chen,

Wang

et al. 2024

Astron Nachr

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There are many rotating spheroids in the universe, and many astronomers and physicists have used theoretical methods to study the characteristics of stellar gravity since Newton's time. This paper derives the solutions of eight scattering states ,, and for the Dirac equation with positive‐energy , and establishes the relationship between the differential scattering cross section and the stellar density . It is found that: (1) For the eight scattering states, their average scattering cross‐sections are proportional to , and depend on the star's radius, and the higher the stellar density , the greater the sensitivity of to the change of ; (2) For the four scattering states , their average scattering amplitudes and depend on the mass of the particles; while for the other four scattering states , , then and are independent of . This study links the gravitational characteristics of stars with the scattering cross section, creating a new method for studying the gravitational characteristics, which helps to reveal the mystery of the gravity of rotating ellipsoidal stars.

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

Cited by 17 publications

References 36 publications

An integral spectral representation of the massive Dirac propagator in the Kerr geometry in Eddington–Finkelstein-type coordinates

An integral spectral representation of the massive Dirac propagator in the Kerr geometry in Eddington–Finkelstein-type coordinates

Scattering‐state solutions to the Dirac spinors with negative‐energy eigenstates in a rotating spheroid

Dirac spinor scattering states with positive‐energy in rotating spheroid models

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