The limiting absorption principle for Dirac operators in the non-extreme Kerr-Newman metric

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(1)立命館大学理工学研究所紀要第76号 2017年 Memoirs of the Institute of Science and Engineering, Ritsumeikan University, Kusatsu, Shiga, Japan. No. 76, 2017. The limiting absorption principle for Dirac operators in the non-extreme Kerr-Newman metric Monika Winklmeier∗ and Osanobu Yamada∗∗. Abstract. We consider the time-dependent Dirac equation in the nonextreme Kerr-Newman metric describing the black hole physics. In order to see the local energy decay at infinity we show a result of the limiting absorption principle and the absolute continuity of the spectrum for the corresponding Dirac operator. Keywords: Limiting absorption principle, Dirac operator, Kerr-Newman metric. 2010 MSC : Primary, 47A40, Secondary, 83C57.. ========================================== * Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia. e-mail : [email protected] ** Faculty of Science and Engineering, Ritsumeikan University, Kusatsu, Japan. e-mail : [email protected]. 1. −15−. This document is provided by JAXA..

(2) Monika Winklmeier, Osanobu Yamada. 1. Introduction. The Kerr-Newman metric is the most general stationary solution of Einstein’s field equations and has the physical interpretation of a massive charged rotating black hole. The physical parameters M , Q and a = J/M have the interpretation as mass, electric charge and angular momentum parameter of the black hole. As in flat spacetime, a spin- 12 particle is described by a Dirac equation. In the Kerr-Newman metric, the Dirac equation is given by the coupled system of partial differential equations =0 (R + A)Ψ where ⎛. imr 0 ⎜ 0 −imr R := ⎜ ⎝ Δ(r) Rt,ϕ − 0 t,ϕ 0 Δ(r) R+. ⎞ Δ(r) Rt,ϕ + 0 t,ϕ 0 Δ(r) R− ⎟ ⎟, ⎠ −imr 0 0 imr. ⎛. ⎞ −D 0 0 Lt,ϕ + ⎜ 0 D −Lt,ϕ 0 ⎟ − ⎟ A := ⎜ t,ϕ ⎝ 0 L+ −D 0 ⎠ −Lt,ϕ 0 0 D − and D := am cos θ,. i ∂ ∂ ∂ t,ϕ 2 2 ± + eQr on (r+ , ∞), R± := (r + a ) i + a i ∂r Δ(r) ∂t ∂ϕ. cot θ ∂ 1 ∂ ∂ t,ϕ + ∓ a sin θ i + i on (0, π). L± := ∂t 2 ∂t sin θ ∂ϕ The physical parameters m and e are the mass and the electric charge of the The functions spin- 12 -particle which is described by the wave function Ψ. Δ(r) and Σ(r, θ) are given by Δ(r) = r2 − 2M r + a2 + Q2 = (r − M )2 + a2 + Q2 − M 2 , Σ(r, θ) = r2 + a2 cos2 θ. According to the black hole parameters M , Q and a, three cases can arise: If + Q2 − M 2 < 0, then the function Δ(r) has exactly two distinct zeros; a2. 2. −16−. This document is provided by JAXA..

(3) The limiting absorption principle for Dirac operators in the non-extreme Kerr-Newman metric. this case is called the non-extreme Kerr-Newman metric. If a2 +Q2 −M 2 = 0, then the function Δ(r) has exactly one zero; this case is referred to as the extreme Kerr-Newman metric. In both cases the metric is interpreted physically as the spacetime generated by a charged rotating massive black hole. If a2 + Q2 − M 2 > 0, then the function Δ(r) has no zeros; in this case the Kerr-Newman metric has no interpretation as the metric of a black hole. We proved in [WY1] the already known result that there are no timeperiodic solutions in the non-extreme case. In the follow-up paper [WY2] we showed that in the extreme case for fixed black hole data there is a sequence of particle masses (mN )N ∈N for which a time-periodic solution of the Dirac equation does exist. A review of earlier and later various topics can be found in [FKSY]. In this note we show that in the non-extreme case any solution of (1) has the local energy decay at t → ±∞ by proposing the limiting absorption principle for the Dirac operator H and the absolute continuity of the spectrum.. 2. The results. Let us consider the non-extreme Kerr-Newman case. Then the function Δ has two different zeros and it can be written as Δ(r) = (r − r+ )(r − r− ) where r− < r+ are positive constants and we consider time-dependent Dirac equations on the Hilbert space 2.

(4). L. r 2 + a2 sin θ dr dθ dϕ (r+ , ∞) × (0, π) × (0, 2π), Δ(r). 4. dr If we introduce the new variable x the variable satisfying dx = rΔ(r) 2 +a2 , trans1 √ form the solution Ψ to sin θ Ψ and and expand it in a Fourier series with respect to ϕ, 1 θ, ϕ, t) = Ψ(r, e−ikϕ √ Ψk (x, θ, t) sin θ k. our problem reduces to solve iE(x, θ). ∂ Ψ (x, θ, t) = LΨ (x, θ, t) = (L0 + V )Ψ (x, θ, t) ∂t.

(5). ∂ = Σ3 −i + g(x)B + V (x, θ) Ψ (x, θ, t) ∂x. (1). 3. −17−. This document is provided by JAXA..

(6) Monika Winklmeier, Osanobu Yamada. in the Hilbert space H := L2 (R × (0, π), dx dθ)4 , where E(x, θ) = I4 + f (x, θ)Σ2 , Δ(r) a sin θ Δ(r) , g(x) = 2 , f (x, θ) = r 2 + a2 r + a2.

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(8). k ∂ σj 0 − Σ2 , Σ j = B = Σ1 −i , 0 −σj ∂θ sin θ .

(9) mr Δ(r) 0 I2 ak + eQr V (x, θ) = − 2 I4 − 2 I2 0 r + a2 r + a2 .

(10) am Δ(r) cos θ 0 i I2 + −i I2 0 r 2 + a2 (σ1 , σ2 , σ3 are the Pauli matrices, k ∈ Z + (1/2), and e, m, Q are positive constants), g(x) = O(1/x), g (x) = O(1/x2 ) as x → +∞, g(x) = O(eγx ), g (x) = O(eγx ) as x → −∞, 1 |f (x, θ)| ≤ . 2 V (x, θ) is a 4 × 4 Hermitian matrix-valued function satisfying.

(11) 0 I2 + O(1/x), Vx (x, θ) = O(1/x2 ) as x → +∞, V (x, θ) = −m I2 0 V (x, θ) = a0 I4 + O(eγx ), Vx (x, θ) = O(eγx ) as x → −∞ ak + eQr+ and γ is a 2 + a2 r+ positive number. We remark (1/2)I4 ≤ E(x, θ) ≤ (3/2)I.. uniformly with respect to 0 < θ < π where a0 = −. Recall the definition of L in (1). Let Ω+ := (−1, ∞), Ω− := (−∞, +1). Let L± be the self-adjoint operator of L on L2 (Ω± × (0, π)) with the domain π d 2 2 2 dx dθ Ψ + g |Ψ | < ∞, dx Ω± 0 the boundary conditions Ψ1 (∓1) + i Ψ2 (∓1) = 0,. Ψ3 (∓1) + i Ψ4 (∓1) = 0, 4. −18−. This document is provided by JAXA..

(12) The limiting absorption principle for Dirac operators in the non-extreme Kerr-Newman metric. respectively, and define H± := E −1/2 L± E −1/2 . Let us consider the eigenvalue equation (H± − z)Φ = (E −1/2 L± E −1/2 − z)Φ = F, where z = z1 + iz2 ∈ C. Setting Ψ := E −1/2 Φ and G = E 1/2 F we have (L± − zS)Ψ = G,.

(13). ∂ Σ3 −i + g(x)B + V (x, θ) − zS Ψ = G in Ω± . ∂x. (2). Introducing.

(14) ∂ 0 mI2 := Σ3 −i + gB − mI2 0 ∂x.

(15) ∂ + gB L− 0 := Σ3 −i ∂x. L+ 0.

(16). (x ∈ Ω+ ), (x ∈ Ω− ),. we obtain from (2) (L− 0. + (L+ 0 + W − z)Ψ = G. +W. −. − (z − a0 ))Ψ = G. (x ∈ Ω+ ). (3). (x ∈ Ω− ),. (4). where

(17). +. W (x, θ) := V (x, θ) + m. 0 I2 I2 0. − zf (x, θ)Σ2 ,. W − (x, θ) := V (x, θ) − a0 − zf (x, θ)Σ2 . We remark that

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(19). 2 ∂2 ∂ 2 ) = Σ + m2 = − 2 + g 2 B 2 − ig Σ3 B + m2 , (L+ −i + gB 3 0 ∂x ∂x

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(21). 2 ∂2 ∂ 2 (L− = Σ3 −i + gB = − 2 + g 2 B 2 − ig Σ3 B, 0) ∂x ∂x which implies also g 2 B 2 − ig Σ3 B ≥ 0. We shall use also the following estimates W + (x, θ) = O(1/x), Wx+ (x, θ) = O(1/x2 ) as x → +∞, W − (x, θ) = O(eγx ), Wx− (x, θ) = O(eγx ) as x → −∞, 5. −19−. This document is provided by JAXA..

(22) Monika Winklmeier, Osanobu Yamada. uniformly with respect to |z| ≤ C for any fixed C > 0. Then it follows that the essential spectrum of H+ := E −1/2 L+ E −1/2 consists of R \ (−m, m), while there may be a sequence of discrete eigenvalues {λn } in (−m, m). On the other hand, the essential spectrum of H− := E −1/2 L− E −1/2 covers the whole real line. Let us consider the equation (3) on Ω+ . + (L0 + W − z)Ψ = (L+ 0 + W − z)Ψ = G.. If we apply (L+ 0 + z) to the equality above, we obtain + + + (L+ 0 + z)(L0 + W − z)Ψ = (L0 + z)G. =⇒ =⇒. + + + + (L+ 0 + z)(L0 − z)Ψ + (L0 + z)W Ψ = (L0 + z)G + + 2 2 + (L+ 0 ) Ψ − z Ψ + (L0 + z)W Ψ = (L0 + z)G. that is, ∂2 + + + 2 2 Ψ +g 2 B 2 Ψ −ig Σ3 B +L+ 0 (W Ψ )+zW Ψ −(z −m )Ψ = (L0 +z)G. ∂x2 (5) We can now apply the limiting absorption principle of second-order differential operators (LAP) to (5) developed by e.g. [IS], [J], [M] and [S].. −. Theorem 1 (LAP for H+ ). Let [a, b] be any closed interval such that [a, b] does not contain m, −m nor any λn . Then for any sufficiently small ε > 0 there exists a positive constant C = C(a, b, ε) such that 2. (H+ − z)−1 F 2−(1+ε)/2 ≤ C F 2(1+ε)/2 + L+ F. 0 (1+ε)/2 for any z = z1 + iz2 such that a ≤ z1 ≤ b and 0 < |z2 | ≤ 1 and any F in a suitable L2 -space. Let us now deal with L− . We start with (4) and apply (L− 0 + z) to it and find − − − (L− 0 + (z − a0 ))(L0 + W − (z − a0 ))Ψ = (L0 + z)G. =⇒. − − 2 2 − (L− 0 ) Ψ − (z − a0 ) Ψ + (L0 + z − a0 )W Ψ = (L0 + z)G,. 6. −20−. This document is provided by JAXA..

(23) The limiting absorption principle for Dirac operators in the non-extreme Kerr-Newman metric. hence −. ∂2 − − 2 Ψ + g 2 B 2 Ψ − ig Σ3 B + L− 0 (W Ψ ) + (z − a0 )W Ψ − (z − a0 ) Ψ ∂x2 = (L− 0 + z − a0 )G.. Theorem 2 (LAP for H− ). Let [a, b] be any closed interval such that [a, b] does not contain a0 . Then for any sufficiently small ε > 0, there exists a positive constant C = C(a, b, ε) such that 2. (H− − z)−1 F 2−(1+ε)/2 ≤ C F 2(1+ε)/2 + L− F. 0 (1+ε)/2 for any z = z1 + iz2 such that a ≤ z1 ≤ b and 0 < |z2 | ≤ 1 and any F . Now let us consider (H − z)Φ = (E −1/2 LE −1/2 − z)Φ = F.. (6). Take C ∞ -functions γ± (x) such that 0 ≤ γ± (x) ≤ 1, γ+ (x) + γ− (x) ≡ 1 and 1 (x ≥ 1) 1 (x ≤ −1) , γ− (x) = . γ+ (x) = 0 (x ≤ −1) 0 (x ≥ 1) The equation (6) gives (H+ − z)(γ+ Φ) = γ+ F − iγ+ Σ3 Φ,. (H+ − z)(γ− Φ) = γ− F − iγ− Σ3 Φ.. Applying Theorem 1, 2 to the above equation and using the elliptic estimate Σ Φ we have for iγ± 3 Theorem 3 (LAP for H). Let [a, b] be any closed interval such that [a, b] does not contain m, −m, a0 nor any λn . Then for any sufficiently small ε > 0 there exists a positive constant C = C(a, b, ε) such that . (H − z)−1 F 2−(1+ε)/2 ≤ C F 2(1+ε)/2 + L0 F 2(1+ε)/2 for any z = z1 + iz2 such that a ≤ z1 ≤ b and 0 < |z2 | ≤ 1 and any F . Corollary 4. H is (spectrally) absolutely continuous in R\{m, −m, a0 , {λn }}. Since H has no eigenvalues (see, e.g. [WY1]), we have 7. −21−. This document is provided by JAXA..

(24) Monika Winklmeier, Osanobu Yamada. Theorem 5. H is purely absolutely continuous in R. Corollary 6 (Local energy decay). The local energy of the solution Ψ (x, θ, t) of (1) tends to zero as t → ±∞, that is, lim. R π. t→±∞ −R 0. |Ψ (x, θ, t))|2 dx dθ = 0. for any R > 0. We will give detailed and complete proofs for the above theorems elsewhere. Acknowledgement. The second author would like to thank Takashi Okaji for valuable discussions.. References [FKSY] F. Finster, N. Kamran, J. Smoller and S. -T. Yau, Linear waves in the Kerr geometry: A mathematical voyage to black hole physics, Bull. AMS, 46 (2009), 635–659. [IS]. T. Ikebe and Y. Sait¯o, Limiting absorption method and absolute continuity for the Schrödinger operator, J. Math. Kyoto Univ., 12 (1972), 513–542.. [J]. W. Jäger, Ein gewöhnlicher Differentialoperator zweiter Ordnung f¨ ur Funktionen mit Werten in einem Hilbertraum, Math. Z., 113 (1970), 68– 98.. [M]. K. Mochizuki, Spectral and scattering theory for second-order partial differential operator, Monographs and Research Notes in Mathematics, CRC Press, 2017.. [S]. Y. Sait¯o, The principle of limiting absorption for second-order differential equations with operator-valued coefficients, Publ. RIMS, Kyoto Univ., 7 (1971/72), 581–619.. [WY1]. M. Winklmeier and O. Yamada, Spectral analysis of radial Dirac operators in the Kerr-Newman metric and its applications to time-periodic solutions, J. Math. Phys., 47 (2006), 102503 (17pp).. [WY2]. M. Winklmeier and O. Yamada, A spectral approach to the Dirac equation in the non-extreme Kerr-Newman metric, J. Phys. A, 42 (2009), 295204 (15pp).. 8. −22−. This document is provided by JAXA..

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