2.1 Teukolsky formalism

In terms of the conventional Boyer–Lindquist coordinates, the metric of a Kerr black hole is expressed as
2 Δ- 2 2 sin2-πœƒ[ 2 2 ]2 ds = − Σ(dt − a sin πœƒdφ ) + Σ (r + a )dφ − a dt Σ + --dr2 + Σd πœƒ2, (3 ) Δ
where Σ = r2 + a2 cos2πœƒ and Δ = r2 − 2M r + a2. In the Teukolsky formalism [106], the gravitational perturbations of a Kerr black hole are described by a Newman–Penrose quantity α β γ δ ψ4 = − C αβγδn m¯ n ¯m  [74, 75], where C αβγδ is the Weyl tensor and
1 nα = ---((r2 + a2),− Δ, 0,a), (4 ) 2Σ m α = √------1-------(iasin πœƒ,0,1,iβˆ•sinπœƒ). (5 ) 2(r + ia cosπœƒ)
The perturbation equation for −4 Ο• ≡ ρ ψ4, −1 ρ = (r − iacos πœƒ), is given by
sπ’ͺ Ο• = 4 πΣTˆ. (6 )
Here, the operator π’ͺ s is given by
( (r2 + a2)2 ) 4M ar ( a2 1 ) sπ’ͺ = − ----------− a2 sin2 πœƒ ∂2t − ------∂t∂Ο• − ---− ---2-- ∂2Ο• Δ Δ ( Δ sin πœƒ ) − s s+1 --1-- a(r −-M-)- icos-πœƒ +Δ ∂r(Δ ∂r) + sin πœƒ∂ πœƒ(sin πœƒ∂πœƒ) + 2s Δ + sin2πœƒ ∂Ο• ( 2 2 ) M-(r--−-a-) 2 +2s Δ − r − iacosπœƒ ∂t − s(s cot πœƒ − 1), (7 )
with s = − 2. The source term ˆT is given by
ˆT = 2 (B ′ + B ∗′), (8 ) 2 2 B ′2 = − 1-ρ8ρˆL −1[ρ −4ˆL0(ρ−2ρ− 1Tnn)] 2 --1-- 8-- 2ˆ − 4-2 ˆ −2-− 2 − 1 -- − 2 √2-ρ ρΔ L− 1[ρ ρ J+(ρ ρ Δ Tmn)], (9 ) 1 B ′2∗= − --ρ8ρΔ2 ˆJ+[ρ−4Jˆ+ (ρ−2ρTmm--)] 4 − --1√--ρ8ρΔ2 ˆJ [ρ−4ρ2Δ − 1Lˆ (ρ−2ρ− 2T--)], (10 ) 2 2 + −1 mn
UpdateJump To The Next Update Information where
i ˆLs = ∂πœƒ − -----∂φ − iasinπœƒ∂t + scot πœƒ, (11 ) sinπœƒ Jˆ+ = ∂r − -1 ((r2 + a2)∂t + a∂φ) , (12 ) Δ
UpdateJump To The Next Update Information and Tnn, Tmn, and Tmm- are the tetrad components of the energy momentum tensor (μ ν Tnn = Tμνn n etc.). The bar denotes the complex conjugation.

If we set s = 2 in Equation (6View Equation), with appropriate change of the source term, it becomes the perturbation equation for ψ0. Moreover, it describes the perturbation for a scalar field (s = 0), a neutrino field (|s| = 1βˆ•2), and an electromagnetic field (|s| = 1) as well.

We decompose ψ4 into the Fourier-harmonic components according to

∑ 1 ∫ ρ− 4ψ4 = √---- dωe−iωt+im φ −2Sβ„“m (πœƒ )R β„“mω(r). (13 ) β„“m 2π
UpdateJump To The Next Update Information The radial function R β„“m ω and the angular function sSβ„“m(πœƒ) satisfy the Teukolsky equations with s = − 2 as
( ) Δ2 d-- 1-dR-β„“m-ω- − V(r)R = T , (14 ) dr Δ dr β„“mω β„“m ω [ ( ) 2 --1--d-- sinπœƒ d-- − a2ω2sin2 πœƒ − (m-−-2-cosπœƒ)-- sin πœƒ dπœƒ dπœƒ sin2 πœƒ ] +4a ω cos πœƒ − 2 + 2ma ω + λ − 2Sβ„“m = 0. (15 )
The potential V (r) is given by
K2 + 4i(r − M )K V (r) = − --------Δ---------+ 8iωr + λ, (16 )
where λ is the eigenvalue of − 2Sa ω β„“m and K = (r2 + a2)ω − ma. The angular function sS β„“m (πœƒ) is called the spin-weighted spheroidal harmonic, which is usually normalized as
∫ π 2 0 |−2Sβ„“m| sin πœƒdπœƒ = 1. (17 )
In the Schwarzschild limit, it reduces to the spin-weighted spherical harmonic with λ → β„“(β„“ + 1). In the Kerr case, however, no analytic formula for λ is known. The source term Tβ„“mω is given by
∫ aω T β„“m ω = 4 dΩdt ρ−5ρ− 1(B ′2 + B′2∗)e−imφ+iωt−√2Sβ„“m-, (18 ) 2π
We mention that for orbits of our interest, which are bounded, T β„“mω has support only in a compact range of r.

We solve the radial Teukolsky equation by using the Green function method. For this purpose, we define two kinds of homogeneous solutions of the radial Teukolsky equation:

{ BtransΔ2e −ikr∗ for r → r+ Rinβ„“m ω → β„“mω ∗ ∗ (19 ) r3Breβ„“fmωeiωr + r−1Biβ„“nmcωe−iωr for r → + ∞, { ∗ ∗ up Cuβ„“pmωeikr + Δ2Crefβ„“m ωe−ikr for r → r+, R β„“m ω → trans 3 iωr∗ (20 ) Cβ„“mω r e for r → + ∞,
where k = ω − ma βˆ•2M r +, and r∗ is the tortoise coordinate defined by
∫ dr∗ r∗ = ---dr dr -2M-r+-- r −-r+- -2M--r−- r −-r−- = r + r+ − r− ln 2M − r+ − r− ln 2M , (21 )
where √ ---2---2- r± = M ± M − a, and where we have fixed the integration constant.

Combining with the Fourier mode e−iωt, we see that Riβ„“nmω has no outcoming wave from past horizon, while Rup has no incoming wave at past infinity. Since these are the properties of waves causally generated by a source, a solution of the Teukolsky equation which has purely outgoing property at infinity and has purely ingoing property at the horizon is given by

( ∫ r ∫ ∞ ) Rβ„“mω = --1--- Rupβ„“mω dr ′Rinβ„“m ωTβ„“mω Δ −2 + Rinβ„“mω dr′Rupβ„“m ωTβ„“mωΔ − 2 , (22 ) W β„“m ω r+ r
where the Wronskian W β„“m ω is given by
trans inc W β„“m ω = 2iωC β„“mω B β„“m ω. (23 )
Then, the asymptotic behavior at the horizon is
BtransΔ2e −ikr∗∫ ∞ ∗ R β„“mω(r → r+) → -β„“m-ωtrans-inc- dr ′Rupβ„“mωT β„“m ωΔ −2 ≡ &tidle;ZHβ„“mω Δ2e− ikr , (24 ) 2iωC β„“m ω Bβ„“mω r+
while the asymptotic behavior at infinity is
r3eiωr∗ ∫ ∞ T (r′)Rin (r′) ∗ Rβ„“mω (r → ∞ ) → -----inc-- dr′--β„“m-ω-----β„“mω----≡ &tidle;Z∞β„“mωr3eiωr . (25 ) 2iωB β„“mω r+ Δ2 (r′)

We note that the homogeneous Teukolsky equation is invariant under the complex conjugation followed by the transformation m → − m and ω → − ω. Thus, we can set in,up ¯R β„“m ω in,up = Rβ„“−m −ω, where the bar denotes the complex conjugation.

We consider Tμν of a monopole particle of mass μ. The energy momentum tensor takes the form

μν -----μ------dz-μdz-ν T = Σ sinπœƒdtβˆ•d τ dτ dτ δ(r − r(t))δ(πœƒ − πœƒ(t))δ (φ − φ(t)), (26 )
where μ z = (t,r(t),πœƒ(t),φ(t)) is a geodesic trajectory, and τ = τ(t) is the proper time along the geodesic. The geodesic equations in the Kerr geometry are given by
[ ( ˆ2 ) ]1βˆ•2 Σ dπœƒ-= ± ˆC − cos2 πœƒ a2(1 − ˆβ„°2) + --lz-- ≡ Θ (πœƒ), dτ sin2πœƒ ( ) ( ) dφ- ˆ --ˆlz--- -a ˆ 2 2 ˆ Σ dτ = − a β„° − sin2πœƒ + Δ β„°(r + a ) − alz ≡ Φ, (27 ) ( ) dt- ˆ --ˆlz--) 2 r2-+-a2( ˆ 2 2 ˆ Σ dτ = − a β„° − sin2πœƒ asin πœƒ + Δ β„°(r + a ) − alz ≡ T, √ -- Σ dr-= ± R, dτ
UpdateJump To The Next Update Information where
2 2 2 2 2 R = [β„°ˆ(r + a ) − aˆlz] − Δ [(ˆβ„°a − ˆlz) + r + ˆC ]. (28 )
and ˆβ„°, ˆl z, and Cˆ are the energy, the z-component of the angular momentum, and the Carter constant of a test particle, respectively. These constants of motion are those measured in units of μ. That is, if expressed in the standard units, they become β„° = μˆβ„°, lz = μˆlz, and C = μ2Cˆ.

Using Equation (27View Equation), the tetrad components of the energy momentum tensor are expressed as

Cnn-- Tnn = μ sin πœƒ δ(r − r(t))δ(πœƒ − πœƒ(t))δ(φ − φ (t)), Cmn Tmn- = μ -----δ(r − r(t))δ(πœƒ − πœƒ(t))δ(φ − φ (t)), (29 ) sin-πœƒ- Tmm--= μ Cm-m-δ(r − r(t)) δ(πœƒ − πœƒ (t))δ(φ − φ(t)), sin πœƒ
where
1 [ dr ]2 Cnn = ---3- β„°ˆ(r2 + a2) − aˆlz + Σ-- , (30 ) 4Σ Λ™t d τ [ ( ) ] ρ [ dr ] ˆlz dπœƒ Cmn = − -√------ β„°ˆ(r2 + a2) − aˆlz + Σ--- isin πœƒ aβ„°ˆ− ---2-- + Σ --- , (31 ) 2 2Σ2tΛ™ dτ sin πœƒ dτ 2 [ ( ) ]2 --- -ρ-- ˆ -ˆlz--- -dπœƒ C mm = 2Σ Λ™t isin πœƒ aβ„° − sin2 πœƒ + Σd τ , (32 )
and Λ™t = dtβˆ•d τ. Substituting Equation (10View Equation) into Equation (18View Equation) and performing integration by part, we obtain
∫ ∫ 4-μ-- ∞ iωt−im φ(t) Tβ„“mω = √2-π dt dπœƒe −{∞ ( ) × − 1-L† ρ−4L †(ρ3S ) C ρ− 2ρ−1δ(r − r(t))δ(πœƒ − πœƒ(t)) 2 1 2 nn 2-2( ) [ -- ] + Δ√--ρ- L†2S + ia(ρ-− ρ)sin πœƒS J+ -C-mn- δ(r − r(t)) δ(πœƒ − πœƒ (t)) 2ρ ρ2ρ2Δ 1 †( 3 --2 −4 ) −2-−2 + -√--L 2 ρ S (ρ ρ ),r Cmn- Δρ ρ δ(r − r(t))δ(πœƒ − πœƒ(t)) 2 2 } 1-3 2 [ −4 (--−2 --- )] − 4ρ Δ SJ+ ρ J+ ρρ C mm δ(r − r(t))δ(πœƒ − πœƒ(t)) , (33 )
where
L †= ∂ − -m---+ aω sinπœƒ + s cotπœƒ, (34 ) s πœƒ sin πœƒ J+ = ∂r + iK βˆ•Δ, (35 )
and S denotes aω −2S β„“m (πœƒ) for simplicity.

For a source bounded in a finite range of r, it is convenient to rewrite Equation (33View Equation) further as

∫ ∞ iωt− im φ(t) 2{ T β„“m ω = μ dte Δ (Ann0 + Amn0 + Amm0-) δ(r − r (t)) −∞ + [(Amn1 + Amm1- )δ(r − r(t))],r } + [Amm2- δ(r − r(t))],rr , (36 )
where
--−-2---− 2-−1 † −4 † 3 Ann0 = √2-πΔ2 ρ ρ CnnL 1[ρ L 2(ρ S)], (37 ) [ ( ) ( ) ] A-- = √-2--ρ− 3C-- L†S iK--+ ρ + ρ- − asin πœƒSK--(ρ − ρ) , (38 ) mn0 πΔ mn 2 Δ Δ [ ( ) 2 ] A --- = − √1--ρ− 3ρC ---S − i K-- − K-- + 2iρK-- , (39 ) mm0 2π mm Δ ,r Δ2 Δ Amn1 = √-2--ρ− 3Cmn [L†S + ia sin πœƒ(ρ − ρ)S], (40 ) πΔ 2 2 -- ( K ) Amm1- = − √---ρ− 3ρCmmS- i---+ ρ , (41 ) 2π Δ --- -1---− 3-- --- A mm2 = − √2-πρ ρC mmS. (42 )
Inserting Equation (36View Equation) into Equation (25View Equation), we obtain &tidle;Zβ„“mω as
∫ ∞ Z&tidle;β„“mω = ----μ---- dteiωt−imφ(t)W β„“m ω, (43 ) 2iωBinβ„“cm ω −∞
where
{ } in dRinβ„“m ω d2Rinβ„“mω W β„“mω = R β„“m ω [Ann0 + Amn0 + Amm0-] − -------[Amn1 + Amm1- ] +----2--Amm2- . (44 ) dr dr r=r(t)

In this paper, we focus on orbits which are either circular (with or without inclination) or eccentric but confined on the equatorial plane. In either case, the frequency spectrum of Tβ„“mω becomes discrete. Accordingly, Z&tidle;β„“m ω in Equation (24View Equation) or (25View Equation) takes the form,

∑ &tidle;Zβ„“mω = δ(ω − ωn )Zβ„“mω. (45 ) n
Then, in particular, ψ4 at r → ∞ is obtained from Equation (13View Equation) as
1 ∑ Saωn ∗ ψ4 = -- Z β„“mωn−-2√-β„“m-eiωn(r −t)+im φ. (46 ) r β„“mn 2 π
At infinity, ψ4 is related to the two independent modes of gravitational waves h+ and h × as
1-¨ ¨ ψ4 = 2(h+ − ih× ). (47 )
From Equations (46View Equation) and (47View Equation), the luminosity averaged over t ≫ Δt, where Δt is the characteristic time scale of the orbital motion (e.g., a period between the two consecutive apastrons), is given by
| |2 ⟨ ⟩ ∑ ||Z β„“mω || ∑ ( ) dE- = -----n---≡ dE- . (48 ) dt 4π ω2n dt β„“mn β„“,m,n β„“,m,n
In the same way, the time-averaged angular momentum flux is given by
⟨ ⟩ 2 ( ) ( ) dJz- ∑ m-|Z-β„“m-ωn|- ∑ dJz- ∑ m-- dE- dt = 4πω3n ≡ dt = ωn dt . (49 ) β„“,m,n β„“,m,n β„“mn β„“,m,n β„“mn

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