Teukolsky formalism

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-𝜃[ 2 2 ]2 ds = − Σ(dt − a sin 𝜃dφ ) + Σ (r + a )dφ − a dt Σ + --dr2 + Σd 𝜃2, (3 ) Δ

where $Σ =$ $r2 + a2 cos2𝜃$ 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 𝜃,0,1,i∕sin𝜃). (5 ) 2(r + ia cos𝜃)

The perturbation equation for $−4 ϕ ≡ ρ ψ4$ , $−1 ρ = (r − iacos 𝜃)$ , is given by

Here, the operator $𝒪 s$ is given by

( (r2 + a2)2 ) 4M ar ( a2 1 ) s𝒪 = − ----------− a2 sin2 𝜃 ∂2t − ------∂t∂ϕ − ---− ---2-- ∂2ϕ Δ Δ ( Δ sin 𝜃 ) − s s+1 --1-- a(r −-M-)- icos-𝜃 +Δ ∂r(Δ ∂r) + sin 𝜃∂ 𝜃(sin 𝜃∂𝜃) + 2s Δ + sin2𝜃 ∂ϕ ( 2 2 ) M-(r--−-a-) 2 +2s Δ − r − iacos𝜃 ∂t − s(s cot 𝜃 − 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

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where

i ˆLs = ∂𝜃 − -----∂φ − iasin𝜃∂t + scot 𝜃, (11 ) sin𝜃 Jˆ+ = ∂r − -1 ((r2 + a2)∂t + a∂φ) , (12 ) Δ

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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 (6 ), 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 ( $|s| = 1∕2$ ), and an electromagnetic field ( $|s| = 1$ ) as well.

We decompose $ψ4$ into the Fourier-harmonic components according to

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The radial function $R ℓm ω$ and the angular function $sSℓm(𝜃)$ satisfy the Teukolsky equations with $s = − 2$ as

( ) Δ2 d-- 1-dR-ℓm-ω- − V(r)R = T , (14 ) dr Δ dr ℓmω ℓm ω [ ( ) 2 --1--d-- sin𝜃 d-- − a2ω2sin2 𝜃 − (m-−-2-cos𝜃)-- sin 𝜃 d𝜃 d𝜃 sin2 𝜃 ] +4a ω cos 𝜃 − 2 + 2ma ω + λ − 2Sℓm = 0. (15 )

The potential $V (r)$ is given by

where $λ$ is the eigenvalue of $− 2Sa ω ℓm$ and $K = (r2 + a2)ω − ma$ . The angular function $sS ℓm (𝜃)$ is called the spin-weighted spheroidal harmonic, which is usually normalized as

In the Schwarzschild limit, it reduces to the spin-weighted spherical harmonic with $λ → ℓ(ℓ + 1)$ . In the Kerr case, however, no analytic formula for $λ$ is known. The source term $Tℓmω$ is given by

We mention that for orbits of our interest, which are bounded, $T ℓmω$ 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+ Rinℓm ω → ℓmω ∗ ∗ (19 ) r3Breℓfmωeiωr + r−1Biℓnmcωe−iωr for r → + ∞, { ∗ ∗ up Cuℓpmωeikr + Δ2Crefℓm ωe−ikr for r → r+, R ℓm ω → trans 3 iωr∗ (20 ) Cℓmω r e for r → + ∞,

where $k = ω − ma ∕2M r +$ , 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 $Riℓnmω$ 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

where the Wronskian $W ℓm ω$ is given by

Then, the asymptotic behavior at the horizon is

while the asymptotic behavior at infinity is

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 $in,up ¯R ℓm ω$ $in,up = Rℓ−m −ω$ , where the bar denotes the complex conjugation.

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

where $μ z = (t,r(t),𝜃(t),φ(t))$ 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 ) ]1∕2 Σ d𝜃-= ± ˆC − cos2 𝜃 a2(1 − ˆℰ2) + --lz-- ≡ Θ (𝜃), dτ sin2𝜃 ( ) ( ) dφ- ˆ --ˆlz--- -a ˆ 2 2 ˆ Σ dτ = − a ℰ − sin2𝜃 + Δ ℰ(r + a ) − alz ≡ Φ, (27 ) ( ) dt- ˆ --ˆlz--) 2 r2-+-a2( ˆ 2 2 ˆ Σ dτ = − a ℰ − sin2𝜃 asin 𝜃 + Δ ℰ(r + a ) − alz ≡ T, √ -- Σ dr-= ± R, dτ

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where

and $ˆℰ$ , $ˆl z$ , 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 $ℰ = μˆℰ$ , $lz = μˆlz$ , and $C = μ2Cˆ$ .

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

Cnn-- Tnn = μ sin 𝜃 δ(r − r(t))δ(𝜃 − 𝜃(t))δ(φ − φ (t)), Cmn Tmn- = μ -----δ(r − r(t))δ(𝜃 − 𝜃(t))δ(φ − φ (t)), (29 ) sin-𝜃- Tmm--= μ Cm-m-δ(r − r(t)) δ(𝜃 − 𝜃 (t))δ(φ − φ(t)), sin 𝜃

where

1 [ dr ]2 Cnn = ---3- ℰˆ(r2 + a2) − aˆlz + Σ-- , (30 ) 4Σ ˙t d τ [ ( ) ] ρ [ dr ] ˆlz d𝜃 Cmn = − -√------ ℰˆ(r2 + a2) − aˆlz + Σ--- isin 𝜃 aℰˆ− ---2-- + Σ --- , (31 ) 2 2Σ2t˙ dτ sin 𝜃 dτ 2 [ ( ) ]2 --- -ρ-- ˆ -ˆlz--- -d𝜃 C mm = 2Σ ˙t isin 𝜃 aℰ − sin2 𝜃 + Σd τ , (32 )

and $˙t = dt∕d τ$ . Substituting Equation (10

) into Equation (18

) and performing integration by part, we obtain

∫ ∫ 4-μ-- ∞ iωt−im φ(t) Tℓmω = √2-π dt d𝜃e −{∞ ( ) × − 1-L† ρ−4L †(ρ3S ) C ρ− 2ρ−1δ(r − r(t))δ(𝜃 − 𝜃(t)) 2 1 2 nn 2-2( ) [ -- ] + Δ√--ρ- L†2S + ia(ρ-− ρ)sin 𝜃S J+ -C-mn- δ(r − r(t)) δ(𝜃 − 𝜃 (t)) 2ρ ρ2ρ2Δ 1 †( 3 --2 −4 ) −2-−2 + -√--L 2 ρ S (ρ ρ ),r Cmn- Δρ ρ δ(r − r(t))δ(𝜃 − 𝜃(t)) 2 2 } 1-3 2 [ −4 (--−2 --- )] − 4ρ Δ SJ+ ρ J+ ρρ C mm δ(r − r(t))δ(𝜃 − 𝜃(t)) , (33 )

where

and $S$ denotes $aω −2S ℓm (𝜃)$ for simplicity.

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

∫ ∞ iωt− im φ(t) 2{ T ℓm 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 𝜃SK--(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 𝜃(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 (36

) into Equation (25

), we obtain $&tidle;Zℓmω$ as

where

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 $Tℓmω$ becomes discrete. Accordingly, $Z&tidle;ℓm ω$ in Equation (24 ) or (25 ) takes the form,

Then, in particular, $ψ4$ at $r → ∞$ is obtained from Equation (13

) as

At infinity, $ψ4$ is related to the two independent modes of gravitational waves $h+$ and $h ×$ as

From Equations (46

) and (47

), 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 ℓmω || ∑ ( ) dE- = -----n---≡ dE- . (48 ) dt 4π ω2n dt ℓmn ℓ,m,n ℓ,m,n

In the same way, the time-averaged angular momentum flux is given by