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2.1 Teukolsky formalism

In terms of the conventional Boyer-Lindquist coordinates, the metric of a Kerr black hole is expressed as
D sin2 h [ ]2 ds2 = - --(dt- a sin2 hdf)2 + ------ (r2 + a2)df - adt S S + S-dr2 + Sdh2, (3) D
where S = r2 + a2cos2h and D = r2 - 2M r + a2. In the Teukolsky formalism [61], the gravitational perturbations of a Kerr black hole are described by a Newman-Penrose quantity a b g d y4 = - Cabgdn m n m  [3839], where Cabgd is the Weyl tensor and
a 1 2 2 n = ---((r + a ),- D, 0,a), (4) 2S ma = V~ -----1--------(ia sin h,0,1,i/sin h). (5) 2(r + ia cosh)
The perturbation equation for -4 f =_ r y4, -1 r = (r - ia cosh), is given by
^ sOf = 4pS T . (6)
Here, the operator sO is given by
( ) ( ) (r2 + a2)2 2 2 2 4M ar a2 1 2 sO = - ----------- a sin h @t - ------@t@f - ---- --2--- @f D D ( D sin h ) -s s+1 --1-- a(r---M-)- i-cosh +D @r(D @r) + sinh @h(sin h@h) + 2s D + sin2 h @f ( 2 2 ) +2s M--(r----a-) - r - iacos h @t- s(s cot2h - 1), (7) D
with s = - 2. The source term ^ T is given by
' ^T = 2(B'2 + B*2 ), (8) 1 -- -- B'2 = - -r8rL -1[r-4L0(r- 2r-1Tnn)] 2 - -1 V~ --r8rD2L [r -4r2J (r- 2r-2D -1T-- )], (9) 2 2 -1 + mn 1 -- -- B'*2 = - -r8rD2J+[r - 4J+(r- 2rTmm)] 4 - -1 V~ --r8rD2J+[r - 4r2D -1L- 1(r- 2r-2Tmn)], (10) 2 2
where
L = @ + -m---- aw sin h + s coth, (11) s h sinh J+ = @r + iK/D, (12) 2 2 K = (r + a )w - ma, (13)
and Tnn, Tmn, and Tmm- are the tetrad components of the energy momentum tensor (Tnn = Tmnnmnn 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 y0. 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 y4 into the Fourier-harmonic components according to

sum integral r-4y4 = dwe- iwt+imf -2Slm(h)Rlmw(r). (14) lm
The radial function Rlmw and the angular function sSlm(h) satisfy the Teukolsky equations with s = -2 as
( ) D2 -d- -1 dRlmw-- - V (r)R = T , (15) dr D dr lmw lmw [ 1 d ( d ) (m - 2cos h)2 ------- sin h--- - a2w2 sin2h - -------2------ sin hdh dh sin h ] +4aw cosh - 2 + 2maw + c -2Slm = 0. (16)
The potential V (r) is given by
K2 + 4i(r - M )K V (r) = ---------D--------- + 8iwr + c, (17)
where c is the eigenvalue of - 2Saw lm. The angular function sSlm(h) is called the spin-weighted spheroidal harmonic, which is usually normalized as
integral p 2 0 |- 2Slm| sinhdh = 1. (18)
In the Schwarzschild limit, it reduces to the spin-weighted spherical harmonic with c --> l(l + 1). In the Kerr case, however, no analytic formula for c is known. The source term Tlmw is given by
integral aw Tlmw = 4 d_O_dtr -5r-1(B'2 + B'*2 )e- imf+iwt - V~ 2Slm, (19) 2p
We mention that for orbits of our interest, which are bounded, T lmw 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:

{ BtransD2e -ikr* for r --> r+ Rinlmw --> lmw * * (20) r3Brelfmweiwr + r- 1Bilnmcwe -iwr for r --> + oo , { * * up Cuplmweikr + D2Crelfmwe -ikr for r --> r+, R lmw --> trans 3 iwr* (21) C lmw r e for r --> + oo ,
where k = w - ma/2M r +, and r* is the tortoise coordinate defined by
integral dr* r* = ----dr dr -2M--r+- r---r+- -2M--r-- r---r-- = r + r+ - r-ln 2M - r+ - r- ln 2M , (22)
where V~ ---2---2- r± = M ± M - a, and where we have fixed the integration constant.

Combining with the Fourier mode e-iwt, we see that Rilnmw 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

( integral r integral oo ) Rlmw = ---1-- Ruplmw dr'RinlmwTlmwD -2 + Rinlmw dr'RulpmwTlmwD -2 , (23) Wlmw r+ r
where the Wronskian Wlmw is given by
trans inc Wlmw = 2iwC lmw Blmw. (24)
Then, the asymptotic behavior at the horizon is
BtransD2e -ikr* integral oo * Rlmw(r --> r+) --> --lmw-trans-inc- dr'RuplmwTlmwD - 2 =_ ~ZHlmwD2e -ikr , (25) 2iwC lmw B lmw r+
while the asymptotic behavior at infinity is
r3eiwr* integral oo T (r')Rin (r') * Rlmw(r --> oo ) --> -----inc-- dr'-lmw------lmw----- =_ Z~o o lmwr3eiwr . (26) 2iwB lmw r+ D2(r')

We note that the homogeneous Teukolsky equation is invariant under the complex conjugation followed by the transformation m --> - m and w --> - w. Thus, we can set in,up R lmw in,up = Rl-m -w, where the bar denotes the complex conjugation.

We consider Tmn of a monopole particle of mass m. The energy momentum tensor takes the form

m dzm dzn T mn = --------------------d(r - r(t))d(h- h(t))d(f - f(t)), (27) S sinhdt/dt dt dt
where zm = (t,r(t),h(t),f(t)) is a geodesic trajectory, and t = t(t) is the proper time along the geodesic. The geodesic equations in the Kerr geometry are given by
dh [ ( l2 )]1/2 S --- = ± C - cos2h a2(1- E2) + --z2-- =_ Q(h), dt ( ) sin h df lz a ( 2 2 ) S --- = - aE - ---2-- + -- E(r + a )- alz= _ P, dt ( sin h D ) (28) dt- --lz--) 2 r2-+-a2( 2 2 S dt = - aE - sin2h asin h + D E(r + a ) - alz =_ T, -- S dr- = ± V~ R, dt
where E, l z, and C are the energy, the z-component of the angular momentum, and the Carter constant of a test particle, respectivelyView Footnote, and
2 2 2 2 2 R = [E(r + a ) - alz] - D[(Ea - lz) + r + C]. (29)

Using Equation (28View Equation), the tetrad components of the energy momentum tensor are expressed as

Tnn = m Cnn--d(r- r(t)) d(h - h(t))d(f - f(t)), sinh -- Cmn-- Tmn = m sinh d(r- r(t)) d(h - h(t))d(f - f(t)), (30) C --- Tmm- = m --mm-d(r - r(t))d(h- h(t)) d(f - f(t)), sinh
where
[ ] 1 2 2 dr 2 Cnn = ---3- E(r + a )- alz + S --- , (31) 4S t [ dt ][ ( ) ] -- ---r---- 2 2 dr- -lz--- dh- C mn = - 2 V~ 2S2t E(r + a )- alz + S dt isinh aE - sin2 h + S dt , (32) 2 [ ( ) ]2 ---- -r-- --lz--- dh- Cm m = 2St isin h aE - sin2 h + S dt , (33)
and t = dt/dt. Substituting Equation (10View Equation) into Equation (19View Equation) and performing integration by part, we obtain
integral integral 4m oo iwt-imf(t) Tlmw = V~ --- dt dhe 2p -{ oo ( ) 1-† -4 † 3 -2--1 × - 2L1 r L 2(r S) Cnnr r dr- r(t))d(h - h(t)) 2-2( ) [ -- ] + D V~ --r- L†S + ia(r-- r) sin hS J -Cmn--d(r - r(t))d(h- h(t)) 2r 2 + r2r2D 1 ( -- ) -- + - V~ -L †2 r3S( r2r-4),r CmnDr -2r-2 d(r- r(t))d(h - h(t)) 2 2 } 1- 3 2 [ -4 (--- 2 ---- )] - 4r D SJ+ r J+ rr C m md(r - r(t))d(h - h(t)) , (34)
where
m L †s = @h- -----+ aw sinh + s coth, (35) sin h
and S denotes - 2Salwm(h) for simplicity.

For a source bounded in a finite range of r, it is convenient to rewrite Equation (34View Equation) further as

integral oo { T = m dteiwt- imf(t)D2 (A + A -- + A ---) d(r- r(t)) lmw - oo nn0 mn0 mm0 + [(A-- + A --- )d(r - r(t))] mn1 mm1 } ,r + [Amm2- d(r- r(t))] , (36) ,rr
where
A = V~ ---2-r- 2r-1C L+ [r -4L+ (r3S)], (37) nn0 2pD2 nn 1 2 2 [( )(iK -) K -- ] Amn0 = V~ ---r- 3Cmn- L+2 S --- + r + r - asinhS ---(r - r) , (38) pD [ D ] D 1 (K ) K2 K Amm0- = - V~ --r- 3rCmmS- - i --- - --2 + 2ir--- , (39) 2p D ,r D D 2 -- Amn1 = V~ ---r- 3Cmn[L+2 S + iasinh(r - r)S], (40) pD ( ) --- -2---- 3-- --- K-- A mm1 = - V~ --r rC mmS iD + r , (41) 2p Amm2- = - V~ 1-r- 3rCmmS. (42) 2p
Inserting Equation (36View Equation) into Equation (26View Equation), we obtain ~ Zlmw as
-------- integral -Z~lmw-=-m------------ (43) 2iwBinclmw oo dteiwt-imf(t)Wlmw, - oo
where
{ dRin d2Rin } Wlmw = Rinlmw [Ann0 + Amn0 + Amm0] - ---lmw--[Amn1 + Amm1] + ----l2mwAmm2- . (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 Tlmw becomes discrete. Accordingly, Z~lmw in Equation (25View Equation) or (26View Equation) takes the form,

sum ~Zlmw = d(w - wn)Zlmw. (45) n
Then, in particular, y4 at r-- > oo is obtained from Equation (14View Equation) as
1 sum Sawn * y4 = -- Zlmwn --2 V~ -lm-eiwn(r -t)+imf. (46) r lmn 2p
At infinity, y4 is related to the two independent modes of gravitational waves h+ and h× as
1-¨ ¨ y4 = 2(h+ - ih× ). (47)
From Equations (46View Equation) and (47View Equation), the luminosity averaged over t» Dt, where Dt is the characteristic time scale of the orbital motion (e.g., a period between the two consecutive apastrons), is given by
< > || ||2 dE sum |Zlmwn | sum (dE ) --- = -----2--- =_ --- . (48) dt l,m,n 4pw n l,m,n dt lmn
In the same way, the time-averaged angular momentum flux is given by
< dJ > sum m |Z |2 sum (dJ ) sum m (dE ) ---z = -----lmwn3--- =_ --z- = --- --- . (49) dt l,m,n 4pw n l,m,n dt lmn l,m,n wn dt lmn


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