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Perturbative analysis of gravitational recoil
Hiroyuki Nakano
Carlos O. Lousto
Yosef Zlochower
Center for Computational Relativity and GravitationRochester Institute of Technology
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1. Introduction
Linear momentum flux for binaries (analytic expression): Kidder (1995), Racine, Buonanno and Kidder (2008) PN approach
Mino and Brink (2008) BHP approach, near-horizon (but low frequency)
Cf.) Sago et al. (2005, 2007) BHP approach [dE/dt, dL/dt, dC/dt for periodic orbits]
* BHP approach in the Schwarzschild background
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2. FormulationMetric perturbation in the Schwarzschild background
Regge-Wheeler-Zerilli formalism
* Gravitational waves in the asymptotic flat gauge:
Zerilli function
Regge-Wheeler function
f_lm, d_lm: tensor harmonics (angular function)
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Tensor harmonics:
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Linear momentum loss:
After the angular integration,
* We calculate the Regge-Wheeler and Zerilli functions.
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Kerr metric in the Boyer-Lindquist coordinates,
in the Taylor expansion with respect to a=S/M .
3. Spin as a perturbation
Z
X
Y
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X
Y
Z Background Schwarzschild + perturbation
S_x = M a
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Tensor harmonics expansion for the perturbation:
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L=1, m=+1/-1 odd parity mode
* This is not the gravitational wave mode.
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Particle falling radially into a Schwarzschild black hole
Slow motion approximation
dR/dt ~ v, M/R ~ v^2, v<<1
4. Leading order
X
Y
Z
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Tensor harmonics expansion of the energy-momentum tensor:
L=2, m=0, even parity mode (GW)
L=3, m=0, even parity mode (GW)
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L=1, m=0, even parity mode (not GW mode)
Zero in the vacuume region.* Center of mass system
“Low multipole contributions” Detweiler and Poisson (2004)
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L=2, m=+1/-1, odd parity mode (2nd order)
Leading order
BH Spin [L=1,m=+1/-1 (odd)] and Particle [L=1,m=0 (even)]
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L=2, m=+1/-1, odd parity mode (particle’s spin, GW)
S_1 and S_2 are parallel. X
Y
Z
S_2
S_1
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Gravitational wave modes:
A.
B.
C.
D.
Linear momentum loss:
(A and C + A and D)
(A and B)
* Consistent with Kidder ‘s results in the PN approach.
X
Y
Z
S_2
S_1
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5. Discussion
Racine et al. have discussed the next order...
* Analytically possible in the BHP approach?
1st order perturbations from local source terms (delta function) O.K. in a finite slow motion order.
2nd order perturbations from extended source terms (not local) ???
* The dipole mode (L=1) is important in our calculation.