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EGM20091 Perturbative analysis of gravitational recoil Hiroyuki Nakano Carlos O. Lousto Yosef Zlochower Center for Computational Relativity and Gravitation Rochester Institute of Technology
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EGM20092 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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EGM20093 2. Formulation Metric 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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EGM20094 Tensor harmonics:
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EGM20095 Linear momentum loss: After the angular integration, * We calculate the Regge-Wheeler and Zerilli functions.
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EGM20096 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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EGM20097 X Y Z Background Schwarzschild + perturbation S_x = M a
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EGM20098 Tensor harmonics expansion for the perturbation:
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EGM20099 L=1, m=+1/-1 odd parity mode * This is not the gravitational wave mode.
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EGM200910 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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EGM200911 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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EGM200912 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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EGM200913 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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EGM200914 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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EGM200915 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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EGM200916 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.
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