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Perturbative analysis of gravitational recoil

Perturbative analysis of gravitational recoil. Hiroyuki Nakano Carlos O. Lousto Yosef Zlochower. C enter for C omputational R elativity and G ravitation R ochester I nstitute of T echnology. 1. Introduction. Linear momentum flux for binaries (analytic expression):

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Perturbative analysis of gravitational recoil

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  1. Perturbative analysis of gravitational recoil Hiroyuki Nakano Carlos O. Lousto Yosef Zlochower Center for Computational Relativity and Gravitation Rochester Institute of Technology EGM2009

  2. 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 EGM2009

  3. 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) EGM2009

  4. Tensor harmonics: EGM2009

  5. Linear momentum loss: After the angular integration, * We calculate the Regge-Wheeler and Zerilli functions. EGM2009

  6. 3. Spin as a perturbation Kerr metric in the Boyer-Lindquist coordinates, in the Taylor expansion with respect to a=S/M . Z Y X EGM2009

  7. Background Schwarzschild + perturbation S_x = M a Z Y X EGM2009

  8. Tensor harmonics expansion for the perturbation: EGM2009

  9. L=1, m=+1/-1 odd parity mode * This is not the gravitational wave mode. EGM2009

  10. 4. Leading order Particle falling radially into a Schwarzschild black hole Slow motion approximation dR/dt ~ v, M/R ~ v^2, v<<1 Z Y X EGM2009

  11. Tensor harmonics expansion of the energy-momentum tensor: L=2, m=0, even parity mode (GW) L=3, m=0, even parity mode (GW) EGM2009

  12. 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) EGM2009

  13. 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)] EGM2009

  14. L=2, m=+1/-1, odd parity mode (particle’s spin, GW) S_1 and S_2 are parallel. Z S_1 Y S_2 X EGM2009

  15. 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. Z S_1 Y S_2 X EGM2009

  16. 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. EGM2009

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