1. EGM20091 Perturbative analysis of gravitational recoil Hiroyuki Nakano Carlos O. Lousto Yosef Zlochower Center for Computational Relativity and Gravitation Rochester Institute of Technology

2. 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

3. 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)

4. EGM20094 Tensor harmonics:

5. EGM20095 Linear momentum loss: After the angular integration, * We calculate the Regge-Wheeler and Zerilli functions.

6. 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

7. EGM20097 X Y Z Background Schwarzschild + perturbation S_x = M a

8. EGM20098 Tensor harmonics expansion for the perturbation:

9. EGM20099 L=1, m=+1/-1 odd parity mode * This is not the gravitational wave mode.

10. 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

11. 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)

12. 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)

13. 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)]

14. 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

15. 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

16. 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.