EGM2009 1

Perturbative analysis of gravitational recoil

Hiroyuki Nakano

Carlos O. Lousto

Yosef Zlochower

Center for Computational Relativity and GravitationRochester 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. 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)

EGM2009 4

Tensor harmonics:

EGM2009 5

Linear momentum loss:

After the angular integration,

* We calculate the Regge-Wheeler and Zerilli functions.

EGM2009 6

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

EGM2009 7

X

Y

Z Background Schwarzschild + perturbation

S_x = M a

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

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

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. X

Y

Z

S_2

S_1

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.

X

Y

Z

S_2

S_1

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.