Radiation reaction for inspiralling binary systems with spin-spin coupling 1 Institute of Theoretical Physics, Friedrich-Schiller-University Jena December 3, 2007 1 H. Wang and C. M. Will, Phys. Rev. D, 064017 (2007)
Post-Newtonian approximation Movitation Method: Post-Newtonian approximation Post-Newtonian approximation is so far the best method to study the inspiral stage of coalescing compact binaries. Assumptions: slow motion and weak gravitational interaction: (M/r << 1, v << 1 ). Newtonian gravity dominants, small modification introduced by general relativity. Solutions are expressed as power series in the post-newtonian parameter ɛ v 2 M/r. Different approaches (PM approach, Hamiltonian approach, DIRE,... ). It was pointed out [Cutler et al. 1993] extremely high order PN results (up to O[ɛ 7/2 ] beyond Newtonian order) for both the equations of motion and GW waveforms are needed for the current GW detectors.
Post-Newtonian approximation Movitation Post-Newtonian equations of motion for binary systems The general structure of the hydrodynamic post-newtonian EOM: dv i /dt = U,i + a i PN + a i 2PN + a i 2.5PN + a i 3PN + a i 3.5PN +..., Equations of motion for non-spinning binaries has been know through 3.5PN order: PN order 1PN derived by Lorentz & Droste (1917), confirmed by Einstein, Infeld & Hoffmann. Damour et al. (1981, 1987), 2 & 2.5PN confirmed by Kopeikin et al., Blanchet et al., Itoh et al. and Pati et al.. 3PN Schaefer et al. (2000, 2001), confirmed by Blanchet et al. and Itoh et al.. 3.5PN Q: What about spin? Pati & Will (2000, 2002), confirmed by Schaefer et al. and Blanchet et al.
Spin effect: why do we care? Post-Newtonian approximation Movitation Most astrophysical objects are spinning bodies. the orbital motion can be very different from the non-spinning case. Spin effects contribute to the gravitational waveform and the overall emission of energy and angular momentum. Including spin increases the computational burden and may affect (Berti et al. 2005, Lang et al. 2006) the accuracy on source parameter estimation. It s important to have a complete and reasonable accurate picture of the spin effects in the binary systems.
Derivation of the spin-spin effects The equations of motion and the evolution of spins Spin makes counting the PN order a non-simple task Post-Newtonian parameter ɛ m r v 2, (G = c = 1) Rapidly rotating compact objects For arbitrary rotating objects v O(1), x O(m) S A O(m 2 ) O(ɛ 2 )r 2 S 1S 2 O(ɛ 4 )r 4 Leading order equation of motion with spin: a SO v S m v 3 1.5PN r 3 r 2 a SS 1 S 2 m v 4 2PN mr 4 r 2 v O(v), x O(r) S A O(mvr) O(ɛ 3/2 )r 2 S 1S 2 O(ɛ 3 )r 4 Leading order equation of motion with spin: a SO v S m v 2 1PN r 3 r 2 a SS 1 S 2 m v 2 1PN mr 4 r 2 For our calculation, we assume arbitrary rotating objects.
Spin effects: much less understood Derivation of the spin-spin effects The equations of motion and the evolution of spins What we (do not) know about spin effects. PN order SO SS N 1PN Barker et al. (1975) Barker et al. (1979) 2PN Blanchet et al. (2006) Porto et al. (2006) 3PN?? 2.5PN 3.5PN Will (2005) This work Formally, the spin effects first appear at 1PN order. Leading order spin contribution to the R. R.: 3.5PN effects. Kidder (1995) calculated the leading order spin contribution to the energy and angular momentum flux in the radiation zone, however a rigorous derivation of the equations of motion at radiation reaction order is still necessary. The spin-orbit and spin-spin coupling may lead to the precession of the orbital plane and the individual spins. To fully describe the dynamics, both dv i /dt and ds i /dt are needed.
Deriving the spin effects Derivation of the spin-spin effects The equations of motion and the evolution of spins We start from the hydrodynamic EOM derived by Pati&Will (2002). dv i /dt = U,i + a i PN + ai 2.5PN + ai 3.5PN +..., We assume the bodies are perfect fluid balls. Definition of spin: S i A = ɛijk R A ρ x j v k d 3 x. It is also useful to define another quantity called the proper spin: SA i Si A (1 + 1 2 v A 2 + 3 m B r ) 1 2 [v A (v A S A )] i (3) SA i I jj +S j (3) A I ij Using expansion parameters x, x, v, v, we expand EOM w.r.t. center of each bodies. By keeping the terms that are proportional to (ρ x v) or (ρ x v ), we obtain the spin-orbit contributions. By keeping terms that are proportional to (ρ x v)(ρ x v ) we obtain the spin-spin contributions. Similarly, we can obtain the evolution of individual spins by calculating Ṡi A = ɛijk R A ρ x j a k d 3 x
Results Introduction Derivation of the spin-spin effects The equations of motion and the evolution of spins At PN order, we recover the spin-spin contributions to equations of motion and of individual spins: a PN SS = 3 µr 4 [n(s 1 S 2 ) + S 1 (n S 2 ) + S 2 (n S 1 ) 5n(n S 1 )(n S 2 )]. (Ṡ 1 ) PN SS = 1 «r 3 S 2 3(n S 2 )n S 1. At 3.5PN order, the EOM and equations of individual spins are given by: OK, you got these lengthy expressions, now what?
Energy balance From our result, the near zone orbital energy loss due to the leading order spin-spin gravitational radiation can be obtained by calculating: Ė Near (SS) = µ (v a 3.5PN SS ) When comparing with the radiation zone energy flux (Kidder 1995), we found our result of the energy loss doesn t exactly balance the flux in the radiation zone. Energy is not conserved? The difference is a meaningless total time derivative. Can be absorbed into the definition of the near zone energy (Gauge effects). Our result of energy loss precisely equals the radiation zone flux. Energy conservation.
Angular momentum balance Near zone orbital angular momentum loss: L Near (SS) = µ (x a 3.5PN SS ). Originally, the L Near (SS) doesn t agree with the radiative flux. No big deal, let us prove the difference is a total time derivative. Wait a second, what about the spin angular momentum? We need to consider the total angular momentum loss J Near (SS) = L Near (SS) + S Near (SS). The difference between the total angular momentum loss in the near zone and the radiation zone flux is also a meaningless total time derivative. Redefine the near zone total angular momentum angular momentum conservation.
Spin precession S A can not be completely eliminated by extracting time derivatives (different from the spin-orbit case!). After absorbing the total time derivative to the definition of spin, the residual contribution is a pure precession. No precession if the companion spin is perpendicular to the orbital plane.
Conclusion We obtained the 3.5 post-newtonian order spin-spin radiation reaction and equation of spin for inspiralling binary systems. Using the our result, we calculated the near zone energy and angular momentum loss, which precisely balance the radiative flux of these quantities in the radiation zone. The spin-spin coupling introduces an additional pure precession to the individual spins.