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10 Gravitational Waves from Compact Binaries

We pointed out that the 3PN equations of motion, Equations (189View Equation, 190View Equation), are merely Newtonian as regards the radiative aspects of the problem, because with that precision the radiation reaction force is at the lowest 2.5PN order. A solution would be to extend the precision of the equations of motion so as to include the full relative 3PN or 3.5PN precision into the radiation reaction force, but, needless to say, the equations of motion up to the 5.5PN or 6PN order are quite impossible to derive with the present technology. The much better alternative solution is to apply the wave-generation formalism described in Part A, and to determine by its means the work done by the radiation reaction force directly as a total energy flux at future null infinity. In this approach, we replace the knowledge of the higher-order radiation reaction force by the computation of the total flux L, and we apply the energy balance equation as in the test of the P of the binary pulsar (see Equations (4View Equation, 5View Equation)):
dE- = - L. (215) dt
Therefore, the result (194View Equation) that we found for the 3.5PN binary’s center-of-mass energy E constitutes only “half” of the solution of the problem. The second “half” consists of finding the rate of decrease dE/dt, which by the balance equation is nothing but finding the total gravitational-wave flux L at the 3.5PN order. Because the orbit of inspiralling binaries is circular, the balance equation for the energy is sufficient (no need of a balance equation for the angular momentum). This all sounds perfect, but it is important to realize that we shall use Equation (215View Equation) at the very high 3.5PN order, at which order there are no proofs (following from first principles in general relativity) that the equation is correct, despite its physically obvious character. Nevertheless, Equation (215View Equation) has been checked to be valid, both in the cases of point-particle binaries [136137] and extended weakly self-gravitating fluids [1418], at the 1PN order and even at 1.5PN (the 1.5PN approximation is especially important for this check because it contains the first wave tails).

Obtaining L can be divided into two equally important steps: (1) the computation of the source multipole moments I L and J L of the compact binary and (2) the control and determination of the tails and related non-linear effects occuring in the relation between the binary’s source moments and the radiative ones UL and VL (cf. the general formalism of Part A).


 10.1 The binary’s multipole moments
 10.2 Contribution of wave tails
 10.3 Orbital phase evolution
 10.4 The two polarization waveforms

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