## 10 Gravitational Waves from Compact Binaries

We pointed out that the 3PN equations of motion, Eqs. (146, 147), 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 , and we apply
the energy balance equation as in the test of the of the binary pulsar (see Eqs. (4, 5)):
Therefore, the result (152) that we found for the 3.5PN binary’s center-of-mass energy constitutes only
“half” of the solution of the problem. The second “half” consists of finding the rate of decrease ,
which by the balance equation is nothing but finding the total gravitational-wave flux 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 the equation (153) 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,
Eq. (153) has been checked to be valid, both in the cases of point-particle binaries [85, 86]
and extended weakly self-gravitating fluids [5, 9], 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 can be divided into two equally important steps: (1) the computation of
the source multipole moments and 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 and (cf. the general formalism of Part A).