Second we split the integral into two parts: a contribution from the body zone , and from elsewhere, . Schematically we evaluate the following two types of integrals (we omit indices of the field),
As for the body zone contribution, we make a multipole expansion using the scaling of the integrand, i.e. in the body zone. For example, the part in Equation (76), , gives[129, 150, 165] for some useful formulas of STF tensors. Also we define . To derive the 3 PN equations of motion, up to and as well as up to are required.
In the above equations we define the multipole moments as7. Also we call the dipole moment of the star , and the quadrupole moment of the star .
Then we transform these moments into more convenient forms. By the conservation law (67), we have
Now, from the above equations, especially Equation (88), we find that the body zone contributions, , are of order . Note that if we can not or do not assume the (nearly) stationarity of the initial data for the stars, then, instead of Equation (88) we have.
To obtain the lowest order , we have to evaluate the surface integrals . Generally, in the moments and appear formally at the order and . Thus and appear as that only terms of contribute to the 3 PN equations of motion. The important thing here is that and are at most in .
Finally, since the order of is higher than that of , we conclude that .
As far as the contribution is concerned, since the integrand is at least quadratic in the small deviation field , we make the post-Newtonian expansion in the integrand. Then, basically, with the help of (super-)potentials which satisfy , denoting the Laplacian, we have for each integral (see e.g. the term in Equation (77))). For the terms in Equation (77), we use potentials many times to convert all the volume integrals into surface integrals and “” terms8.
In fact finding the super-potentials is one of the most formidable tasks especially when we proceed to high post-Newtonian orders. Fortunately, up to 2.5 PN order, all the required super-potentials are available. At 3 PN order, there appear many integrands for which we could not find the required super-potentials. To obtain the 3 PN equations of motion, we devise an alternative method similar to the method employed by Blanchet and Faye . The details of the method will be explained later.
Now the lowest order integrands can be evaluated with the body zone contribution , and since is , we find
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