Super-potential-in-series method

5.2 Super-potential-in-series method

As all what we need to do is to evaluate the surface integrals in the general form of equations of motion (111

), we need an expression for the gravitational field only around the star. In fact, we have developed such a method in [91

] for the source term of the following form

where $a$ and $b$ are integers and $p = 0, 1$ , $q = 0,1$ . Note that $A, A ′ = 1,2$ . Then, we take spatial derivatives out of the Poisson integral,

Note that the integration region is $N∕B$ and therefore $g(⃗x)$ is nonsingular in $N ∕B$ . For this kind of source term, we have given a method in [91

] to find a field $F (m,n) [A,c]$ in the neighborhood of star $A$ in the following sense:

We have checked at 3 PN order that the resulting field from this method is equal to the field obtained from the usual (super-potential) method whenever the super-potentials are available. Unfortunately, however, this method is not perfect and we need another method to derive the equations of motion which we explain now.