Super-potential method

5.1 Super-potential method

Up to 2.5 PN order, we have solved all the Poisson equations necessary to derive the 2.5 PN gravitational field. At 3 PN order, we have found a part of the solutions of the Poisson equations, which we call (super-)potentials¹⁰ . For example,

where $S = r1 + r2 + r12$ , and $f (1,−1)$ which satisfies $Δf (1,− 1) = r1∕r2$ is given in [99

] as

It is possible to add any homogeneous solution to super-potentials. In our formalism, the only place where we use super-potentials is Equation (101 ). In the case above, we could add, say, $1∕r1$ to $f (1,−1)$ . (Note that to evaluate the surface integrals in the general form of equations of motion (111 ) we need super-potentials in the spatial region $N ∕B$ which do not include any singularity due to the point particle limit.) It is easy to see that contribution from a possible additional homogeneous solution cancels out between the “ $− 4πg(⃗x)$ ” term and the surface integral in Equation (101 ).

Useful super-potentials for derivation of the 3 PN field are given in [30 , 27 , 91 , 99 ].