Third Post-Newtonian Gravitational Field

5 Third Post-Newtonian Gravitational Field

In the post-Newtonian approximation, we need to solve Poisson equations to find the metric. Up to 2.5 PN order, explicit forms of the metric have been obtained in [30 ]. However, it seems impossible to derive the 3 PN accurate gravitational field in harmonic coordinates in a closed form completely. The problem is that it seems difficult (if at all possible) to find a particular solution of the Poisson equations for non-compact sources. The works so far overcome this problem by not solving the Poisson equations but keeping the Poisson integral unevaluated. Then to derive the equations of motion, we basically interchange the order of operations; first we evaluate surface integrals with the Poisson integrals as integrands (in the surface integral approach [91 ]) or compute derivatives of the Poisson integrals (when one adopts the geodesic equation [27 ]), and then we evaluate the remaining volume integrals. We first explain the usual method and a method to derive a field just around the star, and then explain the method mentioned above.

5.1 Super-potential method
5.2 Super-potential-in-series method
5.3 Direct-integration method