The theory was formulated in Chapter 6, and now we must get our hands dirty with its implementation. In this chapter we construct the second post-Minkowskian approximation to the metric of a curved spacetime produced by a bounded distribution of matter. For concreteness we choose the matter to consist of a perfect fluid. Our treatment allows the fluid to be of one piece (in the case of a single body), or broken up into a number of disconnected components (in the case of an N-body system).

Although the post-Minkowskian approximation does not require slow motion, we shall nevertheless assume that the fluid is subjected to a slow-motion condition of the sort described in Sec. 6.3.2: if νc is a characteristic velocity within the fluid, we insist that νc/c ≪ 1. This amounts to incorporating a post-Newtonian expansion within the post-Minkowskian approximation. We do this for two reasons. First, our ultimate goal is to describe situations of astrophysical interest, and the virial theorem implies that U ~ ν2 for any gravitationally bound system; weak fields are naturally accompanied by slow motion. Second, any attempt to keep the velocities arbitrary in the post-Minkowskian expansion quickly leads to calculations that are unmanageable, and we prefer to avoid these complications here.

We begin in Sec. 7.1 by assembling the required tools and exploring the general structure of the gravitational potentials in the near and wave zones.