Post-Newtonian theory is the theory of weak-field gravity within the near zone, and of the slowly moving systems that generate it and respond to it. It was first encountered in Chapter 7, where it was embedded within the post-Minkowskian approximation; the idea relies on the slow-motion condition introduced in Sec. 6.3.2. But while post-Minkowskian theory deals with both the near and wave zone, here we focus exclusively on the near zone. In this chapter we develop the post-Newtonian theory systematically.

We begin in Sec. 8.1 by collecting the main ingredients obtained in Chapter 7, including the near-zone metric to 1PN order and the matter's energy-momentum tensor Tαβ. In Sec. 8.2 we present an alternative derivation of the post-Newtonian metric, based on the Einstein equations in their standard form; this is the “classic approach” to post-Newtonian theory, adopted by Einstein, Infeld, and Hoffmann in the 1930s, and by Fock, Chandrasekhar, and others in the 1960s. Although it produces the same results, we will see that the classic approach presents us with a number of ambiguities that are not present in the post-Minkowskian approach. In Sec. 8.3 we explore the coordinate freedom of post-Newtonian theory, and construct the most general transformation that preserves the post-Newtonian expansion of the metric. And in Sec. 8.4 we derive the laws of fluid dynamics in post-Newtonian theory; these will be applied to the motion of an N-body system in Chapter 9.