In this chapter we embark on a general program to specialize the formulation of general relativity to a description of weak gravitational fields. We will go from the exact theory, which governs the behavior of arbitrarily strong fields, such as those of neutron stars and black holes, to a useful approximation that applies to weak fields, such as those of planets, main-sequence stars, and white dwarfs. This approximation will reproduce the predictions of Newtonian theory, but we will formulate a method that can be pushed systematically to higher and higher order to produce an increasingly accurate description of a weak gravitational field. We shall find that the method is so successful that it can actually handle fields that are not so weak. For example, it provides a perfectly adequate description of gravity at a safe distance from a neutron star, and it can be used as a foundation to study the motion of a binary black-hole system, provided that the mutual gravity between bodies is weak.

The foundation for these methods is “post-Minkowskian theory,” the topic of this chapter and the next. In post-Minkowskian theory the strength of the gravitational field is measured by the gravitational constant G, and the Einstein field equations are formally expanded in powers of G. At zeroth post-Minkowskian order there is no field, and one deals with Minkowski spacetime.