In this chapter we apply the results of Chapter 8 to situations in which a fluid distribution breaks up into a collection of separated bodies. Our aim is to go from a fine-grained description involving the fluid variables {ρ*, p, Π, ν} to a coarse-grained description involving a small number of center-of-mass variables for each body. We accomplish this reduction in Sec. 9.1, and in Sec. 9.2 we apply it to a calculation of the spacetime metric in the empty region between bodies; the metric is thus expressed in terms of the mass-energy MA, position rA(t), and velocity νA(t) of each body. In Sec. 9.3 we derive post-Newtonian equations of motion for the center-of-mass positions, and in Sec. 9.4 we show that the same equations apply to compact bodies with strong internal gravity. In Sec. 9.5 we allow the bodies to rotate, and we calculate the influence of the spins on the metric and equations of motion; we also derive evolution equations for the spin vectors. We conclude in Sec. 9.6 with a discussion of how point particles can be usefully incorporated within post-Newtonian theory, in spite of their infinite densities and diverging gravitational potentials.

From fluid configurations to isolated bodies

We consider a situation in which a distribution of perfect fluid breaks up into a number N of separated components. We call each component a “body,” and label each body with the index A = 1, 2, …, N.