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.

In the preceding three chapters we stayed safely in the near zone and ignored all radiative aspects of the motion of bodies subjected to a mutual gravitational interaction. In this chapter we move to the wave zone and determine the gravitational waves produced by the moving bodies. To achieve this goal we must return to the post-Minkowskian approximation developed in Chapters 6 and 7, because the post-Newtonian techniques of Chapter 8 are necessarily restricted to the near zone.

We begin in Sec. 11.1 by reviewing the notion of far-away wave zone, in which the gravitational-wave field can be extracted from the (larger set of) gravitational potentials hαβ; we explain how to perform this extraction and obtain the gravitational-wave polarizations h+ and h×. In Sec. 11.2 we derive the famous quadrupole formula, the leading term in an expansion of the gravitational-wave field in powers of νc/c (with νc denoting a characteristic velocity of the moving bodies); we flesh out this discussion by examining a number of applications of the formula. Section 11.3 is a very long excursion into a computation of the gravitational-wave field beyond the quadrupole formula, in which we add corrections of fractional order (νc/c), (νc/c)2, and (νc/c)3 to the leading-order expression.

The relativistic formulation of the laws of physics developed in Chapter 4 excluded gravitation, and our task in this chapter is to complete the story by incorporating this all-important interaction (our personal favorite!). In Sec. 5.1 we explain why relativistic gravitation must be thought of as a theory of curved spacetime. In Sec. 5.2 we develop the elementary aspects of differential geometry that are required in a study of curved spacetime, and in Sec. 5.3 we show how the special-relativistic form of the laws of physics can be generalized to incorporate gravitation in a curved-spacetime formulation. We describe the Einstein field equations in Sec. 5.4, and in Sec. 5.5 we show how to solve them in the restricted context of small deviations from flat spacetime. We conclude in Sec. 5.6 with a description of spherical bodies in hydrostatic equilibrium, featuring the most famous (and historically the first) exact solution to the Einstein field equations; this is the Schwarzschild metric, which describes the vacuum exterior of any spherical distribution of matter (including a black hole).

Gravitation as curved spacetime

5.1.1 Principle of equivalence

Relativistic gravity

The relativistic Euler equation (4.59), unlike its Newtonian version of Eq. (1.23), does not contain a term that describes a gravitational force acting on the fluid. To insert such a term requires an understanding of how the Newtonian theory of gravitation can be generalized to a relativistic setting.