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.

In this chapter we apply the tools developed in the previous two chapters to an exploration of the orbital dynamics of bodies subjected to their mutual gravitational attractions. Many aspects of what we learned in Chapters 1 and 2 will be put to good use, and the end result will be considerable insight into the behavior of our own solar system. To be sure, the field of celestial mechanics has a rich literature that goes back centuries, and this relatively short chapter will only scratch the surface. We believe, however, that we have sampled the literature well, and selected a good collection of interesting topics. Some of the themes introduced here will be featured in later chapters, when we turn to relativistic aspects of celestial mechanics.

We begin in Sec. 3.1 with a very brief survey of celestial mechanics, from Newton to Einstein. In Sec. 3.2 we give a complete description of Kepler's problem, the specification of the motion of two spherical bodies subjected to their mutual gravity. In Sec. 3.3 we introduce a powerful formalism to treat Keplerian orbits perturbed by external bodies or deformations of the two primary bodies; in this framework of osculating Keplerian orbits, the motion is at all times described by a sequence of Keplerian orbits, with constants of the motion that evolve as a result of the perturbation. We shall apply this formalism to a number of different situations, and highlight a number of important processes that take place in the solar system and beyond.

The central theme of this book is gravitation in its weak-field aspects, as described within the framework of Einstein's general theory of relativity. Because Newtonian gravity is recovered in the limit of very weak fields, it is an appropriate entry point into our discussion of weak-field gravitation. Newtonian gravity, therefore, will occupy us within this chapter, as well as the following two chapters.

There are, of course, many compelling reasons to begin a study of gravitation with a thorough review of the Newtonian theory; some of these are reviewed below in Sec. 1.1. The reason that compels us most of all is that although there is a vast literature on Newtonian gravity – a literature that has accumulated over more than 300 years – much of it is framed in old mathematical language that renders it virtually impenetrable to present-day students. This is quite unlike the situation encountered in current presentations of Maxwell's electrodynamics, which, thanks to books such as Jackson's influential text (1998), are thoroughly modern. One of our main goals, therefore, is to submit the classical literature on Newtonian gravity to a Jacksonian treatment, to modernize it so as to make it accessible to present-day students. And what a payoff is awaiting these students! As we shall see in Chapters 2 and 3, Newtonian gravity is most generous in its consequences, delivering a whole variety of fascinating phenomena.

During the past forty years or so, spanning roughly our careers as teachers and research scientists, Einstein's theory of general relativity has made the transition from a largely mathematical curiosity with limited relevance to the real world to arguably the centerpiece of our effort to understand the universe on all scales.

At the largest scales, those of the universe as a whole, cosmology and general relativity are joined at the hip. You can't do one without the other. At the smallest scales, those of the Planck time, Planck length, and Planck energy, general relativity and particle physics are joined at the hip. String theory, loop quantum gravity, the multiverse, branes and bulk – these are arenas where the geometry of Einstein and the physics of the quantum may be inextricably linked. These days it seems that you can't do one without the other.

At the intermediate scales that interest astronomers, general relativity and astrophysics are becoming increasingly linked. You can still do one without the other, but it's becoming harder. One of us is old enough to remember a time when the majority of astronomers felt that black holes would never amount to much, and that it was a waste of time to worry about general relativity. Today black holes and neutron stars are everywhere in the astronomy literature, and gravitational lensing – the tool that relies on the relativistic bending of light – is used for everything from measuring dark energy to detecting exoplanets.

The preceding chapters were devoted to a Newtonian description of the gravitational interaction, and it is now time to embark on an exploration of its relativistic aspects. As we shall argue in the next chapter, a relativistic theory of gravity that respects the principle of equivalence reviewed in Sec. 1.2 must be a metric theory in which gravitation is a manifestation of the curvature of spacetime. The simplest metric theory of gravitation is Einstein's general relativity, and our task in this chapter and the next is to introduce its essential elements. Subsequent chapters will develop the weak-field limit of general relativity, and in these chapters we will return to notions (such as gravitational potentials and forces) that are familiar from Newtonian physics. But a proper grounding of the weak-field limit must rest on the exact theory, and we shall now work to acquire the required knowledge. It is, of course, unlikely that a mere two chapters will suffice to introduce all relevant aspects of general relativity. What we intend to cover here is a rather minimal package, the smallest required for the development of the weak-field limit.

This chapter is devoted to a description of physics in Minkowski spacetime (also known as flat spacetime), which codifies in a particularly elegant way the kinematical rules of special relativity.

From Chapter 5 until now we have confined our attention to Einstein's general theory of relativity. But general relativity is not the only possible relativistic theory of gravity. Even in the late 1800s, well before Einstein began his epochal work on special and general relativity, there were attempts to devise theories of gravity that went beyond Newtonian theory. Some attempts were modeled on Maxwell's electrodynamics. Some replaced ∇2 with a wave operator in Poisson's equation of Newtonian gravity, in an attempt to formulate a theory that was invariant under Lorentz transformations. None of these attempts was very successful; for example, most theories could not account for the anomalous perihelion advance of Mercury. In 1913, before Einstein completed the general theory of relativity, Nordström proposed a theory involving a curved spacetime; the metric was expressed as gαβ = Φηαβ, with the scalar field Φ satisfying a Lorentz-invariant wave equation. But the theory automatically predicts a zero deflection of light, and ultimately it failed the test of experiment.

Alternative proposals appeared even after the publication of general relativity and the empirical successes with Mercury and the deflection of light. The eminent mathematician and philosopher Alfred North Whitehead formulated such an alternative theory in 1922. Troubled by the fact that in general relativity the causal relationships in spacetime are not known a priori, but only after the metric has been determined for a given distribution of matter, he devised a theory with a background Minkowski metric in order to put causality on a “firmer” ground.