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.

In Chapters 8, 9, and 10 we examined gravitational phenomena that take place in the near zone, the region of space which contains the source of the gravitational field, and which is confined to a radius R that is much smaller than λc, the characteristic wavelength of the emitted radiation. This near-zone physics excluded radiative phenomena, and the dynamics of the system was entirely conservative. In Chapter 11 we moved to the wave zone, situated at a distance R that is much larger than λc, and studied the gravitational waves produced by processes taking place in the near zone; this wave-zone physics is all about radiative phenomena. In the first part of this chapter we continue our exploration of wave-zone physics by describing how gravitational waves transport energy, momentum, and angular momentum away from their source. These radiative losses imply that the near-zone physics cannot be strictly conservative, and in the second part of the chapter we identify the radiation-reaction forces which produce the required dissipation within the system. This chapter, therefore, is all about the linkage between the near and wave zones.

Radiative losses and radiation reaction are subtle topics in general relativity, and the mathematical description of these phenomena is technically demanding. To ease our entry into this subject, in Sec. 12.1 we first review the situation in the simpler context of flat-spacetime electromagnetism. We return to gravity in Sec. 12.2, in which we develop a general description of radiative losses in general relativity.

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.

In November 1915, Einstein completed a calculation whose result so agitated him that he worried that he might be having a heart attack. He later wrote to a friend that “for several days I was beside myself in joyous excitement.” What Einstein calculated was the contribution to the advance of the perihelion of Mercury from the first post-Newtonian corrections to Newtonian gravity provided by his newly completed theory of general relativity. This had been a notorious and unsolved problem in astronomy, ever since Le Verrier pointed out in 1859 that there was a discrepancy of approximately 43 arcseconds per century in the rate of advance between what was observed and what could be accounted for in Newtonian theory from planetary perturbations (refer to Secs. 3.1 and 3.4). Many earlier attempts to devise relativistic theories of gravity, including Einstein's own “Entwurf” (outline) theory of 1913 with Marcel Grossmann, had failed to give the correct answer. Now armed with the correct field equations, Einstein found an approximate vacuum solution that could be applied to the geodesic motion of Mercury around the Sun. He found that the orbit was almost Keplerian, but with a perihelion that advances at a rate that matched Le Verrier's observations.

For Einstein, this success with Mercury was the first concrete evidence that his theory, over which he had struggled so mightily for the past four years, might actually be correct.

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.

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.

In Chapter 1 we introduced the foundations of Newtonian gravity, and presented the equations that govern the gravitational potential of spherical and nearly spherical bodies. We also examined the center-of-mass motion of extended bodies, and witnessed the remarkable near-decoupling of the external dynamics – the motion of each body as a whole – from the internal dynamics – the internal fluid motions within each body. As we saw in Chapter 1, the details of internal structure, encapsulated in multipole moments of the mass distribution, have a limited influence on the motion of the body as a whole. In this chapter we take the focus away from the external dynamics and examine the internal structure and dynamics of extended, self-gravitating bodies. We shall return to the theme of the near-decoupling of the external and internal dynamics, and reveal the limited influence of the center-of-mass motion and the external bodies on the structure of a selected body.

We begin in Sec. 2.1 with a review of the equations of fluid mechanics that are relevant to the internal dynamics; these are best formulated in the moving reference frame of a selected body A in an N-body system. In Sec. 2.2 we examine the simplest models of internal structure, involving spherical symmetry, assuming that the body is non-rotating and not influenced by external bodies.

A Warning

Before describing how to observe and image the Sun, I must give the standard warning. The Sun is the only astronomical object that could cause harm to an observer. If any observing aid, binocular or telescope is used to directly observe the Sun without the use of suitable solar filters, the retina can be irreparably damaged, leaving a blind spot or worse. It is not so much the visible light as the infrared radiation that is the problem. So if a filter is used, perhaps to observe a partial solar eclipse, it is vital that it be opaque to infrared. Any filter used to observe the Sun must be specifically designed for this use and it must be one that will totally block the infrared emission whilst bringing the visible light down to safe levels.

When the Sun is observed (and very great care is taken to use appropriate filters!) it appears to have a sharp edge but there is, of course, no actual surface. We are, in fact, just seeing down through the solar atmosphere to a depth where the gas becomes what is called ‘optically thick’. This deepest visible layer of the atmosphere is called the ‘photosphere’ (as this is where the photons that we see originate) and is about 500 km thick (Figure 12.1). The effective temperature of the photosphere is ~5,800 K. The convective transport of energy from below gives rise to a mottling of the surface − solar granulations that are about 1,000 km across. Each granulation cell lasts about 5–10 minutes as hot gas, having risen from below the surface, radiates energy away, cools and sinks down again.

This chapter will discuss the fundamentals of telescopes and observing which are independent of the telescope type or, as in the case of the contrast of a telescope image, dependent on aspects of the telescope design. Later sections in the chapter will discuss the effects of the atmosphere on image quality due to the ‘atmospheric seeing’ and the faintness of stars that can be seen due to its ‘transparency’. The final sections will give details as to how the stars are charted and named on the celestial sphere and how time, relating to both the Earth (Universal Time) and the stars (sidereal time) are determined.

There is one problem that can cause some confusion: the mixing of two units of length: millimetres and inches. Quite a number of US and Russian telescopes have their apertures defined in inches, and the two common focusers have diameters of 1¼ and 2 inches. However, the focal lengths of telescopes and eyepieces are always specified in millimetres, as are the apertures of more recent US, Japanese and European telescopes. In this book I have used the unit which is appropriate and have not tried to convert inches into millimetres when, for example, referring to a 9.25-inch Schmidt-Cassegrain telescope. In calculations and where no specific telescope is referred to, millimetres are always used.