The idea of this chapter is to go into more detail on a few selected topics in relativity, concentrating on those that are of the most direct interest to the astrophysicist: fluid mechanics, weak fields and orbits therein, gravitational radiation and black holes. A word of warning: so far, all equations have been explicitly dimensional, and contain the correct powers of c; from now on, this will not always be the case. It is common in the literature on relativity to simplify formulae by choosing units where c = 1. Since this can be confusing for the beginner, c will be retained where it does not make the algebra too cumbersome. However, sometimes it will disappear from a derivation temporarily. This should encourage the good habit of always being aware of the dimensional balance of any given equation.

Relativistic fluid mechanics

One of the attractive features of relativity is the economical form that many of its fundamental equations can take. The price paid for this is that the quantities of interest are not always immediately available; the process of ‘unpacking’ some of these expressions can become rather painful and reveal considerable buried complexity. It is worth illustrating this with the example of fluid mechanics, not just for its own sake, but because we will end up with some results that are rather useful in astrophysics and cosmology.

Magnetic fields in the Cosmos

Around 1600 William Gilbert, physician to Queen Elizabeth I of England, proposed a bold hypothesis to explain why a suspended compass needle points in the north–south direction. He suggested that the whole Earth is a huge magnet and attracts the compass needle. This is probably the first time that somebody proposed an astronomical object—the planet Earth—to have a large-scale magnetic field. Initially it was thought that the Earth's magnetism was of ferromagnetic origin. By the end of the nineteenth century, it became clear that a ferromagnetic substance does not retain the magnetism when heated beyond a certain temperature (the Curie point). Since the interior of the Earth is believed to be hotter than the Curie temperature of any known ferromagnetic substance, it was apparent that one has to look for alternative explanations for the Earth's magnetic field.

Until the beginning of the twentieth century, it was not known whether other astronomical objects have magnetic fields as well. When Hale (1908) made the momentous discovery of magnetic fields in sunspots on the basis of the Zeeman splittings of sunspot spectra, the existence of magnetic fields outside the Earth's environment was conclusively established for the first time. Large sunspots can have magnetic fields of the order of 3000 G, which is much stronger than the Earth's field (the maximum value on the Earth's surface is about 0.6 G). One of the major achievements of twentieth century astronomy is to establish that magnetic fields are ubiquitous in the Universe.

We have come to the end of a long journey. Before saying a final goodbye to the reader, we wish to present an assortment of mixed fares in this final chapter. The main purpose of the book has been to develop the fundamentals fully. We now give a glimpse of what lies beyond the horizon.

Often it happens that one does not know the details of the physical conditions inside an astrophysical system, but can make rough estimates of different kinds of energies contained in the system (kinetic, potential, magnetic, etc.). To handle such situations, one can suitably integrate the basic equations to obtain an equation connecting different types of total energies of the system. This equation is known as the virial equation. Since this approach is very general, we present a discussion of it in this final chapter, when the reader should be in a position to possess a broad overview of the whole field. While applying hydrodynamics and magnetohydrodynamics to astrophysical systems, often it becomes necessary to incorporate relativistic corrections or to include the effects of radiation pressure. The subjects of relativistic hydrodynamics and radiation hydrodynamics respectively deal with these problems. These are vast fields of study, and we cannot provide proper treatments of these two fields in this elementary book. Just to give an idea of how one proceeds, we discuss the basic equations of these two fields in §17.2 and §17.3. Compared to the usual style of presentation in this book, these two sections would appear like skeletons without flesh and blood.

Hydrodynamics, magnetohydrodynamics, kinetic theory and plasma physics are becoming increasingly important tools for astrophysics research. Many graduate schools in astrophysics around the world nowadays offer courses to train graduate students in these areas. This was not the case even a few years ago—say around 1980—when it was rare for an astrophysics graduate school to teach these subjects, and the students who needed the knowledge of these subjects for their research were supposed to pick up the tricks of the trade on their own. With increasing applications of these subjects to astrophysics—especially to understand many phenomena discovered in the radio, X-ray or infrared wavelengths—the need is felt to impart a systematic training in these areas to all graduate students in astrophysics.

When I joined the faculty of the Astronomy Programme in Bangalore in 1987, I argued that a course covering these subjects should be introduced. My colleague and friend, Rajaram Nityananda, shared my enthusiasm for it, and we together managed to convince the syllabus committee of the need for it. From then onwards, this course has been taught regularly in our graduate programme, the responsibility of teaching it falling on my shoulders on several occasions. When I taught this course for the first time in 1988, I had to work very hard preparing lectures from different sources. I was lucky to have taken such a course myself as a graduate student in Chicago in 1981—taught by E. N. Parker—although it was somewhat unusual at that time for astronomy departments in the U.S.A. to offer such courses.

The need for a statistical theory

The linear perturbation theory presented in Chapter 7 makes it clear that a fluid configuration can, under certain circumstances, be unstable to perturbations. Once the perturbations grow to sufficiently large amplitudes, the linear theory is no longer applicable. Hence the linear theory is unable to predict what eventually happens to an unstable fluid system.

To understand the effect of an instability on a general dynamical system, let us employ the notion of a phase space introduced in Chapter 1. Figure 8.1 is a schematic representation of the phase space of a dynamical system, within which let P be a point corresponding to an unstable equilibrium. If the state of the system is represented exactly by P, then the state does not change by virtue of equilibrium. If, however, there is some perturbation around the equilibrium, then the state of the system is represented by some point in the neighborhood of P, and as the perturbation grows, the point in the phase space moves away from P. Thus, depending on whether the initial state was exactly at P or slightly away, the final state after some time can lie in very different regions of the phase space. Because of the limited accuracy in any measurement in a realistic situation, one can only assert that the initial state of a system lies in some finite region of the phase region.

Introduction

In the previous chapter, we developed MHD following a pattern somewhat similar to the pattern followed earlier while developing hydrodynamics. After presenting the basic equations, we first considered the possibility of static equilibrium, and afterwards waves and instabilities were discussed. Although the mathematical analysis in the presence of a magnetic field becomes much more complicated than the corresponding analysis in the pure hydrodynamic case and consequently our discussions in Chapter 14 were often less complete than the earlier corresponding discussions in the pure hydrodynamic case, we have seen that the basic techniques and the methodology were the same.

We now wish to look at a class of MHD problems loosely called topological problems. Let us first consider a situation of ideal MHD, where we have a magnetofluid of zero resistivity. Then, according to Alfvén's theorem, the magnetic field is completely frozen in the plasma. We have pointed out one important consequence of Alfvén's theorem in §14.2. If two fluid elements lie on a magnetic field line, then they would always lie on one field line. We may have two far-away fluid elements in the ideal magnetofluid connected by a magnetic field line. No matter what happens to the magnetofluid or how it evolves in time, this connectivity between the two far-away fluid elements remains preserved if the resistivity is zero. The preservation of such connectivities may introduce some constraints on the dynamics of the system.