The concepts of general relativity

special relativity To understand the issues involved in general relativity, it is helpful to begin with a brief summary of the way space and time are treated in special relativity. The latter theory is an elaboration of the intuitive point of view that the properties of empty space should be the same throughout the universe. This is just a generalization of everyday experience: the world in our vicinity looks much the same whether we are stationary or in motion (leaving aside the inertial forces experienced by accelerated observers, to which we will return shortly).

The immediate consequence of this assumption is that any process that depends only on the properties of empty space must appear the same to all observers: the velocity of light or gravitational radiation should be a constant. The development of special relativity can of course proceed from the experimental constancy of c, as revealed by the Michelson-Morley experiment, but it is worth noting that Einstein considered the result of this experiment to be inevitable on intuitive grounds (see Pais 1982 for a detailed account of the conceptual development of relativity). Despite the mathematical complexity that can result, general relativity is at heart a highly intuitive theory; the way in which our everyday experience can be generalized to deduce the large-scale structure of the universe is one of the most magical parts of physics.

The Robertson–Walker metric

To the relativist, cosmology is the task of finding solutions to Einstein's field equations that are consistent with the large-scale matter distribution in the universe. Modern observational cosmology has demonstrated that the real universe is highly symmetric in its large-scale properties, but the evidence for this was not known at the time when Friedmann and Lemaître began their pioneering investigations. Just as Einstein aimed to write down the simplest possible relativistic generalization of the laws of gravity, so cosmological investigation began by considering the simplest possible mass distribution: one whose properties are homogeneous (constant density) and isotropic (the same in all directions).

isotropy implies homogeneity At first sight, one might think that these two terms mean the same thing, but it is possible to construct universes that are homogeneous but anisotropic; the reverse, however, is not possible. Consider an observer who is surrounded by a matter distribution that is perceived to be isotropic; this means not only that the mass density is a function of radius only, but that there can be no preferred axis for other physical attributes such as the velocity field. This has an important consequence if we take the velocity strain tensor ∂vi/∂xj and decompose it into symmetric and antisymmetric parts.

The sequence of galaxy formation

The discussion of galaxy evolution in chapter 13 raised many basic questions about the process of galaxy formation: did bulges form first, and did they accrete disks later? What is the importance of galaxy mergers? What sets the form of the galaxy luminosity function? In addition, we have seen in chapters 12 and 16 that an Ω = 1 universe requires galaxy formation to be biased in favour of high-density environments; how could such a bias have arisen? The purpose of this chapter is to present some of the theoretical tools with which these questions may be tackled. We start with a simple overview of two contrasting ways in which collapsed objects like galaxies could form.

dissipationless collapse What will be the final state of an object that breaks away from the background and undergoes gravitational collapse? If the matter of the object is collisionless (either purely dark matter, or stars), this is a relatively well-posed problem, which should be capable of a clear solution.

The analytical approach has concentrated on gravitational thermodynamics, and sought an equilibrium solution. This has turned out to be a subtle and paradoxical problem, whose main analysis goes back to a classic paper by Lynden-Bell (1967). Imagine initially that the self-gravitating body consists of gas, so that it is reasonable to look for an equilibrium solution in the form of a configuration of constant temperature.

The previous chapter showed how to understand the propagation of light in a homogeneous universe, allowing the intrinsic properties of cosmologically distant objects to be inferred from the observations we make today. The real universe is of course not perfectly homogeneous, and the propagation of light is influenced by the gravitational fields of collapsed objects. This phenomenon of gravitational lensing (known in French literature by the perhaps more appropriate term gravitational mirages) was first studied by Einstein in 1912 (see Renn, Sauer & Stachel 1997), well before the 1919 eclipse verification of Einstein's formula for light deflection. The Sun can deflect light by angles of order 1 arcsec; over a large enough baseline, this deflection can focus light rays, so that the Sun acts as a gravitational telescope. Over cosmological baselines in particular, the distortions produced by this form of gravitational imaging can be large and significant. In 1979, the discipline of gravitational lensing leapt from theoretical speculation to being an observational tool with the discovery of 0957+561, a quasar split into two images 6 arcsec apart by the gravitation of an intervening galaxy (Walsh, Carswell & Weymann 1979). The subject of gravitational lensing has since developed into one of the major tools of cosmology. It is an invaluable way of probing mass distributions directly, irrespective of whether the matter is dark or visible. For (much) more detail, consult the textbook by Schneider, Ehlers & Falco (1992), or the review by Blandford & Narayan (1992).

Advances in observational astronomy during the 1990s have finally allowed direct study of the population of normal galaxies at high redshifts, as discussed in chapter 13. For more than two decades prior to this, the only objects that could be studied out to cosmologically important distances were ‘active’ galaxies such as quasars, where the dominant energy output is not due to stars. This chapter attempts to separate out those aspects of active galaxies that are of especial cosmological interest, although any such division is inevitably blurred. Many of the interesting details of the subject will be omitted: good references for digging deeper into this area are Weedman (1986), Blandford, Netzer & Woltjer (1990), Hughes (1991) and Robinson & Terlevich (1994).

The population of active galaxies

classification Active galaxies come in a variety of species, many of which overlap. The definition of an active galaxy is one where a significant fraction of the energy output in at least some waveband is not contributed by normal stellar populations or interstellar gas; however, all galaxies emit non-thermal radiation at some level, and so classification as active is only a matter of degree. Things are a little more clear cut with AGN (active galactic nuclei), where the non-thermal emission comes mainly from the central few pc of the galaxy.

Elementary particles and fields

The apparatus of quantum field theory is an effective tool for calculating the rates of physical processes. All that is needed is a Lagrangian, and then there is a relatively standard procedure to follow. The trouble with this generality is that there is no way of understanding why certain Lagrangians are found in nature while others are not. The present chapter is concerned with the progress that has been made in solving this problem. The last four decades have seen tremendous developments in understanding: a few simple principles end up specifying much of the way in which elementary particles interact, and point the way to a potential unification of the fundamental interactions. These developments have led to the standard model of particle physics, which in principle allows any observable quantity related to the interactions of particles to be calculated in a consistent way.

The standard model asserts that the building blocks of physics are a certain set of fundamental particles from which the composite particles seen in experiments are constructed; their properties are listed in table 8.1 (see the references to the Particle Data Group in the bibliography). This set is much larger than the electron, proton and photon, which were all that was needed in 1920, but it is believed to be complete.