In Part II of this book, we consider the history of the Universe after it is roughly one second old. It is quite well understood, and is known to begin with an era of rapid expansion dominated by hot radiation. In contrast, we have no certain knowledge about the preceding era that is the subject of Part IV.

We assume general relativity, which seems to accord with observation. This chapter is devoted to the unperturbed Universe. We begin with a general discussion of the main possibilities for particle distribution functions in thermal equilibrium, applying whenever the Universe has a gaseous component. Then we discuss baryon and lepton number. After these preliminaries, we lay down initial conditions, that are assumed to hold when the Universe is slightly less than one second old. It is assumed that there exist, in thermal equilibrium, protons, neutrons, electrons, positrons and all three species of neutrino and antineutrino.

We then see how, as the Universe expands, the initial conditions determine the primordial abundance of the lightest nuclei, in apparent agreement with observation. The success of this Big Bang Nucleosynthesis (BBN) calculation is one of the cornerstones of modern cosmology.

Next we follow the subsequent evolution of the unperturbed Universe. For this evolution to agree with observation, two more components have to be added to the cosmic fluid, leading to what is called the ∧CDM model.

What is cosmology?

The universe in the large

Cosmology is the study of the universe as a whole: its history, evolution, composition, dynamics. The primary aim of research in cosmology is to understand the large-scale structure of the universe, but cosmology also provides the arena, and the starting point, for the development of all the detailed small-scale structure that arose as the universe expanded away from the Big Bang: galaxies, stars, planets, people. The interface between cosmology and other branches of astronomy, physics, and biology is therefore a rich area of scientific research. Moreover, as astronomers have begun to be able to study the evidence for the Big Bang in detail, cosmology has begun to address very fundamental questions of physics: what are the laws of physics at the very highest possible energies, how did the Big Bang happen, what came before the Big Bang, how did the building blocks of matter (electrons, protons, neutrons) get made? Ultimately, the origin of every system and structure in the natural world, and possibly even the origin of the physical laws that govern the natural world, can be traced back to some aspect of cosmology.

In the 23 years between the first edition of this textbook and the present revision, the field of general relativity has blossomed and matured. Upon its solid mathematical foundations have grown a host of applications, some of which were not even imagined in 1985 when the first edition appeared. The study of general relativity has therefore moved from the periphery to the core of the education of a professional theoretical physicist, and more and more undergraduates expect to learn at least the basics of general relativity before they graduate.

My readers have been patient. Students have continued to use the first edition of this book to learn about the mathematical foundations of general relativity, even though it has become seriously out of date on applications such as the astrophysics of black holes, the detection of gravitational waves, and the exploration of the universe. This extensively revised second edition will, I hope, finally bring the book back into balance and give readers a consistent and unified introduction to modern research in classical gravitation.

The first eight chapters have seen little change. Recent references for further reading have been included, and a few sections have been expanded, but in general the geometrical approach to the mathematical foundations of the theory seems to have stood the test of time. By contrast, the final four chapters, which deal with general relativity in the astrophysical arena, have been updated, expanded, and in some cases completely re-written.

Fundamental principles of special relativity (SR) theory

The way in which special relativity is taught at an elementary undergraduate level – the level at which the reader is assumed competent – is usually close in spirit to the way it was first understood by physicists. This is an algebraic approach, based on the Lorentz transformation (§ 1.7 below). At this basic level, we learn how to use the Lorentz transformation to convert between one observer's measurements and another's, to verify and understand such remarkable phenomena as time dilation and Lorentz contraction, and to make elementary calculations of the conversion of mass into energy.

This purely algebraic point of view began to change, to widen, less than four years after Einstein proposed the theory. Minkowski pointed out that it is very helpful to regard (t, x, y, z) as simply four coordinates in a four-dimensional space which we now call spacetime. This was the beginning of the geometrical point of view, which led directly to general relativity in 1914–16. It is this geometrical point of view on special relativity which we must study before all else.

Fluids

In many interesting situations in astrophysical GR, the source of the gravitational field can be taken to be a perfect fluid as a first approximation. In general, a ‘fluid’ is a special kind of continuum. A continuum is a collection of particles so numerous that the dynamics of individual particles cannot be followed, leaving only a description of the collection in terms of ‘average’ or ‘bulk’ quantities: number of particles per unit volume, density of energy, density of momentum, pressure, temperature, etc. The behavior of a lake of water, and the gravitational field it generates, does not depend upon where any one particular water molecule happens to be: it depends only on the average properties of huge collections of molecules.

Nevertheless, these properties can vary from point to point in the lake: the pressure is larger at the bottom than at the top, and the temperature may vary as well. The atmosphere, another fluid, has a density that varies with position. This raises the question of how large a collection of particles to average over: it must clearly be large enough so that the individual particles don't matter, but it must be small enough so that it is relatively homogeneous: the average velocity, kinetic energy, and interparticle spacing must be the same everywhere in the collection. Such a collection is called an ‘element’.