Differentiable manifolds and tensors

The mathematical concept of a curved space begins (but does not end) with the idea of a manifold. A manifold is essentially a continuous space which looks locally like Euclidean space. To the concept of a manifold is added the idea of curvature itself. The introduction of curvature into a manifold will be the subject of subsequent sections. First we study the idea of a manifold, which we can regard as just a fancy word for ‘space’.

Manifolds

The surface of a sphere is a manifold. So is any m-dimensional ‘hyperplane’ in an n-dimensional Euclidean space (m ≤ n). More abstractly, the set of all rigid rotations of Cartesian coordinates in three-dimensional Euclidean space will be shown below to be a manifold. Basically, a manifold is any set that can be continuously parametrized. The number of independent parameters is the dimension of the manifold, and the parameters themselves are the coordinates of the manifold. Consider the examples just mentioned. The surface of a sphere is ‘parametrized’ by two coordinates θ and φ. The m-dimensional ‘hyperplane’ has m Cartesian coordinates, and the set of all rotations can be parametrized by the three ‘Euler angles’, which in effect give the direction of the axis of rotation (two parameters for this) and the amount of rotation (one parameter). So the set of rotations is a manifold: each point is a particular rotation, and the coordinates are the three parameters. It is a three-dimensional manifold.

This book has evolved from lecture notes for a full-year undergraduate course in general relativity which I taught from 1975 to 1980, an experience which firmly convinced me that general relativity is not significantly more difficult for undergraduates to learn than the standard undergraduate-level treatments of electromagnetism and quantum mechanics. The explosion of research interest in general relativity in the past 20 years, largely stimulated by astronomy, has not only led to a deeper and more complete understanding of the theory, it has also taught us simpler, more physical ways of understanding it. Relativity is now in the mainstream of physics and astronomy, so that no theoretical physicist can be regarded as broadly educated without some training in the subject. The formidable reputation relativity acquired in its early years (Interviewer: ‘Professor Eddington, is it true that only three people in the world understand Einstein's theory?’ Eddington: ‘Who is the third?’) is today perhaps the chief obstacle that prevents it being more widely taught to theoretical physicists. The aim of this textbook is to present general relativity at a level appropriate for undergraduates, so that the student will understand the basic physical concepts and their experimental implications, will be able to solve elementary problems, and will be well prepared for the more advanced texts on the subject.

In pursuing this aim, I have tried to satisfy two competing criteria: first, to assume a minimum of prerequisites; and, second, to avoid watering down the subject matter.

This is an account of the discovery and exploration of a sea of thermal radiation that smoothly fills space. The properties of this radiation (which we describe beginning on page 16) show that it is a fossil, a remnant from a time when our universe was denser and hotter and vastly simpler, a very nearly uniform sea of matter and radiation. The discovery of the radiation left from this early time is memorable because, as is often true of fossils, measurements of its properties give insights into the past. The study of this fossil radiation has proved to be exceedingly informative for cosmology, the study of how our universe expanded, cooled, and evolved to its present complicated condition.

The discovery of the fossil radiation grew out of a mix of lines of evidence that were sometimes misinterpreted or overlooked, and of ideas that were in some cases perceptive but ignored and in other cases misleading but entrenched. In the 1960s, it was at last generally recognized that the pieces might fit together and teach us something about the large-scale nature of the universe. We introduce the accounts of how this happened by explaining the lines of research that led up to the situation then. The story of what happened when the pieces were put together in the 1960s is told through the recollections of the people in the best position to know – those involved in the research. We have essays by most who took part in the recognition that this fossil exists, its properties may be measured, and what is measured may inform us about the nature of the physical universe. This did not happen all at once; nor was it done by a single person; nor was it always done knowingly. The collection of essays tell what happened in all the richness and complexity we suppose is typical of any activity that people take seriously.

We want to convey an impression of the broad effort that went into the measurements of the CMBR without obscuring the story with details. To that end we present separately, in this Appendix, tabulations of the experiments that figure in the panoramic illustrations of the progress of the measurements in the figures in Chapter 5. We present first a tabulation of observations of the interstellar CN absorption lines that gave important early measures of the CMBR energy spectrum. The next, longer, tabulation is of the experiments that explored the energy spectrum of the CMBR. The progress on this front is illustrated in Figures 5.2–5.4. The third and still longer tabulation shows the experiments in the enormous effort to map out the intrinsic anisotropy of the CMBR, in the developments illustrated in Figures 5.9–5.18.

Table A.1 lists the progress in measurements of the CMBR energy spectrum at the two energy level separations in the CN molecule that happen to be conveniently located for this purpose. The general method of observing the excitation of interstellar CN and interpreting the excitation in terms of the CMBR spectrum is discussed in Chapter 3 (commencing on page 42). The essays in Chapter 4 (starting on page 74) recall what happened when this interpretation was first generally realized. The observations yield measures of the CMBR spectrum at the two frequencies indicated in the table and plotted as open boxes in Figures 5.2 to 5.4. It was very important to early studies of the CMBR that the CN observation at 113 GHz combined with the measurements at lower frequencies, which could be made from the ground, showed that the CMBR has a Rayleigh–Jeans power-law spectrum with the downward departure from the power law at 113 GHz to be expected for a thermal spectrum as it heads to its peak.

To understand the essays in the next chapter about what happened in the 1960s you have to appreciate the nature of research in cosmology then. To understand the nature of this research you have to consider its history. Figure 3.1 illustrates the major steps leading to one big advance in cosmology, the identification of the CMBR as a fossil remnant from the big bang. This figure was made by members of Princeton Gravity Research Group. David Wilkinson was its main author, he used it in lectures on cosmology starting in 1968, and it is a good illustration of his style. Another version was eventually published (in Wilkinson and Peebles 1983).

The figure maps relations among the topics we discuss in this chapter. The map is complicated because the story is complicated, but there are a few themes. We begin with the first of these, the development of the idea that the abundances of the stable isotopes of the lightest elements, hydrogen and helium, were determined by thermonuclear reactions in the early hot stages of expansion of the universe (with modest adjustments for what happened in stars much later). We consider next the line of thought that led Dicke to persuade Roll and Wilkinson to search for the CMBR. We then turn to the development of the means of detecting and measuring the properties of the radiation left from the hot big bang. We conclude this chapter with an assessment of what people were thinking and doing in cosmology in the early 1960s, at the start of the time surveyed by the essays in the next chapter.

Nucleosynthesis in a hot big bang

Hydrogen is the most abundant of the chemical elements (apart from places like Earth where the heavier elements have collected and condensed), helium amounts to about 25% by mass, and only about 2% of the baryon mass is in heavier elements.

This chapter was written in collaboration with J. Richard Bond

In 1970 we knew that space is filled with a near-uniform sea of microwave radiation, the CMBR. That was interesting, an addition to the list of what is known about the universe. But, as we have been discussing, there was reason to think that this was a particularly important find, a fossil left nearly undisturbed from a time when our universe was very different from now – dense, hot and rapidly expanding. The main piece of evidence was the spectrum – the variation of the intensity or energy in this radiation with wavelength – which was known to be close to the thermal form one would expect for radiation left undisturbed from hot early stages of an expanding universe. It was encouraging also for this hot big bang picture that it offered an explanation of another observation, the large abundance of the element helium, which could be another fossil remnant from the early universe. But in 1970 this big bang interpretation was a large conclusion about the nature of the universe to draw from the exceedingly limited set of evidence we could bring to bear. Perhaps the CMBR originated in some other way, possibly by processes operating in the universe as it is now, as in the steady state cosmology. Or perhaps the radiation came from a time when conditions were different from now but different also from standard ideas about the big bang.

The early exploration of alternatives to the idea of a hot big bang is reviewed beginning on page 34. Debates such as this are a normal and healthy part of science, and there is a standard procedure for resolving them: sift through the evidence, set about gathering more of it, and explore how it all might fit together. The process is a learning curve: by trial and error seek ways to develop improved trials that may on occasion show how to do even better, and then still better.

This is the story of a major advance in science, the discovery of fossil radiation left from the early stages of expansion of the universe – the big bang. Colleagues in informal conversations now only vaguely recalled led us to realize that this story is particularly worth examining because it happened in what was then a small line of research, and one that still is relatively simple compared to many other branches of physical science. That makes it well suited for an examination of how science actually is done, warts and all, in all the details – usually too numerous to mention – recalled by many of the people who did the work.

All the main steps in this story – the prediction, detection, identification, and exploration of the properties of the fossil radiation from the big bang – have been presented in histories of science. But these histories do not have the space (or the aim) to give an impression of what it was like to live through those times. We sense a similar feeling of incompleteness in many histories of science written by physicists, as well as by professional historians and sociologists. And there is a well-established remedy: assemble recollections from those who were involved in the work. An example in the broader field of cosmology – the study of the large-scale structure of the universe – is the collection of interviews in Origins: the Lives and Worlds of Modern Cosmologists (Lightman and Brawer 1990). We follow that path, but in more detail in a more limited line of research.

Early studies of the fossil radiation involved a relatively small number of people in what has proved to be a considerable advance in establishing the physical nature of the universe. This means we could aim for complete coverage of recollections from everyone involved in the early work who is still with us.

Our plan of ordering the essays is to group them by topic, with chronological order within groups. The grouping is by the focus of the research, as indicated by the section headers. Since this focus tends to evolve with time, the result is that these recollections of what happened are presented in a roughly chronological order. For example, in the second half of the 1960s a first order of business on the experimental side was the test of how the energy of the CMBR varies with wavelength, and on the theoretical side it was the exploration of ideas about what a significant departure from a thermal spectrum might mean. These continued to be pressing issues at the end of the 1960s, but there was increasing interest in the experimental search for departures from an exactly isotropic distribution of the radiation, and in the development of the theory of the departures from isotropy that might be expected to accompany the known departures from an exactly homogeneous distribution of the matter. Thus we present the essays whose main focus is the spectrum before those largely concerned with the anisotropy of the CMBR.

Since many of the essays do not fit the headers our plan required arbitrary and debatable decisions on ordering. This is a realistic illustration of what was happening in the 1960s, of course. The confusion extends to the recollections: the stories are not complete and they are not always even consistent with each other. The reader, therefore, must be prepared for a distinct change of style from the linear – but we hope efficient – history of ideas in the previous chapter to the chaos of the real world of science.

Precursor evidence from communications experiments

David C. Hogg: Early low-noise and related studies at Bell Laboratories, Holmdel, NJ

The US National Academy of Engineering cites Hogg's election to the Academy for his “contributions to the understanding of electromagnetic propagation at microwave frequencies through the atmosphere.” A native of saskatchewan, Hogg' current interest is the composition of music.