For over half a century, the general theory of relativity has stood as a monument to the genius of Albert Einstein. It has altered forever our view of the nature of space and time, and has forced us to grapple with the question of the birth and fate of the universe. Yet, despite its subsequently great influence on scientific thought, general relativity was supported initially by very meager observational evidence. It has only been in the last two decades that a technological revolution has brought about a confrontation between general relativity and experiment at unprecedented levels of accuracy. It is not unusual to attain precise measurements within a fraction of a percent (and better) of the minuscule effects predicted by general relativity for the solar system.

To keep pace with these technological advances, gravitation theorists have developed a variety of mathematical tools to analyze the new high precision results, and to develop new suggestions for future experiments made possible by further technological advances. The same tools are used to compare and contrast general relativity with its many competing theories of gravitation, to classify gravitational theories, and to understand the physical and observable consequences of such theories.

The first such mathematical tool to be thoroughly developed was a “theory of metric theories of gravity” known as the Parametrized Post-Newtonian (PPN) formalism, which was suited ideally to analyzing solar system tests of gravitational theories.

The summer of 1974 was an eventful one for Joseph Taylor and Russell Hulse. Using the Arecibo radio telescope in Puerto Rico, they had spent the time engaged in a systematic survey for new pulsars. During that survey, they detected 50 pulsars, of which 40 were not previously known, and made a variety of observations, including measurements of their pulse periods to an accuracy of one microsecond. But one of these pulsars, denoted PSR 1913 + 16, was peculiar: besides having a pulsation period of 59 ms – shorter than that of any known pulsar except the one in the Crab Nebula – it also defied any attempts to measure its period to ± 1 μs, by making “apparent period changes of up to 80 μs from day to day, and sometimes by as much as 8 μs over 5 minutes” (Hulse and Taylor, 1975). Such behavior is uncharacteristic of pulsars, and Hulse and Taylor rapidly concluded that the observed period changes were the result of Doppler shifts due to orbital motion of the pulsar about a companion. By the end of September, 1974, Hulse and Taylor had obtained an accurate “velocity curve” of this “single line spectroscopic binary.” The velocity curve was a plot of apparent period of the pulsar as a function of time.

The Principle of Equivalence has played an important role in the development of gravitation theory. Newton regarded this principle as such a cornerstone of mechanics that he devoted the opening paragraphs of the Principia to a detailed discussion of it (Figure 2.1). He also reported there the results of pendulum experiments he performed to verify the principle. To Newton, the Principle of Equivalence demanded that the “mass” of any body, namely that property of a body (inertia) that regulates its response to an applied force, be equal to its “weight,” that property that regulates its response to gravitation. Bondi (1957) coined the terms “inertial mass” m1 and “passive gravitational mass” mp, to refer to these quantities, so that Newton's second law and the law of gravitation take the forms

F = m1a, F = mPg

where g is the gravitational field. The Principle of Equivalence can then be stated succinctly: for any body mP = m1

An alternative statement of this principle is that all bodies fall in a gravitational field with the same acceleration regardless of their mass or internal structure. Newton's equivalence principle is now generally referred to as the “Weak Equivalence Principle” (WEP).

It was Einstein who added the key element to WEP that revealed the path to general relativity.

Within general relativity, the structure and motion of relativistic, condensed Objects–neutron stars and black holes–are subjects that have attracted enormous interest in the past two decades. The discovery of pulsars in 1967, and of the x-ray source Cygnus XI in 1971, have turned these “theoretical fantasies” into potentially viable denizens of the astrophysical zoo. However, relatively little attention has been paid to the study of these objects within alternative metric theories of gravity. There are two reasons for this. First, as potential testing grounds for theories of gravitation, the observations of neutron stars and black holes are generally thought to be weak, because of the large uncertainties in the nongravitational physics that is inextricably intertwined with the gravitational effects in the structure and interactions of such bodies. Examples are uncertainties in the equation of state for matter at neutronstar densities, and uncertainties in the detailed mechanisms for x-ray emission from the neighborhood of black holes. Second, compared with the simplicity of the post-Newtonian limits of alternative theories and the consequent availability of a PPN formalism, the equations for neutronstar structure and black hole structure are so complex in many theories, and so different from theory to theory, that no systematic study has been possible.

Neutron stars were first suggested as theoretical possibilities within general relativity in the 1930s (Baade and Zwicky, 1934). They are highly condensed stars where gravitational forces are sufficiently strong to crush atomic electrons together with the nuclear protons to form neutrons, raise the density of matter above nuclear density (ρ ∼ 3 x 1014 g cm-3), and cause the neutrons to be quantum-mechanically degenerate. A typical neutron-star model has m ≃ 1m☉, R ≃10 km.

There remains a number of tests of post-Newtonian gravitational effects that do not fit into either of the two categories, classical tests or tests of SEP. These include the gyroscope experiment (Section 9.1), laboratory experiments (Section 9.2), and tests of post-Newtonian conservation laws (Section 9.3). Some of these experiments provide limits on PPN parameters, in particular the conservation-law parameters ζ1, ζ>2, ζ3, ζ4, that were not constrained (or that were constrained only indirectly) by the classical tests and by tests of SEP. Such experiments provide new information about the nature of post-Newtonian gravity. Others, however, such as the gyroscope experiment and some laboratory experiments, all yet to be performed, determine values for PPN parameters already constrained by the experiments discussed in Chapters 7 and 8. In some cases, the prior constraints on the parameters are tighter than the best limit these experiments could hope to achieve. Nevertheless, it is important to carry out such experiments, for the following reasons:

(i) They provide independent, though potentially weaker, checks of the values of the PPN parameters, and thereby independent tests of gravitation theory. They are independent in the sense that the physical mechanism responsible for the effect being measured may be completely different than the mechanism that led to the prior limit on the PPN parameters. An example is the gyroscope test of the Lense–Thirring effect, the dragging of inertial frames produced purely by the rotation of the Earth. It is not a preferred-frame effect, yet it depends upon the parameter α1.

On September 14, 1959, 12 days after passing through her point of closest approach to the Earth, the planet Venus was bombarded by pulses of radio waves sent from Earth. Anxious scientists at Lincoln Laboratories in Massachusetts waited to detect the echo of the reflected waves. To their initial disappointment, neither the data from this day, nor from any of the days during that month-long observation, showed any detectable echo near inferior conjunction of Venus. However, a later, improved reanalysis of the data showed a bona fide echo in the data from one day: September 14. Thus occurred the first recorded radar echo from a planet.

On March 9, 1960, the editorial office of Physical Review Letters received a paper by R. V. Pound and G. A. Rebka, Jr., entitled “Apparent Weight of Photons”. The paper reported the first successful laboratory measurement of the gravitational red shift of light. The paper was accepted and published in the April 1 issue.

In June, 1960, there appeared in volume 10 of the Annals of physics a paper on “A Spinor Approach to General Relativity” by Roger Penrose. It outlined a streamlined calculus for general relativity based upon “spinors” rather than upon tensors.

Later that summer, Carl H. Brans, a young Princeton graduate student working with Robert H. Dicke, began putting the finishing touches on his Ph.D. thesis, entitled “Mach's Principle and a Varying Gravitational Constant”.

Since the publication of the first edition of this book in 1981, experimental gravitation has continued to be an active and challenging field. However, in some sense, the field has entered what might be termed an Era of Opportunism. Many of the remaining interesting predictions of general relativity are extremely small effects and difficult to check, in some cases requiring further technological development to bring them into detectable range. The sense of a systematic assault on the predictions of general relativity that characterized the “decades for testing relativity” has been supplanted to some extent by an opportunistic approach in which novel and unexpected (and sometimes inexpensive) tests of gravity have arisen from new theoretical ideas or experimental techniques, often from unlikely sources. Examples include the use of laser-cooled atom and ion traps to perform ultra-precise tests of special relativity, and the startling proposal of a “fifth” force, which led to a host of new tests of gravity at short ranges. Several major ongoing efforts continued nonetheless, including the Stanford Gyroscope experiment, analysis of data from the Binary Pulsar, and the program to develop sensitive detectors for gravitational radiation observatories.

For this edition I have added chapter 14, which presents a brief update of the past decade of testing relativity. This work was supported in part by the National Science Foundation (PHY 89-22140).

The overwhelming empirical evidence supporting the Einstein Equivalence Principle, discussed in the previous chapter, has convinced many theorists that only metric theories of gravity have a hope of being completely viable. Even the most carefully formulated nonmetric theory – the Belinfante – Swihart theory – was found to be in conflict with the Moscow Eötvös experiment. Therefore, here, and for the remainder of this book, we shall turn our attention exclusively to metric theories of gravity.

In Section 3.1, we review the concept of universal coupling, first defined in Section 2.5. Armed with EEP and universal coupling, we then develop, in Section 3.2, the mathematical equations that describe the behavior of matter and nongravitational fields in curved spacetime. Every metric theory of gravity possesses these equations.

Metric theories of gravity differ from each other in the number and type of additional gravitational fields they introduce and in the field equations that determine their structure and evolution; nevertheless, the only field that couples directly to matter is the metric itself. In Section 3.3, we discuss general features of metric theories of gravity, and present an additional principle, the Strong Equivalence Principle that is useful for classifying theories and for analyzing experiments.

Universal Coupling

The validity of the Einstein Equivalence Principle requires that every nongravitational field or particle should couple to the same symmetric, second rank tensor field of signature –2.

We now breathe some life into the PPN formalism by presenting a chapter full of metric theories of gravity and their post-Newtonian limits. This chapter will illustrate an important application of the PPN formalism, that of comparing and classifying theories of gravity. We begin in Section 5.1 with a discussion of the general method of calculating post-Newtonian limits of metric theories of gravity. The theories to be discussed in this chapter are divided into three classes. The first class is that of purely dynamical theories (see Section 3.3). These include general relativity in Section 5.2; scalar–tensor theories, of which the Brans–Dicke theory is a special case in Section 5.3; and vector–tensor theories in Section 5.4. The second class is that of theories with prior geometry. These include bimetric theories in Section 5.5; and “stratified” theories in Section 5.6. The theories described in detail in these five sections are those of which we are aware that have a reasonable chance of agreeing with present solar system experiments, to be described in Chapters 7, 8, and 9. Table 5.1 presents the PPN parameter values for the theories described in these five sections. The third class of theories includes those that, while perhaps thought once to have been viable, are in serious violation of one or more solar system tests. These will be described briefly in Section 5.7.

With the PPN formalism and its associated equations of motion in hand, we are now ready to confront the gravitation theories discussed in Chapter 5 with the results of solar system experiments. In this chapter, we focus on the three “classical” tests of relativistic gravity, consisting of (i) the deflection of light, (ii) the time delay of light, and (iii) the perihelion shift of Mercury.

This usage of the term “classical” tests is a break with tradition. Traditionally, the term “classical tests” has referred to the gravitational redshift experiment, the deflection of light, and the perihelion shift of Mercury. The reason is largely historical. These were among the first observable effects of general relativity to be computed by Einstein. However, in Chapter 2 we saw that the gravitational red-shift experiment is really not a test of general relativity, rather it is a test of the Einstein Equivalence Principle, upon which general relativity and every other metric theory of gravity are founded. Put differently, every metric theory of gravity automatically predicts the same red-shift. For this reason, we have dropped the red-shift experiment as a “classical” test (that is not to deny its importance, of course, as our discussion in Chapter 2 points out). However, we can immediately replace it with an experiment that is as important as the other two, the time delay of light.

Since the discovery by Hubble and Slipher in the 1920s of the recession of distant galaxies and the inferred expansion of the universe, cosmology has been a testing ground for gravitational theory. That discovery was thought at the time to be a great confirmation of general relativity for two reasons. First, general relativity, in its original form, predicted a dynamical universe that necessarily either expands or contracts. Of course, Einstein had later modified the theory by introducing the “cosmological constant” into the field equations in order to obtain static cosmological solutions in accord with the current, pre-Hubble observations. To his great joy, following Hubble's discovery, Einstein was allowed to drop the cosmological constant.

Second, was simply the fact that general relativity was capable of dealing with the structure and evolution of the universe as a whole, a capability not shared by Newtonian theory (unless special assumptions are made). However, this capability is more a consequence of the Einstein Equivalence Principle (alternatively of the metric-theory postulates) than a property of general relativity. Because of EEP, spacetime is endowed with a metric g which determines the results of observations made using nongravitational equipment (light rays, telescopes, spectrometers, etc.) and the motion of test bodies (galaxies). Via the field equations provided by each metric theory of gravity, the distribution of matter then determines the metric g, and thereby the entire physical spacetime in which observations are made.

Introduction

There will be no attempt in this book to describe the detailed physical properties of the interstellar medium (gas, dust, cosmic rays, magnetic field) in our own and other galaxies; that would require a whole book to itself. It is, however, necessary and desirable to discuss some of its properties for several reasons. In the first place I believe that galaxies were initially composed of gas alone, so that the process of galactic evolution is largely a process of conversion of gas into stars with the subsequent evolution of the stars. In the second place, if even five per cent of the visible mass of a galaxy is in the form of gas at the present time as we have seen to be true for our Galaxy, it is not really possible to discuss the structure of the galaxy whilst ignoring the gas. The gas content of galaxies will be discussed in this and the following two chapters.

Here I discuss the present structure of the gas disk in our Galaxy, with some comments about other galaxies and with some remarks about the possible past behaviour of the disk. In Chapter 7 I shall discuss the chemical evolution of galaxies, about which information is obtained by a study of the chemical composition of stars of different ages and of the present composition of the interstellar gas.

Introduction: the Hubble classification of galaxies

In the preceding chapter I have discussed in considerable detail the properties of one individual galaxy – the Galaxy. In this chapter I discuss other galaxies (external galaxies) and I compare and contrast their properties with those of the Galaxy. The easiest property of a galaxy to discuss is its visual appearance. Soon after the existence of external galaxies had been established in the early 1920s it was realised that galaxies of regular shape could be divided into two main classes, spiral galaxies and elliptical galaxies. Subsequently it was realised that the spirals should be subdivided into ordinary spirals and barred spirals and that a further class known as lenticular galaxies should be introduced. In addition there were irregulars, galaxies possessing no obvious symmetry. In the 1930s Hubble introduced his classification of galactic types which, with some modifications, is still used today.

The simplest version of the Hubble classification is illustrated in fig. 33. At the time that Hubble introduced the classification, he thought that it might represent an evolutionary sequence with galaxies possibly evolving from elliptical to spiral form but, as we shall see later, that is not believed to be true today. There are alternative classifications of galaxies in use but the Hubble system is essentially adequate for the present book. There is one important group of galaxies which was not known to Hubble because its members are rare and the nearest one is a very large distance from the Galaxy.

Introduction

It is not really possible to consider the formation and early evolution of galaxies without also considering cosmology, that is the structure and evolution of the whole Universe. The reason for this is that, as I have mentioned earlier in Chapter 3, we have no clear evidence for more than one epoch of galaxy formation and that epoch appears to have been shortly after the origin of the Universe, if we believe that the Doppler shifts in the spectra of distant galaxies indicate expansion of the Universe from an initially dense state. I am supposing that the galaxies have formed out of intergalactic (or more accurately pregalactic) gas in much the same ways as stars have formed out of the interstellar gas. However, there is one important difference. Once a galaxy has formed, its distance from other remote galaxies increases because of the expansion of the Universe, but there is no reason for believing that the galaxy itself expands. Thus, whereas star formation can be assumed to take place in a system of gas which is stationary apart from internal motions within a galaxy, the formation of a galaxy takes place against a background of general expansion of the pregalactic gas.

It is important to know when condensations of galactic size were first established, because this influences how much gravitational energy was released in galaxy formation.

This book is in effect a second edition of the book Galaxies: Structure and Evolution published by Wykeham Publications in 1978. When copies of the original edition were exhausted, the publishers were unwilling to reprint it. I am grateful to Dr Simon Mitton of the Cambridge University Press for agreeing to take the book over and for encouraging me to undertake the necessary task of revising the text.

The problem of the structure and evolution of galaxies is central to astronomy. On the one hand a galaxy is composed of stars, whose individual properties are known at least in broad outline. However, the process of star formation, which is crucial to the evolution of galaxies, is not at all well understood. On the other hand galaxies and clusters are the main constituents in the Universe and their properties provide important information about the origin and evolution of the Universe. In addition both the origin and present structure of galaxies are influenced by the possibility that the major form of matter in the Universe is not luminous stars but invisible weakly interacting particles.

In this book I discuss in general terms what is known both about the present structure of galaxies and about their past life history. Most of the detailed discussion refers to our own Galaxy. Although the subject is treated precisely where that is possible it will be apparent that, while the main ideas appear to be well-established, there are very considerable detailed uncertainties.