We have not dealt in this book with all possible experiments on gravitation that have been or could be carried out in the laboratory, whether on the ground or in a space vehicle, but have concentrated on those on which most work has been done and from which most results have been obtained. That is because we have been concerned more with questions of experimental design and technique rather than with the bearing of the results on theories of gravitation. Something was said of that in the Introduction and we simply call attention again to recent reviews such as those of Cook (1987b), Will (1987) and others in the Newton Tercentary review of Hawking & Israel (1987). We have restricted our accounts to the weak principle of equivalence, the inverse square law and the measurement of the constant of gravitation partly because in numbers of results they dominate the subject, but more importantly because, having been so frequently and thoroughly studied, it seems that all the significant issues of experimental method and design are brought out when they are considered.

It was observed in the conclusion of the last chapter on the constant of gravitation, that the definition and calculation of the entire attraction upon a detector such as a torsion balance is no simple matter, and that applies equally to experiments on the inverse square law, as may be shown by the details of the calculations that were necessary in the experiments of Chen et al., (1984).

Introduction

In this chapter attention will be given to the various production and absorption mechanisms operating in the celestial settings. Although the value of the subject is bound up with these two topics – production is by way of a variety of very energetic processes, and the low absorption experienced by γ-rays allows us to ‘see’ regions not otherwise accessible – they are not treated in great detail here. The reason is that the processes are rather well known and have been described in detail by a number of authors. Specifically, the books of Stecker (1971), Chupp (1976) and Hillier (1984) give excellent treatments. Our own descriptions, then, are brief.

Starting with production mechanisms, a summary is given in Figure 1.1. Understandably, the relative importance of the various mechanisms depends on the properties of the production region: gas density, temperature, magnetic field, ambient radiation etc. It is often the determination of these conditions that is the end product of the analysis of the γ-ray observations.

Gamma-ray production mechanisms

Gamma-ray lines

γ-ray lines have been observed from a variety of regions: solar flares, the Galactic Centre, Galactic Plane and the object SS 433, and they exhibit a variety of temporal features, from time independence to rapid time variability.

The lines generated in solar flares are proving to be of considerable interest, but in view of our preoccupation with astronomical regions further afield we refer the reader to Ramaty and Lingenfelter (1981), Ramaty, Lingenfelter and Kozlovsky (1982) and references quoted therein.

The next class of solar system experiments that test relativistic gravitational effects may be called tests of the Strong Equivalence Principle (SEP). That principle states that (i) WEP is valid for self-gravitating bodies as well as for test bodies (GWEP), (ii) the outcome of any local test experiment, gravitational or nongravitational, is independent of the velocity of the freely falling apparatus, and (iii) the outcome of any local test experiment is independent of where and when in the universe it is performed. In Section 3.3, we pointed out that many metric theories of gravity (perhaps all except general relativity) can be expected to violate one or more aspects of SEP. In Chapter 6, working within the PPN framework, we saw explicit evidence of some of these violations: violations of GWEP in the equations of motion for massive self-gravitating bodies [Equations (6.33) and (6.40)]; preferred-frame and preferred-location effects in the locally measured gravitational constant GL [Equation (6.75)]; and nonzero values for the anomalous inertial and passive gravitational mass tensors in the semiconservative case [Equation (6.88)].

This chapter is devoted to the study of some of the observable consequences of such violations of SEP, and to the experiments that test for them. In Section 8.1, we consider violations of GWEP (the Nordtvedt effect), and its primary experimental test, the Lunar Laser-Ranging“Eötvös” experiment. Section 8.2 focuses on the preferred-frame and preferredlocation effects in GL. The most precise tests of these effects are obtained from geophysical measurements.

Our discussion of experimental tests of post-Newtonian gravity in Chapters 7, 8, and 9 led to the conclusion that, within margins of error ranging from 1% to parts in 10-7 (and in one case even smaller), the post-Newtonian limit of any metric theory of gravity must agree with that of general relativity. However, in Chapter 5, we also saw that most currently viable theories of gravity could accommodate these constraints by appropriate adjustments of arbitrary parameters and functions and of cosmological matching parameters. General relativity, of course, agrees with all solar system experiments without such adjustments. Nevertheless, in spite of their great success in ruling out many metric theories of gravity (see Sections 5.7, 8.5), it is obvious that tests of post-Newtonian gravity, whether in the solar system or elsewhere, cannot provide the final answer. Such tests probe only a limited portion, the weak-field slow-motion, or post-Newtonian limit, of the whole space of predictions of gravitational theories. This is underscored by the fact that the theories listed in Chapter 5 whose post-Newtonian limits can be close to, or even coincident with, that of general relativity, are completely different in their formulations, One exception is the Brans–Dicke theory, which for large ω, differs from general relativity only by modifications of O(l/ω) both in the post-Newtonian limit and in the full, exact theory.

We have seen that, despite the possible existence of long-range gravitational fields in addition to the metric in various metric theories of gravity, the postulates of metric theories demand that matter and nongravitational fields be completely oblivious to them. The only gravitational field that enters the equations of motion is the metric g. The role of the other fields that a theory may contain can only be that of helping to generate the spacetime curvature associated with the metric. Matter may create these fields, and they, plus the matter, may generate the metric, but they cannot interact directly with the matter. Matter responds only to the metric.

Consequently, the metric and the equations of motion for matter become the primary theoretical entities, and all that distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric.

The comparison of metric theories of gravity with each other and with experiment becomes particularly simple when one takes the slow-motion, weak-field limit. This approximation, known as the post-Newtonian limit, is sufficiently accurate to encompass all solar system tests that can be performed in the foreseeable future. The post-Newtonian limit is not adequate, however, to discuss gravitational radiation, where the slowmotion assumption no longer holds, or systems with compact objects such as the binary pulsar, where the weak-field assumption is not valid, or cosmology, where completely different assumptions must be made. These issues will be dealt with in later chapters.

In this chapter, we present a brief update of the past decade of testing relativity. Earlier updates to which the reader might refer include “The Confrontation between General Relativity and Experiment: An Update” (Will, 1984), “Experimental Gravitation from Newton's Principia to Einstein's General Relativity” (Will, 1987), “General Relativity at 75: How Right Was Einstein?” (Will, 1990a), and “The Confrontation Between General Relativity and Experiment: a 1992 Update” (Will, 1992a). For a popular review of testing general relativity, see “Was Einstein Right?” (Will, 1986).

The Einstein Equivalence Principle

(a) Tests of EEP

Several recent experiments that constitute tests of the Weak Equivalence Principle (WEP) were carried out primarily to search for a “fifth-force” (Section 14.5). In the “free-fall Galileo experiment” performed at the University of Colorado (Niebauer, McHugh and Faller, 1987), the relative free-fall acceleration of two bodies made of uranium and copper was measured using a laser interferometric technique. The “Eöt–Wash” experiment (Heckel et al., 1989; Adelberger, Stubbs et al., 1990) carried out at the University of Washington used a sophisticated torsion balance tray to compare the accelerations of beryllium and copper. The resulting upper limits on η[Equation (2.3)] from these and earlier tests of WEP are summarized in Figure 14.1

Dramatically improved “ mass isotropy” tests of Local Lorentz Invariance (LLI) (Section 2.4(b)) have been carried out recently using lasercooled trapped atom techniques (Prestage et al., 1985; Lamoreaux et al., 1986; Chupp et al., 1989).

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.