Introduction

Thermal noise is unavoidable and sets the fundamental limit to the detectability of the response of an oscillator to any gravitational effect, but it is not the only disturbance to which an oscillator may be subject. Other forces may act on the mass of a torsion pendulum if it is subject to electric or magnetic or extraneous gravitational fields. The point of support of a torsion pendulum or other mechanical oscillator may be disturbed by ground motion. Ground motion is predominantly translational and so might be thought not to affect a torsion pendulum to a first approximation. However, all practical oscillators have parasitic modes of oscillation besides the dominant one, and although in linear theory normal modes are independent, in real non-linear systems modes are coupled. Thus, even if in theory seismic ground motion had no component of rotation about a vertical axis, none the less there would be some coupling between the primary rotational mode of a torsion pendulum and its oscillations in a vertical plane. In practice, therefore, any disturbance of a mechanical oscillator may masquerade as a response to a gravitational signal.

External sources of noise can be avoided with proper design of experiments. In this chapter we shall discuss both the sources of external disturbance and also the ways in which oscillators of different design respond.

Ground disturbance

Sources of ground noise

We begin with a discussion of seismic motions that move the point of support of a pendulum.

Introduction

For reasons concerned with the availability of contemporary γ-ray data, the lower limit for ‘medium energy’ quanta can be taken as 35 MeV (this is the lower limit for the important SAS II satellite experiment). The upper limit again comes from satellite data availability and is rather arbitrarily taken as 5000 MeV, the upper limit of the highest COS B satellite energy band; in fact, the photon flux falls off with energy so rapidly that our knowledge about γ-rays above 1000 MeV from satellite experiments is virtually nil. As will be discussed in Chapter 5, however, knowledge blooms again above 1011 eV, where Cerenkov radiation produced by γ-ray-induced electrons in the atmosphere allows detections to be made.

Although there are some who still believe that unresolved discrete sources contribute considerably to the diffuse γ-ray flux, the majority view is that the sources are responsible for only 10−20% of the γ-ray flux and that the predominant fraction arises from cosmic ray (CR) interactions with gas and radiation in the interstellar medium (ISM). In fact, some 30 years ago, both Hayakawa (1952) and Hutchinson (1952) had made estimates of the CR–ISM-induced γ-ray flux and had shown it to be within the scope of experimental measurement.

The foregoing is not to say that the discrete sources are unimportant, indeed the reverse is true, and there is considerable interest in ways of explaining the observed γ-ray flux from identified sources (the Crab and Vela pulsars) and the unidentified but definite sources such as Geminga (2CG 195 + 04 in the COS B source catalogue of Hermsen 1980, 1981).

Introduction

The essence of the principle of equivalence goes back to Galileo and Newton who asserted that the weight of a body, the force acting on it in a gravitational field, was proportional to its mass, the quantity of matter in it, irrespective of its constitution. This is usually known as the weak principle of equivalence and is the cornerstone of Newtonian gravitational theory and the necessary condition for many other theories of gravitation including the theory of general relativity. In recent times, however, it was found that the weak principle of equivalence was not sufficient to support all theories and the principle has been extended as (1) Einstein's principle of equivalence and (2) the strong principle of equivalence.

Following a brief discussion of the principle of equivalence, this chapter is devoted to an account of the principal experimental studies of the weak principle of equivalence.

Einstein's principle of equivalence

Gravitation is one of the three fundamental interactions in nature and a question at the heart of the understanding of gravitation is whether or how other fundamental physical forces change in the presence of a gravitational force.

Einstein answered this fundamental question with the assertion that in a non-spinning laboratory falling freely in a gravitational field, the non-gravitational laws of physics do not change. That means that the other two fundamental interactions of physics – the electro-weak force and the strong force between nucleons – all couple in the same way with a gravitational interaction, namely: in a freely falling laboratory, the non-gravitational laws of physics are Lorentz invariant as in the theory of special relativity.

Introduction

In tests of the weak principle of equivalence, exact calculations of the attractions of masses are not necessary, but they are essential in experiments to test the inverse square law and to measure the gravitational constant. In fact, the calculation of the gravitational attraction of laboratory masses is usually not at all simple, because the dimensions of the masses are comparable with the separations between them, so that neither the test mass nor the attracting mass can be treated as a point object. In the following sections we discuss the gravitational attractions of laboratory masses with various common geometrical shapes. We present the results in terms of the gravitational efficiency, that is, the ratio of the gravitational attraction of a laboratory mass at a certain separation to that of a point mass with the same mass and separation. Furthermore, the precision demanded in measurements of separations of masses, the most difficult measurements in the determination of G and the test of gravitational law, depends on the geometry of the masses. These effects can have a strong influence on the conduct and final results of an experiment and it is essential to discuss in detail the calculation of potentials and attractions before going on to describe experiments.

Masses of three forms are often used in the laboratory: spheres, cylinders and rectangular prisms. The formula for the gravitational attraction of a sphere is well known and simple, but in practice it is not possible to manufacture an ideal sphere, the practical problem is usually how the real precision of manufacture affects the results; cylinders and prisms can be made very precisely but calculating the attraction is difficult.

Introduction

Although the weak principle of equivalence has been verified for ordinary macroscopic matter to very high precision, two questions remain open:

1. Is the principle valid for antimatter? Although indirect evidence from virtual antimatter in nuclei and short-lived antiparticles suggests that antimatter may have normal gravitational properties, no direct tests of the validity of the weak principle of equivalence for antimatter have been made.

2. Is the principle valid for microparticles? As the test bodies in macroscopic experiments are formed of neutrons, protons and electrons bound in nuclei, there is no doubt about the validity of the weak principle of equivalence for bound particles. However, the possibility of the principle of equivalence being violated for free particles should be studied.

Two main features characterize laboratory tests of the weak principle of equivalence for free elementary particles, both the consequence of their small masses. (1) When forces on substantial masses of bulk material are compared, a null experiment based on comparing different test bodies of two kinds of material can be devised. That is not possible for microscopic particles, and the gravitational accelerations have to be measured directly and subsequently compared with the acceleration of ordinary bulk matter to obtain the Eötvös coefficient. (2) The gravitational forces are very weak, even in the field of the Earth (which is the strongest attractive field), and so the accuracy of any experiment is very poor compared with Eötvös-type experiments using bulk masses.

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).