INTRODUCTION

Those of us who had the privilege of being Dennis Sciama's students during what Hajicek has described as ‘the Golden Age of General Relativity’ can trace many of the current concerns of the subject back to the ideas which he fostered, either directly or indirectly, within his research group in Cambridge. This was the environment in which major contributions to most of the foundational ideas about singularities: from the controversies about the steady state and big bang theories; through the critique of the early Lifshitz-Khalatnikov arguments which at first suggested that the big bang singularity was not generic, leading to definitions of just what constituted a singularity; to the Hawking-Penrose singularity theorems themselves. The issue of cosmic censorship stemmed naturally from this work, and illustrates well the combination of rigorous mathematics with a firm hold on physical relevance which he established at that time. In this talk I shall try to give an outline of the historical work on cosmic censorship, focussing at the end on my own recent work on shell crossing singularities. I shall not be concerned with what George Ellis, in this meeting, has termed the position of the goal posts — the details of exactly what the target is; rather, I shall be arguing that we should in fact be playing a different game.

A fully covariant approach to transfer phenomena by using flux-limiters is presented. Explicit formulas for the radiation flux and radiation stress tensor are given for a wide class of physical situations.

INTRODUCTION

In several areas of cosmology and astrophysics the transfer of radiation through high-speed moving media plays a crucial role (accretion flow into black holes, X-ray bursts on a neutron star, supernova collapse, jets in radio sources, galaxy formation, phase transition in the early universe). If one wants to take into account all the effects associated with these transport processes, the full relativistic transport equation must be used.

Early discussion of radiative viscosity was performed by several authors in a non covariant formulation (Jeans 1925, Rosseland 1926, Vogt 1928, Milne 1929), but the appropriate transfer equation for the case of special relativity was given in a classical paper by Thomas (1930). A manifestly covariant form of the transfer equation was obtained by Hazelhurst and Sargent (1959), by using a geometrical formalism. Finally Lindquist (1966) performed the extension to the general relativistic situation and Mihalas (1983) analyzed in depth the order of magnitude of the various terms which appear in the transport equation.

From a mathematical point of view, the transport equation is an integro—differential equation and the task to solve it is in general very hard.

From the time that Newton first proposed that there was a universal force of gravity inversely proportional to the square of the distance between two point masses, there have been recurrent investigations of how far that rule was correct, and many different alternative forms have been suggested. The other assumption that Newton made, that the force of gravity did not depend on the chemical composition of bodies, has also been questioned from time to time; Newton himself carried out the first experimental test of what has become known as the weak principle of equivalence. It has often been suggested that some apparently anomalous behaviour in celestial mechanics should be ascribed to a failure of the inverse square law; indeed Clairaut developed the first analytical theory of the motion of the Moon because of discrepancies between Newton's theory and observation that might have been due to an inverse-cube component of the force. As with all subsequent studies before general relativity, careful analysis showed that the effects were consistent with the inverse square law. General relativity predicts a small deviation from the inverse square law close to very massive bodies, a deviation that has been confirmed by careful observation.

The motions of celestial bodies about each other are, with very minor exceptions, unable to reveal any departure from the weak principle of equivalence; if such departures are to be detected, they must be sought in laboratory experiments or geophysical observations.

Three hundred years after Newton published Philosophiae Naturalis Principia Mathematica the subject of gravitation is as lively a subject for theoretical and experimental study as ever it has been (Hawking & Israel, 1987). Theorists endeavour to relate gravity to quantum mechanics and to develop theories that will unify the description of gravity with that of all other physical forces. Experimenters have looked for gravitational radiation, for anomalies in the motion of the Moon that would correspond to a failure of the gravitational weak principle of equivalence, for deviations from the inverse square law and for various other effects that would be inconsistent with general relativity. The cosmological implications of general relativity continue to be elaborated and various ways of using space vehicles to test notions of gravitation have been proposed. In particular, the last three decades have seen a considerable effort devoted to applying modern techniques of measurement and detection of small forces to experiments on gravitation that can be done within an ordinary physics laboratory, and it is those that are the subject of this book.

Our scope is indeed quite restricted. It is concerned with experiments where the conditions are under the experimenter's control, in contrast to observation, where they are not. It is concerned with experiments that can be done within a more or less ordinary-sized room, that is to say, the distances between attracting body and attracted body do not usually exceed a few metres and may often be much less, while the masses of gravitating bodies are of the order of kilograms or much less.

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

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).

The launch of Sputnik 1 on 4 October 1957 was a traumatic event for the USA and much of the western world. For years there had been an unspoken assumption that the Russians were dark and backward people, and that all new initiatives in science and technology occurred, almost as a natural law, in ‘the West’. Disbelief was widespread. ‘What I say is truth, and truth is what I say’, that popular saying of the 1980s, had its adherents in the 1950s too, and they assured the world that Sputnik 1 was just propaganda and was not really in orbit at all.

My view of the event was different. For several years we had been showing in theory how ballistic rockets could be turned into satellite launchers by adding a small upper stage to produce the necessary extra velocity. The USSR had launched an intercontinental rocket in August 1957, and little extra velocity would be needed to attain orbit. So it would be quite easy for the USSR to launch a small satellite like Sputnik 1, which was a sphere 58 cm in diameter of mass 84 kg with four long aerials (Fig. 2.1). The real surprise was the final-stage rocket that accompanied Sputnik 1 into orbit. The rocket appeared much brighter than the pole star as it crossed the night sky, and seemed likely to be at least 20 m long, far larger than anything contemplated in our paper-studies of satellites: the final-stage rocket for our reconnaissance satellite was less than 5 m long.

It was in 1953 that the metamorphosis of missiles into satellites began. One important new start was the prospect of rockets for upper-atmosphere research. The impetus came from a group of scientists belonging to the Royal Society's Gassiot Committee, particularly Professor Harrie Massey of University College London, and Professor David Bates of the Queen's University, Belfast. The existence of the Gassiot Committee was an extraordinary stroke of luck for space science, as I came to realize much later. The Royal Society covers all science, and until 1935 the one exception to this rule was the Gassiot Committee, the Society's only specialized ‘inhouse’ committee: it had been formed in 1871, to oversee Kew Observatory, and was expanded during the Second World War to cover atmospheric physics in general. The Gassiot Committee was vitally important for two reasons: first, it was a preconstructed official pathway into space; second, the Royal Society was fully committed from the outset, thus making respectable a subject dismissed by many as ‘utter bilge’.

The Gassiot Committee organized an Anglo-American conference on rocket exploration of the upper atmosphere, at Oxford in August 1953, and this can now be seen as the first British step on the ladder into space which we climbed for nearly twenty years. I cannot remember much about the meeting, except that it was held in a dark medieval lecture-room, lit by a few light bulbs with dusty white shades: it seemed paradoxical that these new ventures into space were being planned in such antiquated surroundings.

A rocket fired up the north face of the Eiger towards the summit might serve as a suitable simile for the worldly aspects of my career in science. From 1957 until about 1970 the upward thrust was strong, and the rocket seemed on course for the stratosphere. During the 1970s the propellant seemed to burn out and the momentum decreased. About 1980 the rocket came to rest on a rather precarious shelf, halfway up the cliff: there was a danger of being pushed off into free fall; on the other hand, the position was a commanding one, from which good work might be done. As it turned out, the danger was averted and the decade was most productive.

In 1980 the researches based on orbit analysis seemed to be in good health. The Earth Satellite Research Unit at Aston University, under Dr Brookes, had moved to a spacious modern building at St Peter's College, Saltley, and the prediction service was transferred from the Appleton Laboratory to ESRU in July, because the Appleton Laboratory was being moved and merged with the Rutherford Laboratory. (Pierre Neirinck retired from Appleton but continued as a keen analyst of satellites.) In September 1980, when a meeting of visual observers was held at St Peter's College, ESRU was thriving, with four staff members working on predictions, four more as Hewitt camera observers, and a strong research team that included Philip Moore and two recently-appointed Research Fellows, Graham Swinerd and Bill Boulton, both working on orbit analysis and popularly known as the heavenly twins.