INTRODUCTION

It is widely though erroneously believed that one can see the Milky Way Galaxy. In fact, one's image of the Milky Way depends more on how one looks at it than on what is available to be seen. For reasons which are related to population biology more than to astrophysics, our eyes are optimised to detect the peak energy output from thermal sources with a surface temperature near 6000K. Thus, unless such an object is typical of the entire contents of the Galaxy, there is no reason why we should be able to see by eye a representative part of whatever may be out there. If we had X-ray or UV sensitive eyes we would ‘see’ only hotter objects, if infrared or microwave eyes only cooler objects.

No single section of the electro-magnetic spectrum provides the ‘best’ view of the Galaxy. Rather, all views are complementary. However, some views are certainly more representative than are others. The most fundamental must be a view of the entire contents of the Galaxy. Such a view would require access to a universal property of matter, which was independent of the state of that matter. This is provided by gravity, since all matter, by definition, has mass. Mass generates the gravitational potential, which in turns defines the size and the shape of the Galaxy. While the most reliable and comprehensive, such a view is also the hardest to derive. Nonetheless, we will repeatedly return to the gravitational picture of the Galaxy in these lectures.

INTRODUCTION

The solution of model Z has been found in many cases with account taken of both viscous and electromagnetic core-mantle coupling (Braginsky 1978; Braginsky & Roberts 1987; Braginsky 1988; Braginsky 1989; Cupal & Hejda 1989). Apart from Braginsky (1989), the time evolution of the solution was used simply as an aid to obtain the steady-state solution. Cupal & Hejda (1992) found numerically a transient solution of model Z having the form of a decaying oscillation. The accuracy of such solutions depends on the numerical method used, on the density of space and time discretization, and for that matter, on the character of the solution itself. An important question is which characteristics of the time behaviour of the solution reflect the real (physical) behaviour of the system and which follow from the limitations of the numerical method.

MOTIVATION

In the absence, so far, of fully hydrodynamic dynamo models representative of the Earth or the planets, the main focus of planetary dynamo theory remains with (axisymmetric) mean-field dynamo models in which the contribution from the nonlinear interaction of the non-axisymmetric components of the problem are parameterized through a prescribed α-effect (see for example Roberts 1993).

MOTIVATION

The interaction between convection and magnetic fields plays a central role in the theory of stellar dynamos. In order to investigate this interaction in detail, we consider a simplified problem: two-dimensional convection in a vertical magnetic field. To represent the astrophysical situation, in which there are no sidewalls, we consider a box with periodic boundary conditions in the horizontal direction, allowing horizontal flows. It is found that convection can be unstable to a horizontal shearing motion.

INTRODUCTION

The study of passive scalar advection provides fundamental understanding of various phenomena such as convection and mixing that are ubiquitous in nature. In particular, the probability distribution of amplitude and its spatial gradients are of vital importance in relation to recent active studies of non-Gaussian probability density functions (PDFs) endemic in turbulence.

Since Castaing et al. (1989) reported exponential-like tails on the PDF of temperature fluctuations in thermal convection at very high Rayleigh numbers, there has been increasing interest in the mechanism of the non-Gaussian tails on PDFs of amplitudes. In a recent paper, Pumir, Shraiman & Siggia (1991) have suggested that the non-Gaussian tails for an advected passive temperature field may be induced by the presence of a mean-temperature profile. A simple physical mechanism for this is proposed in the present paper. The resultant non-Gaussian statistics will be shown by numerical simulations and theoretical analysis for a transport equation without molecular diffusion. In this paper, the result on PDFs is summarized; other details will be presented elsewhere (Kimura & Kraichnan 1993).

This volume contains papers contributed to the NATO Advanced Study Institute ‘Theory of Solar and Planetary Dynamos’ held at the Isaac Newton Institute for Mathematical Sciences in Cambridge from September 20 to October 2 1992. Its companion volume ‘Lectures on Solar and Planetary Dynamos’, containing the texts of the invited lectures presented at the meeting, will appear almost contemporaneously. It is a measure of the recent growth of the subject that one volume has proved insufficient to contain all the material presented at the meeting: indeed, dynamo theory now acts as an interface between such diverse areas of mathematical interest as bifurcation theory, Hamiltonian mechanics, turbulence theory, large-scale computational fluid dynamics and asymptotic methods, as well as providing a forum for the interchange of ideas between astrophysicists, geophysicists and those concerned with the industrial applications of magnetohydrodynamics.

The papers included have all been refereed as though for publication in a scientific journal, and the Editors are most grateful to the referees for helping to get all the papers ready in such a short time. They also wish on behalf of the Scientific Organising Committee to record their appreciation of the dedication of the staff of the Isaac Newton Institute, who coped cheerfully with many bureaucratic complexities, and to give special thanks to the Deputy Director, Peter Goddard, for making the whole meeting possible.

INTRODUCTION

Given the existence of a naturally occurring magnetic field, be it astrophysical or geophysical, it is natural to ask whether the field is generated by dynamo action or if instead it is a fossil field, trapped in the body since its formation. In certain contexts it is possible to give a definitive answer. For example, the Ohmic diffusion time of the Earth's core is of the order of 10 years whereas paleomagnetic records show that the magnetic field of the Earth has existed for 109 years. Consequently, since the field has been maintained for so many Ohmic decay times it must be generated by some sort of dynamo process. For astrophysical bodies on the other hand, for which typically the Ohmic time is comparable to the lifetime of the body itself, it is not so straightforward to assert that a field is dynamo-generated. Of course, there may be other factors suggesting the origin of the field, but simply on the basis of the Ohmic decay time the issue often cannot be decided. What we would like therefore is a test to distinguish between these two possibilities.

INTRODUCTION

My first impressions of Dennis Sciama came from a short introductory astrophysics course he gave to undergraduates in 1964. Then in 1966-7 I took his Cambridge Part III course in relativity, in which he charitably ignored my inadvertent use of Euclidean signature in the examination (an error I spotted just at the very end of the allowed time) and gave me a good mark. In both these courses he showed the qualities of enthusiasm and encouragement of students with which I was to become more familiar later in 1967 when I began as a research student. A project on stellar structure had taught me that I did not want to work on that, and I began under Dennis with the idea of looking at galaxy formation. However, by sharing an office with John Stewart I came to read John's paper with George Ellis (Stewart and Ellis, 1968) and its antecedent (Ellis, 1967) and developed an interest in relativistic cosmological models, which led to George becoming my second supervisor.

I was still in Sciama's group, and I learnt a lot from the tea-table conversations, which seemed to cover all of general relativity and astrophysics. Dennis taught us by example that the field should not be sub—divided into mathematics and physics, or cosmological and galactic and stellar, but that one needed to know about all those things to do really good work.

INTRODUCTION

There has been a tremendous growth in the understanding of General Relativity and of its relation to experiment in the past 30 years, resulting in its transformation from a subject in the doldrums on the periphery of theoretical physics, to a subject with a considerable experimental wing and and many recognised major theoretical achievements to its credit. The main areas of development have been,

1. * solar system tests of gravitational theories,

2. * gravitational radiation theory and detectors,

3. * black holes and gravitational collapse,

4. * cosmology and the dynamics of the early universe.

On the theoretical side, this development is based on understanding exact and inexact solutions of the Field Equations (the latter has three different meanings I will discuss later). In this brief review of theoretical developments, there is not space to give full references to all the original papers. Detailed references can be found in previous surveys, in particular ‘HE’ is Hawking and Ellis (1973), ‘TCE’ is Tipler Clarke and Ellis (1980), ‘HI’ is Hawking and Israel (1987), and ‘GR13’ is the proceedings of the 13th International meeting on General Relativity and Gravitation held in Cordoba, Argentina in 1992. Many of the issues raised here are considered at greater length elsewhere in this book, e.g. in the articles by MacCallum and Tod.

It is a great pleasure to speak at this meeting since it gives me a chance to acknowledge the great influence Dennis Sciama has had on my life. It was Dennis who first introduced me to relativity as an undergraduate at Cambridge in 1968 and it was through a popular lecture he gave to the Cambridge University Astronomical Society in that year that I first learnt about the microwave background radiation. I well recall his remark that he was “wearing sackcloth and ashes” as a result of his previous endorsement of the Steady State theory. This made a great impression on me and was an important factor in my later choosing to do research in Big Bang cosmology. When I was accepted as a PhD student by Stephen Hawking, I was therefore delighted to become Dennis' academic grandson. (Incidentally since Stephen has related how he had originally wanted to do his PhD under Fred Hoyle, having never heard of Dennis, I must confess - with some embarrassment - that, when I applied for a PhD, I had never heard of Stephen!) The subject of my PhD thesis was primordial black holes, so it seems appropriate that I should talk on this topic at this meeting, especially as Dennis was my PhD examiner.

Although I am one of the very few people represented here who was never technically a student of Dennis Sciama's (or a student's student or a student's student's student), I was, on the other hand, very much a student of his in a less formalized sense. He was a close personal friend when I was at Cambridge as a research student, and then a little later as a Research Fellow. Although my Ph.D. topic was in pure mathematics, Dennis took me under his wing, and taught me physics. I recall attending superb lecture courses by Bondi and by Dirac, when I started at Cambridge, which in their different ways were inspirations to me, but it was Dennis Sciama who influenced my development as a physicist far more than any other single individual. Not only did he teach me a great deal of actual physics, but he kept me abreast with everything that was going on and, more importantly, provided the depth of insight and excitement - indeed, passion - that made physics and cosmology into such profoundly worthwhile and thrilling pursuits.

I first encountered Dennis at the Kingswood Restaurant, in Cambridge, somewhat before I went up there as a research student, where I was introduced to him by my brother Oliver.

INTRODUCTION

I started as a research student with Dennis Sciama in 1971 at the beginning of his time at Oxford, after he had transferred there from Cambridge, and was subsequently a post-doc with his groups in Oxford and Trieste. It is a great pleasure to have the opportunity of contributing to this book.

In the renaissance of general relativity and cosmology, which is our subject here, one of the central themes has been the study of relativistic gravitational collapse, black holes and neutron stars. At the beginning of my research work, Dennis emphasized to me the role which was going to be played in this by numerical computing and he pointed me in that direction despite some initial reluctance on my part. Applying general relativity to real problems in the real world is a complicated business but gradually it has entered the mainstream of astrophysics to the extent that it now no longer seems to be an exotic curiosity but has come of age as an equal member of the collection of physical theories which are brought into service in attempting to explain how things work. Computing has played a key role in this, making it possible to move beyond theoretical models which have been simplified to the point where analytical techniques are sufficient for studying them, to the development of more detailed models which probe more deeply into the consequences of the theory and come closer to contact with possible observations.

We review the properties of the cluttered Minkowski vacuum. In particular we discuss the example of a uniformly accelerated quantum oscillator in the Minkowski vacuum showing that it does not radiate. Equivalently, the presence of the oscillator does not lead to decoherence (i.e. the emergence of classical probabilities). Mach's Principle was related originally by Einstein to the non-existence of (classical) vacuum cosmological models. We speculate that Mach's Principle may acquire a quantum role as a condition for decoherence of the universe.

INTRODUCTION

Following Hawking's announcement (Hawking 1974,1975) of his result that black holes radiate a thermal flux, Davies (1975) applied an analogous technique to the spacetime of a uniformly accelerated observer in the Minkowski vacuum in the presence of a reflecting wall. He interpreted the result as a flux of radiation from the wall at a temperature ha/4π2ck, where a is the acceleration of the observer. Unruh (1976) independently showed that the Minkowski vacuum appears as a thermal state to any uniformly accelerated detector, the normal modes of which were defined with respect to its own proper time. There is no flux from the horizon but the detector is raised to an excited state with its levels populated according to a Boltzmann distribution at a temperature ha/4π2ck as it would be in an inertial radiation bath at this temperature.

This is mainly a review of the properties of gravitational galaxy distribution functions. It discusses their theoretical derivation, comparison with N-body simulations, and — perhaps most importantly — their observed features. The observed distribution functions place strong constraints on any theory of galaxy clustering.

INTRODUCTION

The galaxy distribution function f(N, v) is the probability of finding N galaxies in a given size volume of space (or in a projected area of the sky) with velocities between v and v + dv. It is the direct analog of the distribution function in the kinetic theory of gases. For perfect gases, the spatial distribution is provided by a Poisson distribution at low densities and a Gaussian distribution at high densities, along with a Maxwell-Boltzmann distribution for the velocities. It is only in the last few years that we have discovered the comparable distribution for galaxies interacting gravitationally in the expanding universe. There are still many aspects of this problem which need to be understood.

Distribution functions had their origin in the observations and speculations of William Herschel two hundred years ago. In his catalog of nebulae he noticed that their distribution was irregular over the sky. Although we now know that some of these nebulae were galaxies and others resulted from stars, HII regions and planetary nebulae, and that some of the irregularities are intrinsic while others are due to local obscuration by the interstellar matter in our Milky Way, Herschel tended to view them all as a single class of objects.

Quasars offer important clues to the process of galaxy formation and the epoch when it occurred. Although they almost certainly involve relativistic processes close to a collapsed object, quasars have unfortunately not yet given us any real tests of strong-field gravity.

INTRODUCTION

In December 1963, the first Texas Conference on Relativistic Astrophysics was held in Dallas. Quasars had just been discovered, and were already being interpreted as gravitationally-collapsed massive objects. In his after-dinner speech, Thomas Gold said that relativists were “not only magnificent cultural ornaments, but might actually be useful to science …. What a shame it would be if we had to dismiss [them all] again”. We haven't had to do so — on the contrary, ‘relativistic astrophysics’ is a subject with ever-widening scope. It burgeoned with the detection of the microwave background in 1965, of neutron stars in 1967, and of the first stellar-mass black hole candidates in 1971. Dennis Sciama's research group was at the centre of all the key debates throughout that exciting period. I was myself fortunate to begin research in 1964, when these developments were just gaining momentum. It was my great good fortune to have been assigned as one of Dennis' students, and he has been a valued mentor and advisor ever since.

The past 30 years have seen a great revival of General Relativity and Cosmology, and major developments in astrophysics. On the theoretical side this has been centred on the rise of the Hot Big Bang model of cosmology and on our developing understanding of the properties of black holes. On the observational side it has been based on astonishing improvement of detectors and measuring instruments in astronomy and experimental relativity, in particular enabling measurement of the microwave background radiation and extension of astronomical observations to the whole electromagnetic spectrum.

Dennis Sciama has played an important role in these developments, particularly through the research schools he has run at Cambridge, Oxford, and Trieste, supervising and inspiring many research students who have worked on these topics, and challenging his colleagues with penetrating questions about the physics and mathematics involved. The extent of his influence will become apparent on studying the Family Tree of students, and the list of books that have been the product of those who have taken part in these research groups (see below).

Dennis' 65th Birthday was on November 18, 1991. To mark this event, a meeting was held at SISSA, Trieste (Italy) from 13th to 15th April, 1992, under the title The Renaissance of General Relativity and Cosmology: A survey meeting to celebrate the 65th birthday of Dennis Sciama.

INTRODUCTION

Since my first interaction with the Kerr metric, early in 1967, when Dennis Sciama suggested to me that I work on it, I was fascinated by the magic of that solution to reduce whatever mathematical expression to simple terms, and by the richness of the information it provided. After nearly 25 years of intense investigation of the Kerr metric carried out by almost all the relativists around the world, new properties continue to be discussed and perhaps deep information about the very nature of gravity is still to be brought to light.

There are basic questions about gravity which, in my opinion, still need to be answered. Some (and perhaps the most obvious ones) are:

1. i) - Why do the properties of a physical system, like energy and momentum, bend the background geometry?

2. ii) - How are energy and momentum actually transferred to the background geometry, leading to a non zero curvature?

3. iii) - To what extent does energy and momentum of the background geometry contribute to these same properties of a physical system?

Answering these types of question is what I mean by going to the roots of gravity. Evidently, central to this issue is the concept of energy in general, for which we require, at the classical level at least, the fulfillment of the energy conditions.