The clustering of galaxies on scales < 5h-1Mpc1 shows some remarkable scaling properties which somehow arise out of nonlinear gravitational self-organisation. This scaling is characteristic of structures that are referred to as multifractals. There are several ways of looking at these structures each providing their own special insights into the nature of the clustering. Multifractal scaling can be shown to be closely associated with the fact that galaxy counts-in-cells are approximately Lognormally distributed and with hierarchical fragmentation processes. Moreover, the statistical moments of the galaxy distribution scale in a way that is reminiscent of the renormalization group. This may throw light on the nature of the underlying dynamics of the nonlinear gravitational clustering process.

INTRODUCTION

When Dennis Sciama published his book “The Unity of the Universe” in 1959, the great debate was which theory of the Universe was the correct one: the “Big Bang” or the “Steady State”? Dennis had been a member of a group of Steady-State enthusiasts at Cambridge in the early 1950's.

INTRODUCTION

I first met Dennis Sciama in 1974 whilst I was still an undergraduate. At our first meeting he told me about the challenge of explaining the large scale regularity of the Universe, along with other of its unusual features, like the existence of galaxies and its proximity to a state of “zero binding energy” that we now tend to call “flatness”, without making special assumptions about initial conditions. Many of these issues remain a continuing focus of attention in cosmology. Here, my intention is to review a number of cosmological ‘principles’ and their interaction with a variety of cosmological developments that have taken place over the period during which Dennis has worked on cosmology. The talk on which this article is based formed a small part of these Proceedings which celebrate the huge contribution that Dennis has made and continues to make to general relativity, cosmology and astrophysics. Besides Dennis' personal contributions and those of his students, that of so many of his former students (and their students) exhibits the non-linear amplification in their effectiveness that was always created by the collaborations and contacts between them that have been catalysed by their shared associations with Dennis.

THE PERFECT COSMOLOGICAL PRINCIPLE

In 1948 Bondi, Gold and Hoyle (Bondi and Gold, 1948; Hoyle 1948) proposed a powerful cosmological symmetry principle which they called the ‘Perfect Cosmological Principle’.

INTRODUCTION

The organisers have asked us to review the progress of some aspect of general relativity and cosmology in which they have a particular interest and to introduce their remarks by describing its relation to their interaction with Dennis Sciama. It is a great pleasure for me to do so and also to pay tribute to the inspiration that he, and his style of doing physics, has been to me over the years. In particular I have tried to follow his example by asking simple physical questions and trying to answer them with the simplest appropriate tools available. For that reason in what follows I shall not give extensive mathematical details but refer the reader to the references. Moreover because of the personal nature of the review I have made no attempt to include in those references every paper on the subject, especially where the story is widely known and can be read up in standard textbooks. For the same reason I have perhaps erred in including too many papers of my own.

I became a Research student of Dennis Sciama in October 1969 after being enthralled by his marvelously lucid and exciting Part III lectures on General Relativity. When Dennis left for Oxford a year later I transferred to Stephen Hawking, himself a former student of Dennis.

INTRODUCTION

In this talk I will discuss the hypothesis (Sciama 1990a) that most of the dark matter in the Milky Way consists of tau neutrinos whose decay into photons is mainly responsible for the widespread ionisation of hydrogen in the interstellar medium (outside HII regions). I introduced this hypothesis because there are several difficulties with the conventional explanation of the observed ionisation. This explanation involves photons emitted by O and B stars, supernovae etc. The two most important difficulties involve the large opacity of the interstellar medium to ionising photons and the large scale—height of the free electron density. The opacity arises mainly from the widespread distribution of atomic hydrogen in the interstellar medium, which makes it difficult for the ionising photons emitted by widely separated sources to reach the regions where the ionisation is observed. The scale—height of the electron density (as derived from pulsar dispersion measure data by Reynolds (1991)) is about 1 kpc, whereas the scale height of the conventional sources is only about one tenth of this.

Both of these problems would be immediately solved by my neutrino hypothesis since the neutrinos would be smoothly distributed throughout the interstellar medium and their scale—height would be expected to exceed 1 kpc.

Although Kibble's original toy cosmic string model is characterised by longitudinal Lorentz invariance, it is argued that the tacit assumption that this feature would be preserved in a realistic treatment is rather naive. Strict longitudinal Lorentz invariance is incompatible with equilibrium, but its violation allows closed string loops to survive in centrifugally supported states instead of radiating all their energy away. Following the explicit suggestion by Witten of a superconductivity mechanism whereby such a violation would be achieved, it was pointed out by Davis and Shellard that although the ensuing distribution of centrifugally supported string loops would be cosmologically admissible in a “lightweight” (electroweak transition) string scenario, it would imply a highly excessive cosmological mass density ratio, Ω ≫ 1 in a “heavyweight” (G.U.T. transition) string scenario of the kind postulated to account for galaxy formation. In order to salvage such scenarios, it might be hoped that Witten type superconductivity does not occur, except perhaps as an ephemeral phenomenon subject to decay by quantum tunnelling. However such optimism overlooks the point that the Witten mechanism is just one particularly simple example, and that even if it fails to apply, experience shows that there are many other ways by which Lorentz symmetry breaking in extended material systems is usually achieved.

Galaxies are the building blocks of the Universe, and most of what we know about them has been discovered since Dennis Sciama became a research student. In the space available to me it is not possible to cover even in outline all significant developments during this period. So I have tried to concentrate on what seem to me to be the most important themes. My choice must surely be heavily influenced by personal taste and experience; I hope only that my prejudices are not too glaringly evident.

THE STRUCTURE OF THE MILKY WAY

Galactic astronomy in the 1950s was dominated by the discovery (Ewen & Purcell, 1951) of the 21 cm line predicted by H. C. van der Hulst in 1944. This made it possible for the first time to study the large-scale kinematics of the Milky Way. For the most part the 21 cm observations confirmed the picture of a disk in differential rotation developed by Oort more than twenty years before. However, there were surprises — most notably the discovery that the disk is warped rather than being perfectly flat (Burke, 1957; Kerr, 1957).

Extinction of stars by dust had first betrayed the existence of the interstellar medium (Trumpler 1930).

In this contribution, I review the work of Dennis Sciama and his collaborators on Mach's Principle, saying both what Mach's Principle is, and more generally what we should expect a ‘Principle’ to be and to do. Then I review the notion of an isotropic singularity, and the evidence for a connection between isotropic singularities and Mach's Principle. I suggest that a reasonable formulation of the cosmological part of Mach's Principle is that the initial singularity of space-time is an isotropic singularity, and that Mach's Principle may become a ‘theorem’ of quantum gravity.

WHAT IS MACH'S PRINCIPLE?

Mach's Principle is the name usually given to a loose constellation of ideas according to which “the inertia of a body is due to the presence of all the other matter in the universe” (Milne 1952) and “the local inertial frame is determined by some average of the motion of the distant astronomical objects” (Bondi 1952). In Wheeler's aphorism “matter there governs inertia here” (Misner et al. 1973). The aim of Mach's Principle is to explain, without recourse to Absolute Space, the origin of inertia, inertial frames and the standard of non-rotation in Newtonian Mechanics, where the existence of these things is a basic assumption.

I had the privilege of collaborating with Dennis Sciama for a few years here in Trieste in building up the Astrophysical Sector of SISSA; and I am glad to tell him today that it has been for me an enjoyable and wonderful experience.

Now, first of all, I feel in some sense obliged to justify the subject of my contribution by saying that, at an age over eighty, it becomes much easier making some philosophical reflections about science than bringing some significant scientific consideration; that is why, in order to take part actively to this conference, intended to convey to Dennis all our wishes for further important scientific achievements, I have found myself confined to presenting only some epistemological puzzles. I was told by the organizers that this could be considered as tolerable; so that I have now only to ask for kindly forgiving me such a deviation from the main line of this meeting.

The second thing to do is to clarify what I mean in the title by “reality”. If scientists and philosophers are quite aware of the almost endless meanings that can be given to this word, at different levels of philosophical depth, this is not so for plain people, who generally stick to our immediate feeling that reality is what we perceive through our senses in our surroundings.

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.

The Crisium basin (Figure 5.1) was recognized as a multi-ring structure by Baldwin (1949, 1963) and Hartmann and Kuiper (1962), who were struck by its remarkable elliptical appearance. Although similar to other basins in the morphologic elements of ring massifs, ejecta, and secondary craters, Crisium displays several features that suggest it may have undergone a distinctly different style of post-impact modification. I will describe the regional and basin geology of the Crisium area and address the nature and causes of these morphological differences.

Regional geological setting

The Crisium basin (Figure 1.1) is on the eastern edge of the near side of the Moon, north of Mare Fecunditatis and southeast of Mare Serenitatis. The basin appears to have formed within a zone of typical highlands crust and mare volcanism was active in this region prior to the basin impact (Schultz and Spudis, 1979, 1983). The average thickness of the crust here is about 60 km (Bills and Ferrari, 1976). The interior of the Crisium basin is completely mare-flooded, which has obscured the relations of basin materials; this obscuration has resulted in controversy regarding the true topographic rim of the basin (Howard et al., 1974; Wilhelms, 1980b, 1987; Croft, 1981b), as discussed below.

The Crisium basin (Figure 5.1) appears to have had minimal interaction with older basin structures. The nearest pre-Crisium basin is the Fecunditatis basin, whose center is located approximately 700 km to the south of Crisium. This basin is tangential to the outermost Crisium ring of about 1000 km diameter mapped by Wilhelms and McCauley (1971) and Fecunditatis effects on the generation of Crisium topography have probably been relatively minor.

The formation of multi-ring basins is one of the most important geological processes in the early history of the Solar System. These impacts can greatly affect the morphology and observed surface composition of planetary crusts. The formation of basins may influence lithospheric development and growth and thus alter the thermal history of the planet. Basin-forming impacts can catalyze volcanic eruptions and initiate major modifications of crustal structure subsequent to the development of their multi-ring topography.

In this chapter, I will conclude my examination of the geology of multi-ring basins by speculating on the role of basins in the early geological evolution of the planets. Many of the ideas offered in this chapter are subjects of ongoing research and answers to some of the questions raised by such speculation may be forthcoming with additional work.

The building blocks of planetary surfaces

The recognition of regional patterns of landforms on the Moon led to the discovery of multi-ring basins; this pattern recognition of “the big picture” out of the chaos of detail displayed by the lunar surface is well described by Hartmann (1981). The use of such perception techniques in planetary photogeology has shown us that basins are also present on Mercury, Mars, and the icy satellites of the jovian planets. Moreover, such discovery is not yet complete; ongoing analysis of planetary images adds every year to the basin inventory of the Solar System.

From the evidence described and analyzed in this book, I believe that basins are the fundamental building blocks of early, planetary crusts.

Multi-ring basins are the largest impact craters on Solar System bodies. They form in the earliest stages of planetary history by the collision of asteroid-sized bodies with planets and affect the subsequent evolution of these latter objects in many profound ways. Many scientists have expended great effort in attempting to understand these features; a casual glance at the literature of planetary science over the last 30 years reveals no less than several hundred entries dealing with some facet of multiring basins.

Planetary scientists studying the problem of multi-ring basins approach it from many different directions. Some are physicists, describing the mechanics of basin formation on the basis of known theory. Other workers make geological maps from photographs, searching for clues to the processes that have shaped the surface of the planet. Still others study the chemistry and mineralogy of terrestrial and lunar samples, using the rock record to reconstruct the physical extremes of heat and pressure produced during large impacts. The basin problem is multi-disciplinary; answers to the many questions raised by these features require knowledge from geology, chemistry, physics, and other fields of study. No one person has the expertise to understand all aspects of the basin problem: So why this book?

The only other book available on the problems posed by basins is the proceedings of a topical conference held at the Lunar and Planetary Institute, Houston, in November, 1980 (Multi-ring Basins: Formation and Evolution, P.H. Schultz and R.B. Merrill, editors, Supplement 15 of Geochimica et Cosmochimica Acta, Pergamon Press, New York, 1981).

The Nectaris basin is located on the lunar near side (Figure 1.1), south of Mare Tranquillitatis and west of Mare Fecunditatis. An origin by impact for the Nectaris basin was advocated first by Baldwin (1949, 1963), who paid particular attention to the development of the Altai scarp, the southwestern topographic rim of the basin (Figure 4.1). The basin is relatively well preserved and served as a prototype multi-ring basin in the pioneering study of Hartmann and Kuiper (1962). More recent systematic studies of the Nectaris basin are those of Whitford-Stark (1981b), Wilhelms (1987), and Spudis et al., (1989). The Apollo 16 mission to the Descartes highlands in 1972 collected samples and orbital data that are directly applicable to comprehension of Nectaris regional geology. In this chapter, I describe the geology of the Nectaris basin and synthesize various data into a geological model for its origin and subsequent development.

Regional geology and setting

The Nectaris basin formed in typical crust of the near side highlands and interacted with two older basins. The average thickness of the crust in the region is about 70 km (Bills and Ferrari, 1976). The surrounding terrain consists of heavily cratered highlands, except where buried by later mare basalts. Nectaris deposits are well preserved to the south and west of the basin, but have been buried to the north and east by the lavas of Maria Tranquillitatis and Fecunditatis (Figure 4.1).

Ever since their recognition, multi-ring basins have fascinated and vexed scientists attempting to reconstruct the geological history of the Moon. As the other terrestrial planets were photographed at high resolution, it became apparent that basins are an important element in the early development of all planetary crusts. This importance spurred research into the basin-forming process and yielded a plethora of models and concepts regarding basin origin and evolution. In this chapter, I outline the general problem areas of basin formation and describe the approach taken by my own work on lunar basins.

Multi-ring basins and their significance

Multi-ring basins are large impact craters. The exact size at which impact features cease to be “craters” and become “basins” is not clear; traditionally, craters on the Moon larger than about 300 km have been called basins (Hartmann and Wood, 1971; Wilhelms, 1987). Basins are defined here as naturally occurring, large, complex impact craters that initially possessed multiple-ring morphology. This definition purposely excludes simulated, multi-ring structures that result from explosion-crater experiments on the Earth and whose mechanics of formation differ from impact events (e.g., Roddy, 1977), although important insights into the mechanics of ring formation may be gained from these studies. The qualification that basins initially possessed multiple rings is in recognition of the fact that many older, degraded basins display only one or two rings, even though their diameters of hundreds of kilometers indicate that they had multiple rings when they originally formed.