Introduction

Recent work in the theory of non-linear dynamical systems has centred on the concept of chaos, a term that applies to a great variety of situations and configurations. This relatively new subject is fascinating in its own right, and the rapidly growing body of knowledge surrounding it has uncovered a number of characteristics shared by all chaotic systems. The concept has not only achieved an extensive currency throughout the mathematical and scientific communities but has also captured the interest of many in the nonmathematical world, the latter largely due to an excellent popular discussion of the history and basic ideas by James Gleick (1988)

If the dynamics of the solar magnetic field are due to magnetoconvective dynamo action within the Sun, then the activity cycle is governed by the non-linear equations of magnetohydrodynamics, discussed in Chapter 11. A number of investigators have suggested that solar and stellar activity cycles are chaotic phenomena and have begun to explore the implications of cyclic systems which are chaotic.

If stellar activity cycles are indeed examples of chaotic systems, then they will share in the universal characteristics of such systems. In order to discuss the implications for cyclic activity, a brief outline of the relevant concepts in the theory of chaos is called for.

Introduction

The recognition that magnetic fields are an essential component not only of solar and stellar activity but also of the structure of galaxies, quasars, and pulsars has focussed considerable theoretical interest on the origin and maintenance of cosmic magnetic fields. Since the length scales associated with many cosmic magnetic fields are very large, the ohmic decay times (see §4.1 and below) are long, and there is no difficulty in explaining the continued existence of primordial or fossil fields, such as the megagauss fields found in magnetic A-type stars; but the changes observed to occur in many cosmic magnetic fields, over periods which may be short compared with the decay time, entail an interaction between the plasma motions and the existing fields which may also maintain these fields against ohmic decay. This has become known as dynamo action, and, in order to understand evolutionary changes occurring in the solar magnetic cycle, it is necessary to probe further into the underlying theory.

Parker (1970) drew attention to the curious asymmetry throughout the universe between electric and magnetic charge on the one hand, and the corresponding fields on the other.

Basic data

The historical studies traced in the previous chapter provided an introduction to our knowledge of the structure of the Sun and of cyclic activity. We now offer a brief summary of the general state of our knowledge of the physical properties of the Sun and of solar-type stars, together with some basic theory relevant to an understanding of cyclic phenomena. The interested reader who desires further information is referred to the more general accounts listed in the references (e.g. Mihalas 1978, Foukal 1990, Stix 1989, Zirin 1989).

Stars are generally classified according to their luminosity and surface temperature, a classification scheme which has been codified as the Hertzprung-Russell (H-R) diagram. In this diagram the absolute magnitude (or logarithm of the total luminosity) is plotted against the logarithm of the surface temperature. In the Harvard classification scheme the categories O, B, A, F, G, K, M, R, and S represent decreasing surface temperatures and increasingly complex spectra, and the Sun (of type G2) sits squarely in the middle. It is about 4.5 x 109 years old, less than half the age of the oldest stars in our galaxy, and in these, as in many other respects, it is a very ‘ordinary’ star.

The central aim of this book is the development of the results and techniques needed to determine when it is possible to extend a space-time through an “apparent singularity” (meaning, a boundary-point associated with some sort of incompleteness in the space-time). Having achieved this, we shall obtain a characterisation of a “genuine singularity” as a place where such an extension is not possible. Thus we are proceeding by elimination: rather than embarking on a direct study of genuine singularities, we study extensions in order to rule out all apparent singularities that are not genuine. It will turn out, roughly speaking, that the genuine singularities which then remain are associated either with some sort of topological obstruction to the construction of an extension, or with the unboundedness of the Riemann tensor when its size is measured in a suitable norm.

I had at one stage hoped that there would be a single simple criterion for when such an extension cannot be constructed, which would then lay down once and for all what a genuine singularity is. But it seems that this is not to be had: instead one has a variety of possible tools and concepts for constructing extensions, and when these fail one declares the space-time to be singular on pragmatic grounds. The main such tools are the use of Hölder and Sobolev norms of functions, used for measuring the extent to which the metric or the Riemann tensor is irregular.

Introduction

Although many of our considerations will be purely geometrical, treating space-time as a pseudo-Riemannian manifold and asking whether or not this geometrical structure is breaking down, it must always be remembered that we are really working with a physical theory, governed by particular physical equations for fields and particles, and that it is the breakdown of the physics that is primarily of interest. The breakdown of the geometry is simply one possible manifestation of the breakdown of the physics.

Unfortunately there is a conflict between the mathematical contexts appropriate to, on the one hand, geometry and, on the other hand, physically significant differential equations. In differential geometry one deals with geodesies, domains of dependence and so on. For this to be valid one requires that the connection should satisfy a Lipshitz condition, which ensures the existence of unique geodesies and normal coordinate neighbourhoods. Providing this holds, the differentiability of the metric has little geometrical significance and it is customary to require it to be C∞ for convenience. By contrast, in the study of hyperbolic differential equations (a type to which Einstein's equations belong) questions of differentiability are crucial. The differentiability chosen reflects the character of the solutions allowed: by choosing a low differentiability one admits solutions like shock-waves or impulse-waves which may be very significant; conversely, by choosing too high a level of differentiability one will brand as “singular” shock-wave solutions that from the point of view of fluid dynamics may be entirely legitimate.

In the first chapter we defined a singular space-time as one containing incomplete inextendible curves that could not be continued in any extension of the space time. We must now give the definition (at times already anticipated) of the noun “singularity”. The fundamental idea is that space-time itself (the structure (M, g)) consists entirely of regular points at which g is well behaved, while singularities belong to a set ∂M of additional points – “ideal points” – added onto M. We denote the combined set M ∪ ∂M by ClM, the closure of M, and define the topology of this set to be such that phrases like “a continuous curve in M ending at a singularity p in ∂M”, or “The limit of R as x tends to a singularity p is …” all have meanings corresponding to one's intuitive picture of what they ought to mean.

The construction can be carried out in various ways and the set of ideal points, ∂M, could contain points other than singularities. Two important classes of ideal points that are not singularities are

1. endpoints of incomplete inextendible curves that can be continued in some extension of M (such endpoints being called regular boundary points) and

2. points “at infinity” such as I+.

If the construction is carried out in such a way that ClM consists only of singularities and points of type (1) then ∂M will consist precisely of the endpoints of all incomplete curves.

In this chapter I shall describe various situations in which it is possible to extend through a boundary point; in these cases the boundary point is not a singularity. As has been explained, we are proceding by elimination, so that the remaining cases must either be regarded as genuine singularities, or be amenable to extension by more powerful means than used here. There is no absolute criterion for what sorts of extension are “legitimate”, and hence no absolute criterion for what is, and what is not, a singularity.

Spherical symmetry

In this situation (which is of considerable interest because of the ease of obtaining exact solutions) it is possible to prove the existence of extensions under weaker assumptions than is normally the case. The results are thus not only of interest in their own right, but may be an indication of the “best possible” results that might be obtainable in the general case.

Definition of the problem

We shall be dealing with a space-time in which the rotation group SO(3,ℝ) acts transitively on spacelike 2-surfaces. So through every point p of the space-time there exists in a neighbourhood of p a totally geodesic timelike 2-surface S orthogonal to the orbits of the group; the surfaces maximal with respect to these properties through a given group orbit are equivalent.

In section 6.2.5 we introduced the strong cosmic censorship hypothesis, which implies that singularities other than the big bang would be unobservable. If this were literally true, then it might be thought that the considerations of this book were physically irrelevant. We shall see, however, that the situation is more complex than this. The main thrust of this book has been the attempt to establish a relation between the curvature strength of singularities (in the sense of section 6.1) and their ‘genuineness’ - i.e. whether or not there is an extension through them. The arguments for cosmic censorship suggest that only sufficiently strong singularities might be censored, and so the crucial question becomes, whether or not the censored singularities are precisely the genuine ones. In view of the importance of this to the whole study of singularities, I give here a more extended account of the cosmic censorship hypothesis.

The weak hypothesis

The strong cosmic censorship hypothesis was preceded by the weak cosmic censorship hypothesis, first formulated by Penrose (1969), who asked: “does there exist a ‘cosmic censor’ who forbids the appearance of naked singularities, clothing each one in an absolute event horizon?”

This last term was subsequently given more precision in terms of future null infinity J+. A full account of this would take us well beyond the scope of this book. In outline, however, J+ is a boundary attached to a conformal extension of a given space-time (M,g).

Introduction

In this chapter we introduce the basic idea of our neutrino decay theory. According to this idea (Sciama 1990a) the widespread ionization of the Milky Way is mainly due to photons emitted by dark matter neutrinos pervading the Galaxy. This idea was proposed because it would immediately solve all the problems described in chapter 5, which arise from the conventional hypothesis that the ionisation sources are bright stars or supernovae. In particular, the ubiquity of the neutrinos could compensate for the small mean free path (≲ 1 pc) of the ionising photons in the intercloud medium, and their halo distribution could account for the large scale height (∼ 670 pc) of the ionised gas in the Reynolds layer.

Of course we can exploit these structural features of the basic idea only if the neutrino decay lifetime τ that would be required is otherwise reasonable. We shall find that we need τ ∼ 2 to 3 × 1023 sees. This value is (just) compatible with the lower limits derived in chapter 8, and with certain particle physics theories which are described there. Adopting this lifetime would also have major implications for a large variety of phenomena in astronomy and cosmology other than the ionisation of the Galaxy, and would enable several puzzling problems to be solved.

The most remarkable consequence of the resulting theory is that its domain of validity is highly constrained. As we shall see, it can be correct only if the decay photon energy Eγ, the rest mass mv of the decaying neutrinos, and the Hubble constant H0 each has a value specified with a precision τ 1 per cent.