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.

Introduction

The interstellar medium of our Galaxy contains a widespread component of ionised gas with fairly well-determined average properties. The source of the ionisation has puzzled astrophysicists for many years (e.g. Mathis 1986, Kulkarni and Heiles 1987, Reynolds 1991, 1992, Walterbos 1991, Heiles 1991). There are five major problems which contribute to the mystery. They are the following:

1. (i) The scale height of the ionised gas ∼ 670 pc (Nordgren et al. 1992), whereas the sources usually considered (e.g. ionizing radiation from 0 stars or supernovae) have a much smaller scale height (∼ 100 pc).

2. (ii) The power requirements which the sources must satisfy in order to maintain the ionisation are rather large and probably rule out any conventional source except 0 stars (Reynolds 1990b).

3. (iii) The interstellar medium is normally regarded as being highly opaque to hydrogen-ionising radiation, so that it is not clear how this radiation can travel hundreds of parsecs from the parent 0 stars to produce the diffuse ionised gas (Mathis 1986, Reynolds 1984, 1987, 1992, Heiles 1991).

4. (iv) The same opacity problem arises when one studies in detail (Reynolds 1990a) the ionisation along the line segments to two pulsars with accurately known parallactic distances.

5. (v) The mean electron density in opaque intercloud regions within a few hundred parsecs of the sun is remarkably constant in different directions (Reynolds 1990a, Sciama 1990c). If the opaque gas has a sufficiently tortuous distribution to explain problem (iii), it is surprising that the resultant electron density is so uniform.

In this chapter we shall elaborate on these problems.

Introduction

The intergalactic flux of hydrogen-ionising photons plays a crucial role in the neutrino decay theory described in part 3 of this book. Accordingly in the present chapter we shall consider various observational estimates of this flux, evaluated at different cosmic epochs. While these estimates are rather uncertain, we shall find that they generally exceed the most recent determinations of the integrated contribution from quasars, which is usually regarded as the main source of the intergalactic ionising flux. Various other photon sources at high red shifts have been proposed to fill the gap, and these are discussed at the end of the chapter. In chapter 11 we shall find that the neutrino decay theory can account for the unexplained flux, but only if various parameters both of the theory and of the universe possess highly constrained values. These values are in general agreement with other estimates of them, and in some cases this agreement is rather precise.

The Density of Intergalactic Neutral Hydrogen

As soon as quasars of red shift ∼ 2 were identified by Schmidt (1965), various authors pointed out that they could be used to probe the density of intergalactic neutral hydrogen. This density could then be used in turn to constrain the intergalactic flux of ionising photons. Consider a layer of neutral hydrogen lying at a red shift z along the line of sight to a quasar of greater red shift Z. Photons emitted by the quasar and reaching the layer with the wavelength of Lyman α would be able to excite neutral hydrogen in the layer to its first excited state.

I started writing Modern Cosmology in 1969, just four years after the discovery of the 3 K cosmic microwave background. The significance of that remarkable discovery was rapidly appreciated by cosmologists, and it naturally dominated a large part of my book. Now, nearly a quarter of a century later, a new topic has come to dominate cosmology, namely, the dark matter problem. This problem, however, is not at all well understood. According to modern estimates some of the dark matter is in the form of ordinary particles — protons, neutrons and electrons — while some of it has a more exotic character. We do not know what form the ordinary dark matter takes, and we do not even know the identity of the exotic dark matter. Yet together they are a pervasive and indeed dominating constituent of the universe, in galaxies, groups and clusters of galaxies and intergalactic space. I therefore thought it desirable to update my book by writing a connected account of what has now become the single most important problem in astronomy and cosmology.

I must confess that I have a second reason for writing this book. In 1990 I proposed the idea that most of the widespread ionization of hydrogen observed in our Galaxy is produced by photons emitted by decaying dark matter neutrinos of non-zero rest-mass. The original motivation for this proposal was that the observed ionisation was puzzling astronomers because it seemed difficult to account for in terms of known sources of ionisation.

This theory, once proposed, rapidly took on a life of its own.