This is a corrected version of Chapters I–III of my Mathematical Introduction to Celestial Mechanics (Prentice-Hall, Inc., 1966). The acknowledgements made in the preface to that book apply equally well to this one. In addition, I am especially indebted to Professor D. G. Saari of Northwestern University for his thorough criticism of the original version.

The subject of this book is the structure of space–time on length-scales from 10–13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe. For reasons explained in chapters 1 and 3, we base our treatment on Einstein's General Theory of Relativity. This theory leads to two remarkable predictions about the universe: first, that the final fate of massive stars is to collapse behind an event horizon to form a ‘black hole’ which will contain a singularity; and secondly, that there is a singularity in our past which constitutes, in some sense, a beginning to the universe. Our discussion is principally aimed at developing these two results. They depend primarily on two areas of study: first, the theory of the behaviour of families of timelike and null curves in space–time, and secondly, the study of the nature of the various causal relations in any space–time. We consider these subjects in detail. In addition we develop the theory of the time-development of solutions of Einstein's equations from given initial data. The discussion is supplemented by an examination of global properties of a variety of exact solutions of Einstein's field equations, many of which show some rather unexpected behaviour.

This book is based in part on an Adams Prize Essay by one of us (S. W. H.). Many of the ideas presented here are due to R. Penrose and R. P. Geroch, and we thank them for their help. We would refer our readers to their review articles in the Battelle Rencontres (Penrose (1968)), Midwest Relativity Conference Report (Geroch (1970c)), Varenna Summer School Proceedings (Geroch (1971)), and Pittsburgh Conference Report (Penrose (1972b)).

In order to discuss the occurrence of singularities and the possible breakdown of General Relativity, it is important to have a precise statement of the theory and to indicate to what extent it is unique. We shall therefore present the theory as a number of postulates about a mathematical model for space–time.

In §3.1 we introduce the mathematical model and in §3.2 the first two postulates, local causality and local energy conservation. These postulates are common to both Special and General Relativity, and thus may be regarded as tested by the many experiments that have been performed to check the former. In §3.3 we derive the equations of the matter fields and obtain the energy–momentum tensor from a Lagrangian.

The space–time manifold

The third postulate, the field equations, is given in §3.4. This is not so well established experimentally as the first two postulates, but we shall see that any alternative equations would seem to have one or more undesirable properties, or else require the existence of extra fields which have not yet been detected experimentally.

The mathematical model we shall use for space–time, i.e. the collection of all events, is a pair (ℳ, g) where ℳ is a connected four-dimensional Hausdorff C∞ manifold and g is a Lorentz metric (i.e. a metric of signature + 2) on ℳ.

In this chapter, we use the results of chapters 4 and 6 to establish some basic results about space–time singularities. The astrophysical and cosmological implications of these results are considered in the next chapters.

In §8.1, we discuss the problem of defining singularities in space–time. We adopt b-incompleteness, a generalization of the idea of geodesic incompleteness, as an indication that singular points have been cut out of space–time, and characterize two possible ways in which b-incompleteness can be associated with some form of curvature singularity. In §8.2, four theorems are given which prove the existence of incompleteness under a wide variety of situations. In §8.3 we give Schmidt's construction of the b-boundary which represents the singular points of space–time. In §8.4 we prove that the singularities predicted by at least one of the the theorems cannot be just a discontinuity in the curvature tensor. We also show that there is not only one incomplete geodesic, but a three-parameter family of them. In §8.5 we discuss the situation in which the incomplete curves are totally or partially imprisoned in a compact region of space–time. This is shown to be related to non-Hausdorff behaviour of the b-boundary. We show that in a generic space–time, an observer travelling on one of these incomplete curves would experience infinite curvature forces. We also show that the kind of behaviour which occurs in Taub–NUT space cannot happen if there is some matter present.