When you're in a train, you can't say whether it is really moving or not–that's an everyday experience from which our theory begins.

You can, of course, see the telephone poles flashing by. The naive interpretation of what you see would be that the poles are moving and you are not. Actually, all you can justifiably assert is the relative motion of the poles and yourself. And for further confirmation, think about all those films showing the inside of a railway compartment travelling at high speed – think about how the illusion of motion is actually produced (Remember: bold type means work for you–page 5).

From this sort of experience we've learnt that watching objects around us merely gives information about relative motion. It can never tell us whether they are moving or we are (or both).

Well then, would some test done inside the train tell us whether it is moving or not? For the present let's stick to trains in steady motion-at constant speed on a straight bit of track. I suggest that everything in this moving train happens in exactly the same way as if it were stationary. For example, the steady motion of the train doesn't make any difference to the problem of keeping your balance. You stand upright just as you would in a stationary train. What happens if you drop something?

It falls straight down as you see it, and lands right beside your feetjust as it would do if the train were motionless.

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.

The space–time structure discussed in the next chapter, and assumed through the rest of this book, is that of a manifold with a Lorentz metric and associated affine connection.

In this chapter, we introduce in §2.1 the concept of a manifold and in §2.2 vectors and tensors, which are the natural geometric objects defined on the manifold. A discussion of maps of manifolds in §2.3 leads to the definitions of the induced maps of tensors, and of sub-manifolds. The derivative of the induced maps defined by a vector field gives the Lie derivative defined in §2.4; another differential operation which depends only on the manifold structure is exterior differentiation, also defined in that section. This operation occurs in the generalized form of Stokes' theorem.

An extra structure, the connection, is introduced in §2.5; this defines the covariant derivative and the curvature tensor. The connection is related to the metric on the manifold in §2.6; the curvature tensor is decomposed into the Weyl tensor and Ricci tensor, which are related to each other by the Bianchi identities.

In the rest of the chapter, a number of other topics in differential geometry are discussed. The induced metric and connection on a hypersurface are discussed in §2.7, and the Gauss–Codacci relations are derived. The volume element defined by the metric is introduced in §2.8, and used to prove Gauss' theorem.

In this chapter we consider the effect of space–time curvature on families of timelike and null curves. These could represent flow lines of fluids or the histories of photons. In §4.1 and §4.2 we derive the formulae for the rate of change of vorticity, shear and expansion of such families of curves; the equation for the rate of change of expansion (Raychaudhuri's equation) plays a central role in the proofs of the singularity theorems of chapter 8. In §4.3 we discuss the general inequalities on the energy–momentum tensor which imply that the gravitational effect of matter is always to tend to cause convergence of timelike and of null curves. A consequence of these energy conditions is, as is seen in §4.4, that conjugate or focal points will occur in families of non-rotating timelike or null geodesics in general space–times. In §4.5 it is shown that the existence of conjugate points implies the existence of variations of curves between two points which take a null geodesic into a timelike curve, or a timelike geodesic into a longer timelike curve.

Timelike curves

In chapter 3 we saw that if the metric was static there was a relation between the magnitude of the timelike Killing vector and the Newtonian potential. One was able to tell whether a body was in a gravitational field by whether, if released from rest, it would accelerate with respect to the static frame defined by the Killing vector.

The expansion of the universe is in many ways similar to the collapse of a star, except that the sense of time is reversed. We shall show in this chapter that the conditions of theorems 2 and 3 seem to be satisfied, indicating that there was a singularity at the beginning of the present expansion phase of the universe, and we discuss the implications of space–time singularities.

In §10.1 we show that past-directed closed trapped surfaces exist if the microwave background radiation in the universe has been partially thermalized by scattering, or alternatively if the Copernican assumption holds, i.e. we do not occupy a special position in the universe. In §10.2 we discuss the possible nature of the singularity and the breakdown of physical theory which occurs there.

The expansion of the universe

In §9.1 we showed that many stars would eventually collapse and produce closed trapped surfaces. If one goes to a larger scale, one can view the expansion of the universe as the time reverse of a collapse. Thus one might expect that the conditions of theorem 2 would be satisfied in the reverse direction of time on a cosmological scale, providing that the universe is in some sense sufficiently symmetrical, and contains a sufficient amount of matter to give rise to closed trapped surfaces. We shall give two arguments to show that this indeed seems to be the case. Both arguments are based on the observations of the microwave background, but the assumptions made are rather different.

In this chapter, we shall show that stars of more than about 1½ times the solar mass should collapse when they have exhausted their nuclear fuel. If the initial conditions are not too asymmetric, the conditions of theorem 2 should be satisfied and so there should be a singularity. This singularity is however probably hidden from the view of an external observer who sees only a ‘black hole’ where the star once was. We derive a number of properties of such black holes, and show that they probably settle down finally to a Kerr solution.

In §9.1 we discuss stellar collapse, showing how one would expect a closed trapped surface to form around any sufficiently large spherical star at a late stage in its evolution. In §9.2 we discuss the event horizon which seems likely to form around such a collapsing body. In §9.3 we consider the final stationary state to which the solution outside the horizon settles down. This seems to be likely to be one of the Kerr family of solutions. Assuming that this is the case, one can place certain limits on the amount of energy which can be extracted from such solutions.

For further reading on black holes, see the 1972 Les Houches summer school proceedings, edited by B. S. de Witt, to be published by Gordon and Breach.