The subject of this book is the structure of space-time on length-scales from 10~13cm, the radius of an elementary particle, up to 1028cm, 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)), Varcnna Summer School Proceedings (Geroch (1971)), and Pittsburgh Conference Report (Penrose (19726)). We have benefited from discussions and suggestions from many of our colleagues, particularly B. Carter and D. W. Sciama. Our thanks are due to them also.

January 1973 G. F. R. Ellis strong and weak interactions have a very short range (~ 10_13cm or less), and although electromagnetism is a long range interaction, the repulsion of like charges is very nearly balanced, for bodies of macroscopic dimensions, by the attraction of opposite charges. Gravity on the other hand appears to be always attractive. Thus the gravitational fields of all the particles in a body add up to produce a field which, for sufficiently large bodies, dominates over all other forces.

Not only is gravity the dominant force on a large scale, but it is a force which affects every particle in the same way. This universality was first recognized by Galileo, who found that any two bodies fell with the same velocity. This has been verified to very high precision in more recent experiments by Eotvos, and by Dicke and his collaborators (Dicke (1964)). It has also been observed that light is deflected by gravitational fields. Since it is thought that no signals can travel faster than light, this means that gravity determines the causal structure of the universe, i.e. it determines which events of space-time can be causally related to each other.

These properties of gravity lead to severe problems, for if a sufficiently large amount of matter were concentrated in some region, it could deflect light going out from the region so much that it was in fact dragged back inwards. This was recognized in 1798 by Laplace, who pointed out that a body of about the same density as the sun but 250 times its radius would exert such a strong gravitational field that no light could escape from its surface. That this should have been predicted so early is so striking that we give a translation of Laplace's essay in an appendix.

One can express the dragging back of light by a massive body more precisely using Penrose's idea of a closed trapped surface. Consider a sphere ¡T surrounding the body. At some instant let 5T emit a flash of light. At some later time t, the ingoing and outgoing wave fronts from ST will form spheres and ^ respectively. In a normal situation, the area of ^ will be less than that of^ (because it represents ingoing light) and the area of will be greater than that of ST (because it represents outgoing light; see figure 1). However if a sufficiently large amount of matter is enclosed within 5T, the areas of ^ and ^ will both be less than that of^". The surfaced" is then said to be a closed trapped surface. As t increases, the area of ^ will get smaller and smaller provided that gravity remains attractive, i.e. provided that the energy density of the matter does not become negative. Since the matter inside 5T cannot travel faster than light, it will be trapped within a region whose boundary decreases to zero within a finite time. This suggests that something goes badly wrong. We shall in fact show that in such a situation a space-time singularity must occur, if certain reasonable conditions hold.

One can think of a singularity as a place where our present laws of physics break down. Alternatively, one can think of it as representing part of the edge of space-time, but a part which is at a finite distance instead of at infinity. On this view, singularities are not so bad, but one still has the problem of the boundary conditions. In other words, one does not know what will come out of the singularity.

There are two situations in which we expect there to be a sufficient concentration of matter to cause a closed trapped surface. The first is in the gravitational collapse of stars of more than twice the mass of the sun, which is predicted to occur when they have exhausted their nuclear fuel. In this situation, we expect the star to collapse to a singularity which is not visible to outside observers. The second situation is that of the whole universe itself. Recent observations of the microwave background indicate that the universe contains enough matter to cause a time-reversed closed trapped surface. This implies the existence of a singularity in the past, at the beginning of the present epoch of expansion of the universe. This singularity is in principle visible to us. It might be interpreted as the beginning of the universe.

In this book we shall study the large scale structure of space-time on the basis of Einstein's General Theory of Relativity. The predictions of this theory are in agreement with all the experiments so far performed. However our treatment will be sufficiently general to cover modifications of Einstein's theory such as the Brans-Dicke theory.

While we expect that most of our readers will have some acquaintance with General Relativity, we have endeavoured to write this book so that it is self-contained apart from requiring a knowledge of simple calculus, algebra and point set topology. We have therefore devoted chapter 2 to differential geometry. Our treatment is reasonably modern in that we have formulated our definitions in a manifestly coordinate independent manner. However for computational convenience we do use indices at times, and we have for the most part avoided the use of fibre bundles. The reader with some knowledge of differential geometry may wish to skip this chapter.

In chapter 3 a formulation of the General Theory of Relativity is given in terms of three postulates about a mathematical model for space-time. This model is a manifold with a metric g of Lorentz signature. The physical significance of the metric is given by the first two postulates: those of local causality and of local conservation of energy-momentum. These postulates are common to both the General and the Special Theories of Relativity, and so are supported by the experimental evidence for the latter theory. The third postulate, the field equations for the metric g, is less well experimentally established. However most of our results will depend only on the property of the field equations that gravity is attractive for positive matter densities. This property is common to General Relativity and some modifications such as the Brans-Dicke theory.

In chapter 4, we discuss the significance of curvature by considering its effects on families of timelike and null geodesies. These represent the paths of small particles and of light rays respectively. The curvature can be interpreted as a differential or tidal force which induces relative accelerations between neighbouring geodesies. If the energy-momentum tensor satisfies certain positive definite conditions, this differential force always has a net converging effect on non-rotating families of geodesies. One can show by use of Raychaudhuri's equation (4.26) that this then leads to focal or conjugate points where neighbouring geodesies intersect.

To see the significance of these focal points, consider a one-dimensional surface «5" in two-dimensional Euclidean space (figure 2). Let p be a point not on<9". Then there will be some curve from<9" to p which is shorter than, or as short as, any other curve from to p. Clearly this curve will be a geodesic, i.e. a straight line, and will intersect if orthogonally. In the situation shown in figure 2, there are in fact three geodesies orthogonal to<9" which pass through^. The geodesic through the point r is clearly not the shortest curve from SP to p. One way of recognizing this (Milnor (1963)) is to notice that the neighbouring

Figure 2. The line pr cannot be the shortest line from p to S?, because there is a focal point q between p and r. In fact either px or py will be the shortest line from p to geodesies orthogonal to 6? through u and v intersect the geodesic through r at a focal point q between«5" and p. Then joining the segment uq to the segment qp, one could obtain a curve from£f top which had the same length as a straight lino rp. However as uqp is not a straight line, one could round off the corner at q to obtain a curve from S? to p which was shorter than rp. This shows that rp is not the shortest curve from «5" to p. In fact the shortest curve will be either xp or yp.

One can carry these ideas over to the four-dimensional space-time manifold with the Lorentz metric g. Instead of straight lines, one considers geodesies, and instead of considering the shortest curve one considers the longest timelike curve between a point p and a spacelike surface {? (because of the Lorentz signature of the metric, there will be no shortest timelike curve but there may be a longest such curve). This longest curve must be a geodesic which intersects^ orthogonally, and there can be no focal point of geodesies orthogonal to<9" between

Figure 2. The line pr cannot be the shortest line from p to S?, because there is a focal point q between p and r. In fact either px or py will be the shortest line from p to r u r

& and p. Similar results can be proved for null geodesies. These results are used in chapter 8 to establish the existence of singularities under certain conditions.

In chapter 5 we describe a number of exact solutions of Einstein's equations. These solutions are not realistic in that they all possess exact symmetries. However they provide useful examples for the succeeding chapters and illustrate various possible behaviours. In particular, the highly symmetrical cosmological models nearly all possess space-time singularities. For a long time it was thought that these singularities might be simply a result of the high degree of symmetry, and would not be present in more realistic models. It will be one of our main objects to show that this is not the case.

In chapter 6 we study the causal structure of space-time. In Special Relativity, the events that a given event can be causally affected by, or can causally affect, are the interiors of the past and future light cones respectively (see figure 3). However in General Relativity the metric g which determines the light cones will in general vary from point to point, and the topology of the space-time manifold Jl need not be that of Euclidean space B*. This allows many more possibilities. For instance one can identify corresponding points on the surfaces Sf^ and in figure 3, to produce a space-time with topology R3 x S1. This would contain closed timelike curves. The existence of such a curve would lead to causality breakdowns in that one could travel into one's past. We shall mostly consider only space-times which do not permit such causality violations. In such a space-time, given any spacelike surface there is a maximal region of space-time (called the Cauchy development of^) which can be predicted from knowledge of data onA Cauchy development has a property ('Global hyper-bolicity') which implies that if two points in it can be joined by a timelike curve, then there exists a longest such curve between the points. This curve will be a geodesic.

The causal structure of space-time can be used to define a boundary or edge to space-time. This boundary represents both infinity and the part of the edge of space-time which is at a finite distance, i.e. the singular points.

In chapter 7 we discuss the Cauchy problem for General Relativity. We show that initial data on a spacelike surface determines a unique solution on the Cauchy development of the surface, and that in a certain sense this solution depends continuously on the initial data. This chapter is included for completeness and because it uses a number

of results of the previous chapter. However it is not necessary to read it in order to understand the following chapters.

In chapter 8 we discuss the definition of space-time singularities. This presents certain difficulties because one cannot regard the singular points as being part of the space-time manifold Jl.

We then prove four theorems which establish the occurrence of space-time singularities under certain conditions. These conditions fall into three categories. First, there is the requirement that gravity shall be attractive. This can be expressed as an inequality on the energy-momentum tensor. Secondly, there is the requirement that there is enough matter present in some region to prevent anything escaping from that region. This will occur if there is a closed trapped surface, or if the whole universe is itself spatially closed. The third requirement is that there should be no causality violations. However this requirement is not necessary in one of the theorems. The basic idea of the proofs is to use the results of chapter 6 to prove there must be longest timelike curves between certain pairs of points. One then shows that if there were no singularities, there would be focal points which would imply that there were no longest curves between the pairs of points.

We next describe a procedure suggested by Schmidt for constructing a boundary to space-time which represents the singular points of space-time. This boundary may be different from that part of the causal boundary (defined in chapter 0) which represents singularities.

In chapter ft, we bIiow that the Rocond condition of theorem 2 of chaptor 8 should bo satisfied near stars of more than times tho solar mass in the final stages of their evolution. The singularities which occur are probably hidden behind an event horizon, and so are not visible front otitcltlo. To itn oxlttt'intl iibcurvoi', llittt'tt itpjjtmr« to Litt a ' Lilitck hole' where the star once was. We discuss the properties of such black holes, and show that they probably settle down finally to one of the Kerr family of solutions. Assuming this to be the case, one can place certain upper bounds on the amount of energy which can be extracted from black holes. In chapter 10 we show that the second conditions of theorems 2 and 3 of chapter 8 should be satisfied, in a time-reversed sense, in the whole universe. In this case, the singularities are in our past and constitute a beginning for all or part of the observed universe.

The essential part of the introductory material is that in § 3.1, § 3.2 and § 3.4. A reader wishing to understand the theorems predicting the existence of singularities in the universe need read further only chapter 4, § 6.2-§ 6.7, and § 8.1 and § 8.2. The application of these theorems to collapsing stars follows in § 9.1 (which uses the results of appendix B); the application to the universe as a whole is given in § 10.1, and relies on an understanding of the Robertson-Walker universe models (§ 5.3). Our discussion of the nature of the singularities is contained in § 8.1, § 8.3-§ 8.5, and § 10.2; the example of Taub-NUT space (§ 5.8) plays an important part in this discussion, and the Bianchi I universe model (§5.4) is also of some interest.

A reader wishing to follow our discussion of black holes need read only chapter 4, § 6.2-§ 6.6, § 6.9, and § 9.1, § 9.2 and § 9.3. This discussion relies on an understanding of the Schwarzschild solution (§ 5.5) and of the Kerr solution (§5.6).

Finally a reader whose main interest is in the time evolution properties of Einstein's equations need read only §6.2-§6.6 and chapter 7. He will find interesting examples given in §5.1, §5.2 and §5.5.

We have endeavoured to make the index a useful guide to all the definitions introduced, and the relations between them.

Was this article helpful?

## Post a comment