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.

The view of physics that is most generally accepted at the moment is that one can divide the discussion of the universe into two parts. First, there is the question of the local laws satisfied by the various physical fields. These are usually expressed in the form of differential equations. Secondly, there is the problem of the boundary conditions for these equations, and the global nature of their solutions. This involves thinking about the edge of space–time in some sense. These two parts may not be independent. Indeed it has been held that the local laws are determined by the large scale structure of the universe. This view is generally connected with the name of Mach, and has more recently been developed by Dirac (1938), Sciama (1953), Dicke (1964), Hoyle and Narlikar (1964), and others. We shall adopt a less ambitious approach: we shall take the local physical laws that have been experimentally determined, and shall see what these laws imply about the large scale structure of the universe.

There is of course a large extrapolation in the assumption that the physical laws one determines in the laboratory should apply at other points of space–time where conditions may be very different. If they failed to hold we should take the view that there was some other physical field which entered into the local physical laws but whose existence had not yet been detected in our experiments, because it varies very little over a region such as the solar system.

We wish to consider Einstein's equations in the case of a spherically symmetric space–time. One might regard the essential feature of a spherically symmetric space–time as the existence of a world-line ℒ such that the space–time is spherically symmetric about ℒ. Then all points on each spacelike two-sphere d centred on any point p of ℒ, defined by going a constant distance d along all geodesies through p orthogonal to ℒ, are equivalent. If one permutes directions at p by use of the orthogonal group SO(3) leaving ℒ invariant, the space–time is, by definition, unchanged, and the corresponding points of d are mapped into themselves; so the space–time admits the group SO(3) as a group of isometries, with the orbits of the group the spheres d. (There could be particular values of d such that the surface d was just a point p′; then p′ would be another centre of symmetry. There can be at most two points (p′ and p itself) related in this way.)

However, there might not exist a world-line like ℒ in some of the space–times one would wish to regard as spherically symmetric. In the Schwarzschild and Reissner–Nordström solutions, for example, space–time is singular at the points for which r = 0, which might otherwise have been centres of symmetry. We shall therefore take the existence of the group SO(3) of isometries acting on two-surfaces like d as the characteristic feature of a spherically symmetric space–time. Thus we shall say that space–time is spherically symmetric if it admits the group SO(3) as a group of isometries, with the group orbits spacelike two-surfaces.

In this chapter we shall give an outline of the Cauchy problem in General Relativity. We shall show that, given certain data on a space like three-surface there is a unique maximal future Cauchy development D+() and that the metric on a subset of D+() depends only on the initial data on J–() ∩. We shall also show that this dependence is continuous if has a compact closure in D+(). This discussion is included here because of its intrinsic interest, because it uses some of the results of the previous chapter, and because it demonstrates that the Einstein field equations do indeed satisfy postulate (a) of §3.2 that signals can only be sent between points that can be joined by a non-spacelike curve. However it is not really needed for the remaining three chapters, and so could be skipped by the reader more interested in singularities.

In §7.1, we discuss the various difficulties and give a precise formulation of the problem. In §7.2 we introduce a global background metric ĝ to generalize the relation which holds between the Ricci tensor and the metric in each coordinate patch to a single relation which holds over the whole manifold. We impose four gauge conditions on the covariant derivatives of the physical metric g with respect to the background metric ĝ.

THE PROCESS OF FISSION

The motion of our hypothetical mass of nebulous matter has now been traced out through its earlier stages in which it formed a rotating nebula, and through its later stages in which this nebula condensed into stars. In the last chapter we considered the general nature of the motion to be expected in the cluster of stars so formed; the present chapter will be devoted to the further history of individual stars.

We have supposed that an individual star comes into existence as a condensation in a nebular arm. In this earliest period of its existence its mean density is very low, being perhaps of the order of 10-17 grammes per cubic centimetre, and its surrounding atmosphere is contiguous with that of the neighbouring stars. At this stage it shares in the rotation of the nebula of which it forms part, the period of this rotation being perhaps of the order of 160,000 years.

As time proceeds the arms of the nebula expand while individual stars contract, so that the stars become continually more distinct from one another until finally they may be regarded as entirely separate bodies, each describing its independent orbit under the gravitational attractions of the other stars.

The best-known configurations of equilibrium of a rotating homogeneous mass, namely Maclaurin's spheroids and Jacobi's ellipsoids, are both of the ellipsoidal form, and this form will prove to be of primary importance in all the cosmogonical problems we shall attempt to solve. We accordingly devote a chapter to the subject of ellipsoidal configurations.

Looked at merely from the point of view of convenience in the development of the subject, the ellipsoidal form has the advantage that the potential of an ellipsoidal mass is known and is comparatively simple, and that the ellipsoidal configurations provide admirably clear examples of Poincaré's theory of linear series and stability. These reasons alone might justify our studying ellipsoidal configurations in some detail, but there are weightier reasons, as we shall soon see.

Throughout this chapter and the three succeeding chapters the matter under discussion will be supposed homogeneous and incompressible; the more complicated problems presented by non-homogeneous and compressible masses will be attacked in Chapter VII.

We shall deal in turn with three distinct problems–the first, that of a mass of liquid rotating freely under its own gravitational forces; the second, that of a mass devoid of rotation but acted on tidally by another mass; the third that of two masses rotating round one another and acting tidally on one another.

The present essay is primarily an attempt to follow up a line of research initiated by Laplace and Maclaurin, and extended in various directions by Roche, Lord Kelvin, Jacobi, Poincaré and Sir G. Darwin. Within two years of the close of his life, Darwin remarked that the way to further progress in cosmogony was blocked by our ignorance of the figures of equilibrium of rotating gaseous masses. He wrote as follows (Darwin and Modern Science, p. 563, and Tides, 3rd edition, p. 401):

“As we have seen, the study of the forms of equilibrium of rotating liquids is almost complete, and a good beginning has been made in the investigation of the equilibrium of gaseous stars, but much more remains to be discovered.”

“As a beginning we should like to know how a moderate degree of compressibility would alter the results for liquid, and…to understand more as to the manner in which rotation affects the equilibrium and stability of rotating gas. The field for the mathematician is a wide one, and in proportion as the very arduous exploration of that field is attained, so will our knowledge of the processes of cosmical evolution increase….

“Human life is too short to permit us to watch the leisurely procedure of cosmical evolution, but the celestial museum contains so many exhibits that it may become possible, by the aid of theory, to piece together bit by bit the processes through which stars pass in the course of their evolution.”