Neutron stars are truly marvelous objects. They represent the end result of the evolution of very massive stars and are the “left-overs” of the enormous explosion that accompanies the death of these stars – namely, a supernova explosion. In a radius of a dozen of kilometers only, these stars can accumulate as much mass as twice that of the Sun, reach temperatures of tens of millions of degrees and magnetic fields that millions of billions larger than those on Earth. More importantly, by being so compact, these stars produce enormous gravitational fields, the largest gravitational fields for an object with a hard surface. This chapter will explain how neutron stars have been discovered and how we have learnt about their incredible properties. It will also stress that, although we now know quite a lot about neutron stars, they still represent a significant mystery in physics, since we have only a rather vague idea of what is inside neutron stars and how they can be built in nature.

In §8.1, we discuss the problem of defining singularities in spacetime. We adopt b-incompleteness as an indication that singular points have been cut out of spacetime, and characterize two ways in which b-incompleteness can be associated with some form of curvature singularity. In §8.2, four theorems are given to 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 spacetime. 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 spacetime, shown to be related to non-Hausdorff behaviour of the b-boundary. We show that in a generic spacetime, 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.

In § 5.1 and § 5.2 we consider the simplest Lorentz metrics: those of constant curvature. The spatially isotropic and homogeneous cosmological models are described in §5.3, and their simplest anisotropic generalizations are discussed in § 5.4. It is shown that all such simple models will have a singular origin provided that A does not take large positive values. The spherically symmetric metrics which describe the field outside a massive charged or neutral body are examined in §5.5, and the axially symmetric metrics describing the field outside a special class of massive rotating bodies are described in §5.6. It is shown that some of the apparent singularities are simply due to a bad choice of coordinates. In §5.7 we describe the Godel universe and in §5.8 the Taub-NUT solutions. These probably do not represent the actual universe but they are of interest because of their pathological global properties. Finally some other exact solutions of interest are mentioned in §5.9.

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 spacetime 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.

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 spacetime 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.

In this chapter we consider the effect of spacetime 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 spacetimes. 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.