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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.