3 - General Relativity

Summary

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.