7 - The Cauchy problem in General Relativity

Summary

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.