In this chapter we give an outline of the Cauchy problem in general relativity and 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 also show that this dependence is continuous if has a compact closure in D+(). This discussion is included here because of its intrinsic interest, its use of some of the results of the previous chapter, and its demonstration 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.

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.

This chapter completes the description of the Einstein equation by finding the correct measure of energy density and the more general measure of spacetime curvature. A density is a quantity per unit spatial volume, such as rest-mass density, charge density, number density, energy density, and so on. The chapter begins by discussing how densities are represented in special and general relativity – for example, the densities of energy and momentum, and their conservation.

The basic principles of general relativity are reviewed, in particular the different forms of the equivalence principle: the weak, Einstein, and strong equivalence principles. The concept of a metric is introduced within special relativity. The Einstein equations are derived in an heurisitic manner including the Christoffel symbols, the Ricci tensor, and the Ricci scalar. The Schwarzschild as the solution of Einstein‘s equation in vacuum are explicitly derived. The notion of the energy–momentum tensor, as the source term of the Einstein equations, is discussed in terms of the four-momentum of particles. For bulk matter, the definition of an ideal fluid is given. The conservation of the energy–momentum tensor in curved space-time is discussed. The Einstein equations are solved for a sphere of an ideal fluid to arrive at the Tolman–Oppenheimer–Volkoff equations, the central equations for the investigation of compact stars. Finally the analytically known solution for a sphere of an incompressible fluid, the Schwarzschild solution, is derived and used to set the Buchdahl limit on the compactness of a compact star.