1 Introduction

Systems of partial differential equations are of central importance in physics. Only the simplest of these equations can be solved by explicit formulae. Those that cannot are commonly studied by means of approximations. There is, however, another approach that is complementary. This consists in determining the qualitative behaviour of solutions, without knowing them explicitly. The first step in doing this is to establish the existence of solutions under appropriate circumstances. Unfortunately, this is often hard, and obstructs the way to obtaining more interesting information. When partial differential equations are investigated with a view to applications, proving existence theorems should not become an end in itself. It is important to remember that, from a more general point of view, it is only a starting point.

The basic partial differential equations of general relativity are Einstein’s equations. In general, they are coupled to other partial differential equations describing the matter content of spacetime. The Einstein equations are essentially hyperbolic in nature. In other words, the general properties of solutions are similar to those found for the wave equation. It follows that it is reasonable to try to determine a solution by initial data on a spacelike hypersurface. Thus the Cauchy problem is the natural context for existence theorems for the Einstein equations. The Einstein equations are also nonlinear. This means that there is a big difference between the local and global Cauchy problems. A solution evolving from regular data may develop singularities.

A special feature of the Einstein equations is that they are diffeomorphism invariant. If the equations are written down in an arbitrary coordinate system, then the solutions of these coordinate equations are not uniquely determined by initial data. Applying a diffeomorphism to one solution gives another solution. If this diffeomorphism is the identity on the chosen Cauchy surface up to first order then the data are left unchanged by this transformation. In order to obtain a system for which uniqueness in the Cauchy problem holds in the straightforward sense that it does for the wave equation, some coordinate or gauge fixing must be carried out.

Another special feature of the Einstein equations is that initial data cannot be prescribed freely. They must satisfy constraint equations. To prove the existence of a solution of the Einstein equations, it is first necessary to prove the existence of a solution of the constraints. The usual method of solving the constraints relies on the theory of elliptic equations.

The local existence theory of solutions of the Einstein equations is rather well understood. Section 2 points out some of the things that are not known. On the other hand, the problem of proving general global existence theorems for the Einstein equations is beyond the reach of the mathematics presently available. To make some progress, it is necessary to concentrate on simplified models. The most common simplifications are to look at solutions with various types of symmetry and solutions for small data. These two approaches are reviewed in Sections 3 and 5, respectively. A different approach is to prove the existence of solutions with a prescribed singularity structure or late-time asymptotics. This is discussed in Section 6. Section 9 collects some miscellaneous results that cannot easily be classified. Since insights about the properties of solutions of the Einstein equations can be obtained from the comparison with Newtonian theory and special relativity, relevant results from those areas are presented in Section 4.

The sections just listed are to some extent catalogues of known results, augmented with some suggestions as to how these could be extended in the future. Sections 7 and 8 complement this by looking ahead to see what the final answer to some interesting general questions might be. They are necessarily more speculative than the other sections but are rooted in the known results surveyed elsewhere in the article. Section 7 also summarizes various results on cosmological models with accelerated expansion.

The area of research reviewed in the following relies heavily on the theory of differential equations, particularly that of hyperbolic partial differential equations. For the benefit of readers with little background in differential equations, some general references that the author has found to be useful will be listed. A thorough introduction to ordinary differential equations is given in [170 ]. A lot of intuition for ordinary differential equations can be obtained from [186 ]. The article by Arnold and Ilyashenko [29 ] is full of information, in rather compressed form. A classic introductory text on partial differential equations, where hyperbolic equations are well represented, is [197 ]. Useful texts on hyperbolic equations, some of which explicitly deal with the Einstein equations, are [335 , 202 , 269 , 239 , 327 , 198 , 137 ].

An important aspect of existence theorems in general relativity that one should be aware of is their relation to the cosmic censorship hypothesis. This point of view was introduced in an influential paper by Moncrief and Eardley [248 ]. An extended discussion of the idea can be found in [114 ].

This article is descriptive in nature and equations have been kept to a minimum. A collection of relevant equations together with the background necessary to understand the notation can be found in [301 ].