Global properties of solutions to Einstein's field equations

The methods invented so far to develop conjectures on and to work out the details of the time evolution of gravitational fields fall into two classes. The first class comprises ''physical considerations,'' techniques of differential geometry and topology which do not use field equations, while others such as study of explicit solutions, of the Cauchy problem local in time of formal expansion type solutions, of approximation procedures, etc., have been used to derive information about the evolution more or less directly from the field equations. In spite of all the ingenuity with which both methods have been employed, the important open problems of classical general relativity, e.g. the development and (causal) structure of singularities and the formation of horizons etc. for space-times arising from regular data, the asymptotic behavior of gravitational fields and the relation between the far fields and the structure of the sources etc., remained unsolved, indicating these techniques are insufficient. The main failure is in the understanding of longtime evolution of the fields and in deriving general statements about solutions of the field equations on the large and near singularities. Research in this area is discussed. (author)