Classifying Einstein's field equations with applications to cosmology and astrophysics

The field equations for spacetimes with finite-dimensional Hamiltonian dynamics are discussed. Examples of models belonging to this class are the cosmological spatially homogeneous models, the astrophysically interesting static spherically symmetric models, static cylindrically symmetric models, and certain cosmological self-similar models. A number of different sources are considered. Although these models arise from quite different physical contexts, their field equations all share a common mathematical structure. This motivates a classification of Einstein's field equations. Several classification schemes, based on properties under various variable transformations, are presented. It is shown how these schemes can be used to classify dynamical properties of the models and how one can thereby obtain qualitative information. It is also shown how one scheme can be used in order to find symmetries and exact solutions.