Spacetime Groups

We give a classification of spacetime groups: simply connected four-dimensional spacetimes which admit a simply transitive four-dimensional isometry group. Our classification scheme, which is based upon the Newman-Penrose formalism and standard Lie algebraic invariants, yields 29 distinct classes of spacetime groups. We give a general solution to the equivalence problem for spacetime groups, that is, given a spacetime group we show how to uniquely determine where it belongs in our classification. We have made an extensive study of the Einstein equations with various matter sources for each spacetime group and we have closed a number of open problems in the exact solution literature.