Exact Hypersurface - Homogeneous Solutions in Cosmology

Exact solutions have always played a central role in the investigation of physical theories whose content is encoded in a complicated set of di erential equations. The theory of general relativity is indeed an example of this. A number of exact solutions of Einstein's eld equations have been of key importance in the discussion of physical problems. Solutions have been found which describe black holes, stellar interiors, gravitational waves, and even the large scale structure of the universe itself. Exact solutions have also served as a guide to point out mathematical features of the theory. The Taub-NUT-M (TaubNewman-Unti-Tamburino-Misner [1,2]) solution, for example, has been of crucial importance for the very definitions one uses in describing the singularities of the full theory. Thus exact solutions may point out features which are not just special to themselves but characterize in some way properties of a wider class of solutions. They may also play a role as \building blocks" for more general solutions. For example, in certain ways the general spatially homogeneous cosmological model near an initial singularity can be understood in terms of very special exact solutions, notably the Kasner and the vacuum Bianchi type II solutions, which to some extent also describe aspects of general cosmological singularities [3]. Sometimes exactly solvable problems are even used as a guide in developing ideas for the construction of more general theories, e.g., quantum gravity. For example, solvable problems in spatially homogeneous (SH) cosmology have been used to implement a number of di erent quantization schemes. Thus there is ample motivation to try to nd exact solutions. Indeed, the book by Kramer et al [4] is largely dedicated to the listing of exact solutions. Several chapters of that book deal with hypersurface-homogeneous (HH) solutions, a class for which the Einstein equations reduce to more manageable ordinary di erential equations. Within the class of HH solutions there are several subclasses of considerable physical interest, the cosmological SH models and the astrophysical static spherically symmetric spacetimes being the most prominent ones. Since the birth of general relativity nearly 80 years ago, an overwhelming number of HH solutions have been produced. A look at physics abstracts shows that this production continues even today at a considerable pace. However, often this search is undertaken as an end in itself without attempting to understand how particular successes t into a larger scheme and without employing any systematic method of attack revealing possible underlying mechanisms. Exceptions do exist though, as illustrated very nicely by the numerous approaches to the problem of nding vacuum solutions for spacetimes with one or two commuting Killing vector elds [4,5]. However, these techniques have not contributed much to the problem of nding HH solutions. It is the purpose of this article