Spatially homogeneous solutions
In this case it is assumed that the four-dimensional spacetime manifold admits a symmetry group with three-dimensional spacelike orbits. After reduction to the quotient manifold this only leaves one dimension (time) and the equations reduce to ordinary differential equations (at least in the simplest cases). In this situation the theory of dynamical systems can be applied to great advantage. I have done recent work in this area in collaboration with Paul Tod ( gr-qc/9811051 ) and Claes Uggla ( gr-qc/0005116 ). I later sharpened the results of these two papers using centre manifold theory ( gr-qc/0112040 ). The kinetic models used in this work mean that reduction to ordinary differential equations is not automatic, due to the presence of velocity variables and dealing with this presents a challenge for the future. This work takes a step beyond the fluid models which were traditionally studied. The work on models with collisionless matter has recently been taken further by Mark Heinzle and Claes Uggla ( gr-qc/0512031 ). Some of the simplest cases with collisional kinetic matter (Einstein-Boltzmann system) have been studied in the theses of two students in Yaounde (Etienne Takou and David Dongo).
Cohomogeneity two solutions
In this case the existence of a symmetry group with two-dimensional spacelike orbits is assumed. The equations reduce to partial differential equations in one space dimension. In some cases, such as spherical symmetry, techniques of ordinary differential equations can still be applied. More generally, hyperbolic equations occur, requiring other approaches. Nevertheless, hyperbolic equations in one space dimension are much simpler than in higher dimensions. In investigating the structure of singularities as well as geodesic completeness when it is true, it is useful to have certain geometrically defined time coordinates. One possibility, that of constant mean curvature (CMC) time coordinates has been studied in work which I did partly in collaboration with Gregory Burnett ( gr-qc/9411011 , gr-qc/9508001 , gr-qc/9605022 ). It has been extended by Oliver Henkel in his PhD done under my supervision. The results are presented in three papers ( gr-qc/0108003 , gr-qc/0110081 , gr-qc/0110082 ) which develop and apply the concept of prescribed mean curvature (PMC) foliations. In further developments Håkan Andréasson , Gerhard Rein , Marsha Weaver and myself were able to extend the results on CMC time coordinates and relate them to results on other types of geometric time coordinates in the literature, thus obtaining a unified picture ( gr-qc/0110089 , gr-qc/0211063 ). Generalizations of some of these results when a cosmological constant is added to the source terms in the Einstein equations were obtained by Blaise Tchapnda in his thesis (cf. gr-qc/0403098 , gr-qc/0305059 , gr-qc/0407062 ). The addition of a scalar field was studied by David Tegankong in his PhD work as part of my collaboration with the University of Yaounde I (cf. gr-qc/0405039 , gr-qc/0501062 , gr-qc/0506131 ).
We are still a long way from having full control of singularities and situations with geodesic completeness under this symmetry assumption. A promising approach to understanding the structure of singularities in this and more general cases is the theory of Fuchsian equations, applied to Gowdy spacetimes in a paper by Satyanad Kichenassamy and myself. In another direction, I have constructed asymptotically flat geodesically complete spacetimes with a kinetic matter model in collaboration with Gerhard Rein ( gr-qc/9306020 ). The generalization of this to the case of charged matter has been investigated by Pierre Noundjeu in his PhD (collaboration with the University of Yaounde I, cf. gr-qc/0303064 , gr-qc/0311081 ). Solutions constructed by the direct application of Fuchsian techniques have the property that the spatial derivatives of the unknowns have roughly the same order of divergence as the unknowns themselves as the singularity is approached. There are solutions, even of cohomogeneity two, where this is not the case. Examples were constructed by Marsha Weaver and myself ( gr-qc/0103102 ). These exhibit strongly inhomogeneous features called spikes.
General solutions
Recently Lars Andersson and I applied Fuchsian techniques to analyse the singularities of a class of spacetimes without any symmetry assumptions ( gr-qc/0001047 ). This was aided by a judicious choice of matter model (scalar field) which apparently simplifies the structure of the singularities. With Marc Henneaux, Thibault Damour and Marsha Weaver, I generalized these results to other matter fields and to spacetimes of various dimensions ( gr-qc/0202069 ). In particular solutions of the Einstein vacuum equations in dimension at least eleven were treated.