In [192] solutions of the vacuum Einstein equations with symmetry and controlled singularity structure were constructed. They are required to satisfy some extra conditions, being polarized or half-polarized. Without these conditions oscillations are expected. The result was generalized to a larger class of topologies in [92].

Anguige [21] has obtained results on solutions with perfect fluid that are general under the condition of plane symmetry, which is stronger than Gowdy symmetry. He also extended this to polarized Gowdy symmetry in [22].

Work related to these Fuchsian methods was done earlier in a somewhat simpler context by Moncrief [247], who showed the existence of a large class of analytic vacuum spacetimes with Cauchy horizons.

As a result of the BKL picture, it cannot be expected that the singularities in general solutions of the Einstein equations in vacuum or with a non-stiff fluid can be handled using Fuchsian techniques (cf. Section 8.1). However, things look better in the presence of a massless scalar field or a stiff fluid. For these types of matter it has been possible [13] to prove a theorem analogous to that of [205] without requiring symmetry assumptions. The same conclusion can be obtained for a scalar field with mass or with a potential of moderate growth [299].

The results included in this review concern the Einstein equations in four spacetime dimensions. Of course, many of the questions discussed have analogues in other dimensions and these may be of interest for string theory and related topics. In [130] Fuchsian techniques were applied to the Einstein equations coupled to a variety of field theoretic matter models in arbitrary dimensions. One of the highlights is the result that it is possible to apply Fuchsian techniques without requiring symmetry assumptions to the vacuum Einstein equations in spacetime dimension at least eleven. Many new results are also obtained in four dimensions. For instance, the Einstein-Maxwell-dilaton and Einstein-Yang-Mills equations are treated. The general nature of the results is that, provided certain inequalities are satisfied by coupling constants, solutions with prescribed singularities can be constructed that depend on the same number of free functions as the general solution of the given Einstein-matter system. Other results on models coming from string theory have been obtained by Fuchsian methods in [256, 254, 255].

http://www.livingreviews.org/lrr-2005-6 |
© Max Planck Society and the author(s)
Problems/comments to |