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6.2 Fuchsian equations

The singular equations that arise in the study of isotropic singularities are closely related to what Kichenassamy [202Jump To The Next Citation Point] calls Fuchsian equations. He has developed a rather general theory of these equations (see [202201200], and also the earlier papers [31203204]). In [205Jump To The Next Citation Point] this was applied to analytic Gowdy spacetimes on T 3 to construct a family of vacuum spacetimes depending on the maximum number of free functions (for the given symmetry class) whose singularities can be described in detail. The symmetry assumed in that paper requires the two-surfaces orthogonal to the group orbits to be surface-forming (vanishing twist constants). In [188] a corresponding result was obtained for the class of vacuum spacetimes with polarized U (1) × U (1) symmetry and non-vanishing twist. The analyticity requirement on the free functions in the case of Gowdy spacetimes on T 3 was reduced to smoothness in [300]. There are also Gowdy spacetimes on S3 and S2 × S1, which have been less studied than those on T 3. The Killing vectors have zeros, defining axes, and these lead to technical difficulties. In [325] Fuchsian techniques were applied to Gowdy spacetimes on 3 S and 2 1 S × S. The maximum number of free functions was not obtained due to difficulties on the axes.

In [192] solutions of the vacuum Einstein equations with U (1) 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 [13Jump To The Next Citation Point] 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 [256254255].

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