Anguige [11] 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 [10].
Work related to these Fuchsian methods was done earlier in a somewhat simpler context by Moncrief [177], 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 [6] to prove a theorem analogous to that of [150] without requiring symmetry assumptions. The same conclusion can
be obtained for a scalar field with mass or with a potential of
moderate growth [216].
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 [92] 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.
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Theorems on Existence and Global Dynamics for the
Einstein Equations
Alan D. Rendall http://www.livingreviews.org/lrr-2002-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |