Another vacuum cosmological model whose nonlinear stability
has been investigated is the Bianchi III form of flat spacetime.
To obtain this model, first do the construction described in the
last section with the difference that the starting solution is
three-dimensional Minkowski space. Then, take the metric product
of the resulting three-dimensional Lorentz manifold with a
circle. This defines a flat spacetime that has one Killing
vector, which is the generator of rotations of the circle. It has
been shown by Choquet-Bruhat and Moncrief [65] that this solution is stable under small vacuum perturbations
preserving the one-dimensional symmetry. More precisely, the
result is proved only for the polarized case, but the authors
suggest that this restriction can be lifted at the expense of
doing some more work. As in the case of the Milne model, a
natural task is to generalize this result to spacetimes with
suitable matter content. The reasons it is necessary to restrict
to symmetric perturbations in this analysis, in contrast to what
happens with the Milne model, are discussed in detail in [65].
One of the main techniques used is a method of modified energy estimates that is likely to be of more general applicability. The Bel-Robinson tensor plays no role. The other main technique is based on the fact that the problem under study is equivalent to the study of the 2+1-dimensional Einstein equations coupled to a wave map (a scalar field in the polarized case). This helps to explain why the use of the Dirichlet energy could be imported into this problem from the work of [5] on 2+1 vacuum gravity.
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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 |