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  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 .
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  on 2+1 vacuum gravity.
|Theorems on Existence and Global Dynamics for the
Alan D. Rendall
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