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5.3 Stability of the (compactified) Milne model

The interior of the light cone in Minkowski space foliated by the spacelike hypersurfaces of constant Lorentzian distance from the origin can be thought of as a vacuum cosmological model, sometimes known as the Milne model. By means of a suitable discrete subgroup of the Lorentz group it can be compactified to give a spatially compact cosmological model. With a slight abuse of terminology the latter spacetime will also be referred to here as the Milne model. The stability of the latter model has been proved by Andersson and Moncrief (see [811]). The result is that, given data for the Milne model on a manifold obtained by compactifying a hyperboloid in Minkowski space, the maximal Cauchy developments of nearby data are geodesically complete in the future. Moreover, the Milne model is asymptotically stable in the sense that any other solution in this class converges towards the Milne model in terms of suitable dimensionless variables.

The techniques used by Andersson and Moncrief are similar to those used by Christodoulou and Klainerman. In particular, the Bel-Robinson tensor is crucial. However, their situation is much simpler than that of Christodoulou and Klainerman, so that the complexity of the proof is not so great. This has to do with the fact that the fall-off of the fields towards infinity in the Minkowksi case is different in different directions, while it is uniform in the Milne case. Thus it is enough in the latter case to always contract the Bel-Robinson tensor with the same timelike vector when deriving energy estimates. The fact that the proof is simpler opens up a real possibility of generalizations, for instance by adding different matter models.


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