6 Prescribed singularities5 Global existence for small 5.2 Stability of Minkowski space

5.3 Stability of the 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. A proof of the stability of the latter model by Andersson and Moncrief has been announced in  [1]. 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 in the Minkowksi case towards infinity 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.



6 Prescribed singularities5 Global existence for small 5.2 Stability of Minkowski space

image Local and Global Existence Theorems for the Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr-2000-1
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