5.3 Stability of the Milne 5 Global existence for small 5.1 Stability of de Sitter

5.2 Stability of Minkowski space

  The other result on global existence for small data is that of Christodoulou and Klainerman on the stability of Minkowski space [54] The formulation of the result is close to that given in section 5.1 but now de Sitter space is replaced by Minkowski space. Suppose then that initial data are given which are asymptotically flat and sufficiently close to those induced by Minkowski space on a hyperplane. Then Christodoulou and Klainerman prove that the maximal Cauchy development of these data is geodesically complete. They also provide a wealth of detail on the asymptotic behaviour of the solutions. The proof is very long and technical. The central tool is the Bel-Robinson tensor which plays an analogous role for the gravitational field to that played by the energy-momentum tensor for matter fields. Apart from the book of Christodoulou and Klainerman itself some introductory material on geometric and analytic aspects of the proof can be found in [25] and [53] respectively.

In the original version of the theorem initial data had to be prescribed on all of tex2html_wrap_inline1372 . A generalization described in [111] concerns the case where data need only be prescribed on the complement of a compact set in tex2html_wrap_inline1372 . This means that statements can be obtained for any asymptotically flat spacetime where the initial matter distribution has compact support, provided attention is confined to a suitable neighbourhood of infinity. The proof of the new version uses a double null foliation instead of the foliation by spacelike hypersurfaces previously used and leads to certain conceptual simplifications.



5.3 Stability of the Milne 5 Global existence for small 5.1 Stability of de Sitter

image Local and Global Existence Theorems for the Einstein Equations
Alan D. Rendall
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