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5.2 Stability of Minkowski space

Another result on global existence for small data is that of Christodoulou and Klainerman on the stability of Minkowski space [108Jump To The Next Citation Point]. 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 for the vacuum Einstein equations are prescribed that 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 [65107], respectively. The result for the vacuum Einstein equations was generalized to the case of the Einstein-Maxwell system by Zipser [354].

In the original version of the theorem, initial data had to be prescribed on all of R3. A generalization described in [211] concerns the case where data need only be prescribed on the complement of a compact set in 3 R. 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. A detailed treatment of this material can be found in the book of Klainerman and Nicolò [212].

An aspect of all this work which seemed less than optimal was the following. Well-known heuristic analyses by relativists produced a detailed picture of the fall-off of radiation fields in asymptotically flat solutions of the Einstein equations, known as peeling. It says that certain components of the Weyl tensor decay at certain rates. The analysis of Christodoulou and Klainerman reproduced some of these fall-off rates but not all. More light was shed on this discrepancy by Klainerman and Nicolò [213] who showed that if the fall-off conditions on the initial data assumed in [108] are strengthened somewhat then peeling can be proved.

A much shorter proof of the stability of Minkowski space has been given by Lindblad and Rodnianski [233]. It uses harmonic coordinates and so is closer to the original local existence proof of Choquet-Bruhat. The fact that this approach was not used earlier is related to the fact that the null condition, an important structural condition for nonlinear wave equations which implies global existence for small data, is not satisfied by the Einstein equations written in harmonic coordinates. Lindblad and Rodnianski formulated a generalization called the weak null condition [232]. This is only one element which goes into the global existence proof but it does play an important role. The result of Lindblad and Rodnianski does not give as much detail about the asymptotic structure as the approach of Christodoulou and Klainerman. On the other hand it seems that the proof generalizes without difficulty to the case of the Einstein equations coupled to a massless scalar field.

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