Another result on global existence for small data is that of
Christodoulou and Klainerman on the stability of Minkowski
space [79]. 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 BelRobinson tensor, which plays an analogous role for the
gravitational field to that played by the energymomentum 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 [44,
78], respectively. More recently, the result for the vacuum
Einstein equations has been generalized to the case of the
EinsteinMaxwell system by Zipser [246].
In the original version of the theorem, initial data had to be
prescribed on all of
. A generalization described in [156] concerns the case where data need only be prescribed on the
complement of a compact set in
. 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.

Theorems on Existence and Global Dynamics for the
Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr20026
© MaxPlanckGesellschaft. ISSN 14338351
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