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 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 [25] and [53] respectively.
In the original version of the theorem initial data had to be
prescribed on all of
. A generalization described in [111] 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.

Local and Global Existence Theorems for the Einstein
Equations
Alan D. Rendall
http://www.livingreviews.org/lrr20001
© MaxPlanckGesellschaft. ISSN 14338351
Problems/Comments to
livrev@aeipotsdam.mpg.de
