### 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 [108]. 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 [65, 107], 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 . A generalization
described in [211] 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. 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.