In Section
2.1, hyperboloidal initial data were mentioned. They can be thought
of as generalizations of the data induced by Minkowski space on a
hyperboloid. In the case of Minkowski space the solution admits a
conformal compactification where a conformal boundary, null
infinity, can be added to the spacetime. It can be shown that in
the case of the maximal development of hyperboloidal data a piece
of null infinity can be attached to the spacetime. For small
data,
i.e.
data close to that of a hyperboloid in Minkowski space, this
conformal boundary also has completeness properties in the future
allowing an additional point
to be attached there (see [102] and references therein for more details). Making contact
between hyperboloidal data and asymptotically flat initial data
is much more difficult and there is as yet no complete picture.
(An account of the results obtained up to now is given in [106].) If the relation between hyperboloidal and asymptotically flat
initial data could be understood it would give a very different
approach to the problem treated by Christodoulou and Klainerman
(Section
5.2). It might well also give more detailed information on the
asymptotic behaviour of the solutions.