Go to previous page Go up Go to next page

9.1 Evolution of hyperboloidal data

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 i+ to be attached there (see [142] 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 [146].) 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 (see Section 5.2). It might well also give more detailed information on the asymptotic behaviour of the solutions.

The results on the hyperboloidal initial value problem rely on the conformal field equations, a reformulation of the Einstein equations which only works in dimension four. There is an alternative method which works in all even dimensions not less than four and gives a new approach in four dimensions. This has been used in [7] to generalize some of the above results to higher even dimensions.


  Go to previous page Go up Go to next page