Description
The seminal result by H. Friedrich on the semiglobal stability of the Minkowski spacetime shows that hyperboloidal initial data for the Einstein field equations which is suitably close to data for the Minkowski spacetime and conformally smooth gives rise to a future development which is future geodesically complete and has the same global structure as the comparable region in the Minkowski spacetime. Moreover, the resulting spacetime is conformally smooth and the generators of null infinity intersect at a point describing timelike infinity. In this talk I will show how a gauge based on the properties of a particular type of conformal invariants, the so-called conformal geodesics, gives rise to an alternative conformal representation of this class of spacetimes in which timelike infinity is described by a hyperboloid describing the endpoints of timelike geodesics. This representation provides an implementation of “scri-fixing” —that is, a gauge in which the location of null infinity is described as the locus of points with a fixed spatial coordinate. Finally, it will be shown how this conformal representation is naturally adapted to the use of ideas and methods of Melrose’s school of microlocal analysis. This observation opens the possibility of obtaining generalisations of Friedrich’s semiglobal results for spacetimes with polyhomogeneous asymptotics.