On solutions of the vacuum Einstein equation in the radiation regime

We review recent results by the author, in collaboration with Erwann Delay, Olivier Lengard, and Rafe Mazzeo, on existence and properties of space-times with controlled asymptotic behavior at null infinity. The standard description of gravitational radiation proceeds as follows: one considers space-times (M , g) which can be conformally completed by adding a boundary ∂M = I , so that an appropriate conformal rescaling of g leads to a metric which extends by continuity to a Lorentzian metric g̃ on the new manifold with boundary M̃ := M ∪ I . This raises several questions: do there exist non-trivial vacuum space-times in which this can be done? does this prescription cover all radiating space-times? or at least all interesting ones? is this the right way to proceed anyway? In this talk I will review some recent progress concerning those questions. Recall [24] that a space-time is called asymptotically simple if the above conformal completion (M̃ , g̃) is smooth, and if every null geodesic of (M , g) has precisely two end points on I . It has been an open question whether there exist any vacuum asymptotically space-times other than the Minkowski one. A celebrated theorem of Christodoulou and Klainerman [4] proves existence of a large family of space-times which are close to being asymptotically simple: The Christodoulou-Klainerman metrics are geodesically complete, and admit conformal completions. However, the differentiability properties of the conformally rescaled metrics are poorly controlled. This last issue does play a role in the theory, as the properties of I ’s with low differentiability are rather different from those of the smooth ones. For instance, the peeling properties of the gravitational field, which are sometimes considered as a characteristic feature of gravitational radiation, are different for conformal completions which are, or which are not, of C3 differentiability class (cf., e.g. [1, 26]). Further, while it is rather likely that null geodesics will also have precisely two end points on I for the Christodoulou-Klainerman space-times, no analysis of this question ∗Partially supported by a Polish Research Committee grant; email [email protected].