Initial data for stationary spacetimes near spacelike infinity (Q2771200)

By definition, a stationary (vacuum) spacetime \((\widetilde{M}, \widetilde{g})\) is one for which there exists a timelike Killing vectorfield \(\xi^a\). The collection of all trajectories of \(\xi^a\) gives rise to an abstract three-dimensional manifold \(\widetilde{X}\). In general \(\widetilde{X}\) can only be identified with a hypersurface \(\widetilde{S}\) of \(\widetilde{M}\) everywhere orthogonal to \(\xi^a\) if, more restrictively, \((\widetilde{M}, \widetilde{g})\) is even static. The paper under review investigates the situation that this is not necessarily the case. It is assumed that the Riemannian metric corresponding to \(\widetilde{X}\) is asymptotically flat in a certain sense. Then a theorem of Beig and W.~Simon implies that (in a particular chart) several quantities, including the conformally rescaled metric of \(\widetilde{X}\), are analytic. The author discusses questions related to analyticity, if the metric is at the same time expressed in terms of a \(3+1\)-decomposition with respect to a hypersurface \(\widetilde{S}=\{t= \text{const.}\}\). It turned out that the fields considered in \(\widetilde{S}\) are in general not analytic at spacelike infinity. Some decay properties of the fields, expressed in radial coordinates, are also obtained.