Characterization of Schwarzschildean initial data

A theorem providing a characterization of Schwarzschildean initial data sets on slices with an asymptotically Euclidean end is proved. This characterization is based on the proportionality of the Weyl tensor and its D’Alambertian that holds for some vacuum Petrov type D spacetimes (e.g. the Schwarzschild spacetime, the C-metric, but not the Kerr solution). The $3+1$ decomposition of this proportionality condition renders necessary conditions for an initial data set to be a Schwarzschildean initial set. These conditions can be written as quadratic expressions of the electric and magnetic parts of the Weyl tensor, and thus involve only the freely specifiable data. In order to complete our characterization, a study of which vacuum static Petrov type D spacetimes admit asymptotically Euclidean slices is undertaken. Furthermore, a discussion of the Arnowitt-Deser-Misner (ADM) 4-momentum for boost-rotation symmetric spacetimes is given. As a by-product of our analysis a certain characterization of the Schwarzschild spacetime is obtained. Finally, a generalization of our characterization, valid for Schwarzschildean hyperboloidal initial data sets is put forward.

Figure 1

Schematic description of types of hypersurfaces covered by our main theorem: (1) a time symmetric Cauchy hypersurface, (2) a hyperboloid which also reaches one of the two spatial infinities, (3) a boosted slice, (4) a generic nontime symmetric Cauchy hypersurface.