Existence of non-trivial, vacuum, asymptotically simple spacetimes

We construct non-trivial vacuum spacetimes with a global J(+). The construction proceeds by proving extension results for initial data sets across compact boundaries, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is the existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. Finally, the construction allows one to add Einstein-Rosen bridges to time-symmetric initial data sets at points satisfying a local parity condition, with the perturbation of the metric localized in an arbitrarily small neighbourhood of the bridge.