Initial data rigidity implies spacetime rigidity

In this article, we revisit the initial data rigidity theorem of Eichmair, Galloway and Mendes (arxiv:2009.09527). The goal is to strengthen their result by showing that the initial data sets concerned carry a vector field that is lightlike and parallel in an ambient sense. This will be used in a second step to show that among the spacetimes satisfying the dominant energy condition there exists locally essentially one spacetime extending these initial data sets. This local uniqueness theorem also applies in the context of other initial data rigidity theorems. Notably, the one in the spin case due the author (arxiv:2304.02331) and a recent study of the mass zero case in the positive energy theorem due to Hirsch and Zhang (arxiv:2403.15984).