Initial data rigidity implies spacetime rigidity
Jonathan Glöckle
TL;DR
The paper tackles local rigidity of DEC spacetimes by proving the ambient existence of a lightlike parallel vector field derived from DEC initial-data $(g,k)$ with $ρ \ge |j|_g$, enabling a Killing-development construction that yields locally unique DEC extensions. It shows how a leafwise construction on MOTS foliations, with $V = u(e_0 + ν)$ and $u = φ^{-1}$ for $φ = ds(ν)^{-1}$, combined with curvature vanishing under Ricci-flat leaves, produces a globally ambient-parallel field. This ambient field underpins a rigidity theorem: if DEC initial data is invariant outside a neighborhood, then the extension is locally geometric unique, with corollaries for spin and MOTS scenarios and connections to asymptotically flat data via Hirsch–Zhang. Altogether, the work provides a robust toolkit for initial-data rigidity, yielding local uniqueness results across multiple DEC regimes and linking to pp-wave and Killing-development frameworks. ($ρ$ denotes energy density and $j$ the momentum current; conventions follow $ρ \ge |j|_g$ and related DEC statements.)
Abstract
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).