Gluing Small Black Holes into Initial Data Sets

Abstract

We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in \(n\ge 3\) spatial dimensions. We glue an \(\epsilon \)-rescaling of an asymptotically flat data set \(({\hat{\gamma }},{\hat{k}})\) into the neighborhood of a point \(\mathfrak {p}\in X\) inside of another initial data set \((X,\gamma ,k)\), under a local genericity condition (non-existence of KIDs) near \(\mathfrak {p}\). As the scaling parameter \(\epsilon \) tends to 0, the rescalings \(\frac{x}{\epsilon }\) of normal coordinates x on X around \(\mathfrak {p}\) become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of \(\mathfrak {p}\) on the other hand, the glued initial data converge back to \((\gamma ,k)\). The initial data we construct enjoy polyhomogeneous regularity jointly in \(\epsilon \) and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets \((X,\gamma ,k)\) and appropriate boosted Kerr initial data sets \(({\hat{\gamma }},{\hat{k}})\) produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass \(\epsilon \) black hole. The proof combines a variant of the gluing method introduced by Corvino and Schoen with geometric singular analysis techniques originating in Melrose’s work. On a technical level, we present a fully geometric microlocal treatment of the solvability theory for the linearized constraints map.