Gluing Small Black Holes into Initial Data Sets

Abstract

AbstractWe prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in $$n\ge 3$$ n ≥ 3 spatial dimensions. We glue an $$\epsilon $$ ϵ -rescaling of an asymptotically flat data set $$({\hat{\gamma }},{\hat{k}})$$ ( γ ^ , k ^ ) into the neighborhood of a point $$\mathfrak {p}\in X$$ p ∈ X inside of another initial data set $$(X,\gamma ,k)$$ ( X , γ , k ) , under a local genericity condition (non-existence of KIDs) near $$\mathfrak {p}$$ p . As the scaling parameter $$\epsilon $$ ϵ tends to 0, the rescalings $$\frac{x}{\epsilon }$$ x ϵ of normal coordinates x on X around $$\mathfrak {p}$$ p become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of $$\mathfrak {p}$$ p on the other hand, the glued initial data converge back to $$(\gamma ,k)$$ ( γ , k ) . The initial data we construct enjoy polyhomogeneous regularity jointly in $$\epsilon $$ ϵ and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets $$(X,\gamma ,k)$$ ( X , γ , k ) and appropriate boosted Kerr initial data sets $$({\hat{\gamma }},{\hat{k}})$$ ( γ ^ , k ^ ) produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass $$\epsilon $$ ϵ 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.