Gluing small black holes into initial data sets

TL;DR

This paper proves a localized gluing technique for Einstein's constraint equations, enabling the insertion of small black hole data into larger initial data sets, with potential applications to modeling extreme mass ratio inspirals.

Contribution

It introduces a new geometric microlocal gluing method for Einstein's constraints, allowing insertion of small black holes into initial data sets under generic conditions.

Findings

Constructed initial data with small black holes and controlled asymptotics.

Demonstrated convergence of glued data to original data away from the insertion point.

Applied method to generate initial data for extreme mass ratio inspirals.

Abstract

We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in $n\geq 3$ 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.