Gluing small black holes along timelike geodesics III: construction of true solutions and extreme mass ratio mergers

TL;DR

This paper constructs exact solutions to Einstein's equations describing small black holes moving along geodesics, and models black hole mergers with detailed control over the spacetime geometry, extending previous approximate solutions.

Contribution

It provides a method to correct approximate solutions to true solutions of Einstein's equations involving small black holes along geodesics, including mergers.

Findings

Construction of true solutions with small black holes along geodesics.

Demonstration of black hole merger models with controlled spacetime behavior.

Development of linear analysis tools for Kerr spacetimes.

Abstract

Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime $(M,g)$ satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic $\mathcal{C}$, we construct, on any compact subset of $M$, solutions $g_\epsilon$ of the Einstein equations which describe a mass $\epsilon$ Kerr black hole traveling along $\mathcal{C}$. More precisely, away from $\mathcal{C}$ one has $g_\epsilon\to g$ as $\epsilon\to 0$, while the $\epsilon^{-1}$-rescaling of $g_\epsilon$ around every point of $\mathcal{C}$ tends to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes which do not admit nontrivial Killing vector fields in a neighborhood of a point on the geodesic. As an application, we construct spacetimes which model the merger of a very light subextremal Kerr black hole with a slowly rotating unit mass Kerr(-de Sitter) black hole, followed by the relaxation of the resulting black hole to its final Kerr(-de Sitter) state. In Part I, we constructed approximate solutions $g_{0,\epsilon}$ of the gluing problem which satisfy the Einstein equations only modulo $\mathcal{O}(\epsilon^\infty)$ errors. Part II introduces a framework for obtaining uniform control of solutions of linear wave equations on $\epsilon$-independent precompact subsets of the original spacetime $(M,g)$. In this final part, we show how to correct $g_{0,\epsilon}$ to a true solution $g_\epsilon$ by adding a metric perturbation of size $\mathcal{O}(\epsilon^\infty)$ which solves a carefully chosen gauge-fixed version of the Einstein vacuum equations. The main novel ingredient is the proof of suitable mapping properties for the linearized gauge-fixed Einstein equations on subextremal Kerr spacetimes.