Gluing small black holes along timelike geodesics II: uniform analysis on glued spacetimes

TL;DR

This paper develops a uniform analytical framework for wave equations on spacetimes with small black holes, enabling the correction of approximate solutions to exact Einstein vacuum solutions in a controlled manner.

Contribution

It introduces uniform Sobolev estimates for tensorial wave equations on glued spacetimes with small black holes, foundational for constructing exact Einstein solutions.

Findings

Established spectral theory for wave equations on Kerr backgrounds.

Proved uniform microlocal estimates for wave propagation near small black holes.

Constructed solutions to a toy nonlinear wave equation with uniform control as black hole size tends to zero.

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 constructed in Part I a family of metrics $g_\epsilon$ on the complement $M_\epsilon\subset M$ of an $\epsilon$-neighborhood of $\mathcal{C}$ with the following behavior: 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; and $g_\epsilon$ solves the Einstein vacuum equation modulo $\mathcal{O}(\epsilon^\infty)$ errors. The ultimate goal, achieved in Part III, is to correct $g_\epsilon$ to a true solution on any fixed precompact subset of $M$ by addition of a size $\mathcal{O}(\epsilon^\infty)$ metric perturbation which needs to satisfy a quasilinear wave equation (the Einstein vacuum equations in a suitable gauge). The present paper lays the necessary analytical foundations. We develop a framework for proving estimates for solutions of (tensorial) wave equations on $(M_\epsilon,g_\epsilon)$ which, on a suitable scale of Sobolev spaces, are uniform on $\epsilon$-independent precompact subsets of the original spacetime $M$. These estimates are proved by combining two ingredients: the spectral theory for the corresponding wave equation on Kerr; and uniform microlocal estimates governing the propagation of regularity through the small black hole, including radial point estimates reminiscent of diffraction by conic singularities and long-time estimates near perturbations of normally hyperbolic trapped sets. As an illustration of this framework, we construct solutions of a toy nonlinear scalar wave equation on $(M_\epsilon,g_\epsilon)$ for uniform timescales and with full control in all asymptotic regimes as $\epsilon\to 0$.