Gluing small black holes along timelike geodesics I: formal solution

TL;DR

This paper constructs a family of approximate solutions to Einstein's vacuum equations that model small black holes along timelike geodesics, providing a rigorous foundation for gravitational self-force analysis.

Contribution

It introduces a method to glue small black holes into a spacetime along geodesics, with metrics approximating Kerr solutions at small scales, extending previous formal perturbation approaches.

Findings

Constructed metrics solve Einstein vacuum equations up to arbitrary order in epsilon.

Metrics approximate original spacetime away from the geodesic as epsilon approaches zero.

Near the geodesic, rescaled metrics tend to a Kerr solution.

Abstract

Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime satisfying the Einstein vacuum equations (possibly with cosmological constant) and an inextendible timelike geodesic, we construct a family of metrics depending on a small parameter $\epsilon>0$ with the following properties. (1) They solve the Einstein vacuum equations modulo $\mathcal{O}(\epsilon^\infty)$. (2) Away from the geodesic they tend to the original metric as $\epsilon\to 0$. (3) Their $\epsilon^{-1}$-rescalings near every point of the geodesic tend to a fixed subextremal Kerr metric. Our result applies on all spacetimes with noncompact Cauchy hypersurfaces, and also on spacetimes without nontrivial Killing vector fields in a neighborhood of a point on the geodesic. If $(M,g)$ is a neighborhood of the domain of outer communications of subextremal or extremal Kerr(-anti de~Sitter) spacetime, our metrics model extreme mass ratio mergers if we choose the timelike geodesic to cross the event horizon. The metrics which we construct here depend on $\epsilon$ and the (rescaled) coordinates on the original spacetime in a log-smooth fashion. This in particular justifies the formal perturbation theoretic setup in work of Gralla-Wald on gravitational self-force in the case of small black holes.