Gluing small black holes along timelike geodesics I: formal 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.

arXiv comment165 pages, 14 figures. The main change compared to v1 is that Hypothesis 9.12 from v1 is now proved (Theorem 9.12), and thus we can unconditionally glue in any subextremal Kerr black hole. Typos corrected, bibliography updated