Gluing small black holes along timelike geodesics I: formal solution

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 $\mathcal{O}({\epsilon }^{\mathrm{\infty }})$. (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.