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

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 C, we construct, on any compact subset of M, solutions g_ε of the Einstein equations which describe a mass ε Kerr black hole traveling along C. More precisely, away from C one has g_ε→ g as ε→ 0, while the ε⁻¹-rescaling of g_ε around every point of 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 g0,ε of the gluing problem which satisfy the Einstein equations only modulo O(ε^∞) errors. Part II introduces a framework for obtaining uniform control of solutions of linear wave equations on ε-independent precompact subsets of the original spacetime $(M,g)$. In this final part, we show how to correct g0,ε to a true solution g_ε by adding a metric perturbation of size O(ε^∞) 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.