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Given a smooth globally hyperbolic $(3+1)$-dimensional spacetime satisfyingthe Einstein vacuum equations (possibly with cosmological constant) and aninextendible timelike geodesic, we construct a family of metrics depending on asmall parameter $\epsilon>0$ with the following properties. (1) They solve theEinstein vacuum equations modulo $\mathcal{O}(\epsilon^\infty)$. (2) Away fromthe 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 fixedsubextremal Kerr metric (assuming a condition on the mode stability at zerofrequency which we verify in the very slowly rotating case). Our results applyon spacetimes which do not admit nontrivial Killing vector fields in aneighborhood of a point on the geodesic. They also apply in a neighborhood ofthe domain of outer communications of subextremal Kerr and Kerr-de Sitterspacetimes, in which case our metrics model extreme mass ratio inspirals if wechoose 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 inparticular justifies the formal perturbation theoretic setup in work ofGralla-Wald on gravitational self-force in the case of small black holes.

Gluing small black holes along timelike geodesics I: formal solution