  Extreme mass-ratio inspirals, in which solar-mass compact bodies spiral into
supermassive black holes, are an important potential source for gravitational
wave detectors. Because of the extreme mass-ratio, one can model these systems
using perturbation theory. However, in order to relate the motion of the small
body to the emitted waveform, one requires a model that is accurate on
extremely long timescales. Additionally, in order to avoid intractable
divergences, one requires a model that treats the small body as asymptotically
small rather than exactly pointlike. Both of these difficulties can be resolved
by using techniques of singular perturbation theory. I begin this dissertation
with an analysis of singular perturbation theory on manifolds, including the
common techniques of matched asymptotic expansions and two-timescale
expansions. I then formulate a systematic asymptotic expansion in which the
metric perturbation due to the body is expanded while a representative
worldline is held fixed, and I contrast it with a regular expansion in which
both the metric and the worldline must be expanded. This results in an
approximation that is potentially uniformly accurate on long timescales. The
equation of motion for the body's fixed worldline is determined by performing a
local-in-space expansion in the neighbourhood of the body. Using this local
expansion as boundary data, I construct a global solution to the perturbative
Einstein equation. To concretely characterize orbits, I next devise a
relativistic generalization of the Newtonian method of osculating orbits.
Making use of this method and two-timescale expansions, I examine the utility
of adiabatic approximations that can forgo an explicit calculation of the
force.
