  In this series we construct an effective field theory (EFT) in curved
spacetime to study gravitational radiation and backreaction effects. We begin
in this paper with a derivation of the self-force on a compact object moving in
the background spacetime of a supermassive black hole. The EFT approach
utilizes the disparity between two length scales, which in this problem are the
size of the compact object and the radius of curvature of the background
spacetime, to treat the orbital dynamics of the compact object, described as an
effective point particle, separately from its tidal deformations. Ultraviolet
divergences are regularized using Hadamard's {\it partie finie} to isolate the
non-local finite part from the quasi-local divergent part. The latter is
constructed from a momentum space representation for the graviton retarded
propagator and is evaluated using dimensional regularization in which only
logarithmic divergences are relevant for renormalizing the parameters of the
theory. As a first important application of this framework we explicitly derive
the first order self-force given by Mino, Sasaki, Tanaka, Quinn and Wald. Going
beyond the point particle approximation, to account for the finite size of the
object, we demonstrate that for extreme mass ratio inspirals the motion of a
compact object is affected by tidally induced moments at $O(\epsilon^4)$, in
the form of an Effacement Principle. The relatively large radius-to-mass ratio
of a white dwarf star allows for these effects to be enhanced until the white
dwarf becomes tidally disrupted, a potentially $O(\epsilon^2)$ process, or
plunges into the supermassive black hole. This work provides a new foundation
for further exploration of higher order self force corrections, gravitational
radiation and spinning compact objects.
