  When a small, uncharged, compact object is immersed in an external background
spacetime, at zeroth order in its mass it moves as a test particle in the
background. At linear order, its own gravitational field alters the geometry
around it, and it moves instead as a test particle in a certain effective
metric satisfying the linearized vacuum Einstein equation. In the letter [Phys.
Rev. Lett. 109, 051101 (2012)], using a method of matched asymptotic
expansions, I showed that the same statement holds true at second order: if the
object's leading-order spin and quadrupole moment vanish, then through second
order in its mass it moves on a geodesic of a certain smooth, locally causal
vacuum metric defined in its local neighbourhood. Here I present the complete
details of the derivation of that result. In addition, I extend the result,
which had previously been derived in gauges smoothly related to Lorenz, to a
class of highly regular gauges that should be optimal for numerical self-force
computations.
