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

Nonlinear gravitational self-force: second-order equation of motion