  The hierarchical three-body problem has many applications in relativistic
astrophysics, and can play an important role in the formation of the binary
black hole mergers detected by LIGO/Virgo. However, many studies have only
included relativistic corrections responsible for the precession of pericenter
of the inner and outer binaries, neglecting relativistic interactions between
the three bodies. We revisit this problem and develop a fully consistent
derivation of the secular three-body problem to first post-Newtonian order. We
start with the Einstein-Infeld-Hoffman equations for a three-body system and
expand the accelerations as a power series in the ratio of the semi-major axes
of the inner ($a_1$) and outer ($a_2$) binary. We then perform a
post-Keplerian, two-parameter expansion of the single-orbit-averaged Lagrange
planetary equations in $\delta = v^2/c^2$ and $\epsilon = a_1/a_2$ using the
method of multiple scales. Using this method, we derive previously-indentified
secular effects at $\delta \epsilon^{5/2}$ order that arise directly from the
equations of motion. We also calculate new secular effects through $\delta
\epsilon^4$ order that can lead to eccentricity growth over many Lidov-Kozai
cycles when the tertiary is much more massive than the inner binary. In such
cases, inclusion of these effects can substantially alter the evolution of
three-body systems as compared to an analysis in which they are neglected.
Careful analysis of post-Newtonian three-body effects will be important to
understand the formation and properties of coalescing binaries that form via
three-body dynamical processes.
