  We present a new approach to the problem of binary black holes in the
pre-coalescence stage, i.e. when the notion of orbit has still some meaning.
Contrary to previous numerical treatments which are based on the initial value
formulation of general relativity on a (3-dimensional) spacelike hypersurface,
our approach deals with the full (4-dimensional) spacetime. This permits a
rigorous definition of the orbital angular velocity. Neglecting the
gravitational radiation reaction, we assume that the black holes move on closed
circular orbits, which amounts to endowing the spacetime with a helical Killing
vector. We discuss the choice of the spacetime manifold, the desired properties
of the spacetime metric, as well as the choice of the rotation state for the
black holes. As a simplifying assumption, the space 3-metric is approximated by
a conformally flat one. The problem is then reduced to solving five of the ten
Einstein equations, which are derived here, as well as the boundary conditions
on the black hole surfaces and at spatial infinity. We exhibit the remaining
five Einstein equations and propose to use them to evaluate the error induced
by the conformal flatness approximation. The orbital angular velocity of the
system is computed through a requirement which reduces to the classical virial
theorem at the Newtonian limit.
