In this chapter we wish to construct initial data for quasiequilibrium binary black holes. That is, we seek solutions corresponding to two black holes in stable, nearly circular orbit about each other. In contrast to Newtonian theory, a stellar binary in general relativity can never be in strict equilibrium, with the companions moving in exactly circular orbits at constant separation for all time. Instead, gravitational radiation emission inevitably leads to loss of orbital energy and angular momentum, causing the orbit to decay. The resulting trajectory then traces out an inspiral rather than a perfect circle. For sufficiently large separations, the binary motion is nearly Newtonian, hence the orbit is nearly circular, decaying very little during one orbital period. If isolated from outside perturbations (e.g., gravitational encounters with other stars), it is expected that astrophysical binaries composed of compact stars (i.e., compact binaries) will ultimately evolve to a quasiequilibrium state following their formation at large separation. The reason is that gravitational radiation loss drives orbital circularization as well as decay, as we will discuss in the next section. Only when the orbits become very close and highly relativistic, just prior to radial plunge and binary merger, do the deviations from circular motion become large.

The construction of quasiequilibrium binary initial data poses a number of conceptual challenges. Getting started, however, is fairly straightforward. To find solutions we shall follow the approaches outlined in Chapter 3.