  We consider compact binary systems, modeled in general relativity as vacuum
or perfect-fluid spacetimes with a helical Killing vector k^\alpha,
heuristically, the generator of time-translations in a corotating frame.
Systems that are stationary in this sense are not asymptotically flat, but have
asymptotic behavior corresponding to equal amounts of ingoing and outgoing
radiation. For black-hole binaries, a rigidity theorem implies that the Killing
vector lies along the horizon's generators, and from this one can deduce the
zeroth law (constant surface gravity of the horizon). Remarkably, although the
mass and angular momentum of such a system are not defined, there is an exact
first law, relating the change in the asymptotic Noether charge to the changes
in the vorticity, baryon mass, and entropy of the fluid, and in the area of
black holes.
  Binary systems with M\Omega small have an approximate asymptopia in which one
can write the first law in terms of the asymptotic mass and angular momentum.
Asymptotic flatness is precise in two classes of solutions used to model binary
systems: spacetimes satisfying the post-Newtonian equations, and solutions to a
modified set of field equations that have a spatially conformally flat metric.
(The spatial conformal flatness formalism with helical symmetry, however, is
consistent with maximal slicing only if replaces the extrinsic curvature in the
field equations by an artificially tracefree expression in terms of the shift
vector.) For these spacetimes, nearby equilibria whose stars have the same
vorticity obey the relation \delta M = \Omega \delta J, from which one can
obtain a turning point criterion that governs the stability of orbits.
