Periodic solutions of the Einstein equations for binary systems

Solutions of the Einstein equations which are periodic and have standing gravitational waves, in the weak-field zone, are valuable approximations to more physically realistic solutions with outgoing waves. A variational principle for the periodic solutions is found which has the power to provide, for binary systems with weak gravitational radiation, an accurate estimate of the relationship between the mass and angular momentum of the system, the masses and angular momenta of the components, the rotational frequency of the frame of reference in which the system is periodic, the frequency of the periodicity of the system, and the amplitude and phase of each multipole component of gravitational radiation. Examination of the boundary terms of the variational principle leads to definitions of the effective mass and effective angular momentum of a periodic geometry which capture the concepts of mass and angular momentum of the source alone with no contribution from the gravitational radiation. These effective quantities are surface integrals in the weak-field zone which are independent of the surface over which they are evaluated, through second order in the deviations of the metric from flat space. The variational principle provides a powerful method to examine the evolution of, say, a binary black hole system from the time when the holes are far apart, through the stage of slow evolution caused by gravitational radiation reaction, up until the moment when the radiation reaction time scale, is comparable to the dynamical time scale.