Formulation for the internal motion of quasi-equilibrium configurations in general relativity

We present a formulation for the internal motion of equilibrium configurations with a rotational Killing vector in general relativity. As an approximation, this formulation is applicable to investigation of the internal motion of quasi-equilibrium configurations such as binary neutron stars. Based on this simple formulation, a condition for the general relativistic counter rotation has been obtained, though in the recent work by Bonazzola, Gourgoulhon and Marck, their condition for the counter rotation is not enough to specify the internal velocity field. Under the condition given in this paper, the internal velocity field can be determined completely. Indeed, in the counter-rotating case, we have also derived Poisson equations for the internal velocity, which take tractable forms in numerical implementation.