New numerical scheme to compute three-dimensional configurations of quasiequilibrium compact stars in general relativity: Application to synchronously rotating binary star systems

We develop a new numerical scheme to obtain quasiequilibrium structures of binary neutron star systems and nonaxisymmetric compact stars as well as the space time around those systems in general relativity. Although, strictly speaking, there are no equilibrium states for binary configurations in general relativity, the time scale of changes in orbital motion due to gravitational wave radiation is long compared with the orbital period. Thus, we can assume that binary neutron star systems, and nonaxisymmetric systems in general are in “quasiequilibrium” states. Concerning the quasiequilibrium states of binary systems in general relativity, several investigations have been already carried out by assuming conformal flatness of the spatial part of the metric. However, the validity of the conformally flat treatment has not been fully analyzed except for axisymmetric configurations. Therefore, it is desirable to solve for the quasiequilibrium states by developing totally different methods from the conformally flat scheme. In this paper, we present a new numerical scheme to solve the Einstein equations for three-dimensional configurations directly, without assuming conformal flatness, although we use the simplified metric for the space time. This new formulation is an extension of the scheme which has been successfully applied for structures of axisymmetric rotating compact stars in general relativity. It is based on the integral representation of the Einstein equations, and takes into account the boundary conditions at infinity. We have checked our numerical scheme by computing equilibrium sequences of binary polytropic star systems in Newtonian gravity and those of axisymmetric polytropic stars in general relativity. We have applied this numerical code to binary star systems in general relativity and have succeeded in obtaining several equilibrium sequences of synchronously rotating binary polytropes with the polytropic indices $N=0.0,$ $0.5,$ and $\mathrm{1.0.}$ It should be noted that our equilibrium sequences are not those of constant baryon mass star models because there is no unique choice of parameters to keep the baryon mass constant for our polytropic relation.