Deriving formulations for numerical computation of binary neutron stars in quasicircular orbits

Two relations, the virial relation $M_{\mathrm{ADM}}{=M}_{\mathrm{K}}$ and the first law in the form $\delta M_{\mathrm{ADM}}=\Omega \delta J,$ should be satisfied by a solution and a sequence of solutions describing binary compact objects in quasiequilibrium circular orbits. Here, $M_{\mathrm{ADM}},$ $M_{\mathrm{K}},$ J, and $\Omega$ are the Arnowitt-Deser-Misner (ADM) mass, Komar mass, angular momentum, and orbital angular velocity, respectively. $\delta$ denotes an Eulerian variation. These two conditions restrict the allowed formulations that we may adopt. First, we derive relations between $M_{\mathrm{ADM}}$ and $M_{\mathrm{K}}$ and between $\delta M_{\mathrm{ADM}}$ and $\Omega \delta J$ for general asymptotically flat spacetimes. Then, to obtain solutions that satisfy the virial relation and sequences of solutions that satisfy the first law at least approximately, we propose a formulation for computation of quasiequilibrium binary neutron stars in general relativity. In contrast to previous approaches in which a part of the Einstein equation is solved, in the new formulation, the full Einstein equation is solved with maximal slicing and in a transverse gauge for the conformal three-metric. Helical symmetry is imposed in the near zone, while in the distant zone, a waveless condition is assumed. We expect the solutions obtained in this formulation to be excellent quasiequilibria as well as initial data for numerical simulations of binary neutron star mergers.