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 $\ensuremath{\delta}{M}_{\mathrm{ADM}}=\ensuremath{\Omega}\ensuremath{\delta}J,$ 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 $\ensuremath{\Omega}$ are the Arnowitt-Deser-Misner (ADM) mass, Komar mass, angular momentum, and orbital angular velocity, respectively. $\ensuremath{\delta}$ 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 $\ensuremath{\delta}{M}_{\mathrm{ADM}}$ and $\ensuremath{\Omega}\ensuremath{\delta}J$ 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.

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