Post-Newtonian hydrodynamic equations using the (3+1) formalism in general relativity

Using the (3+1) formalism in general relativity, we perform the post-Newtonian (PN) approximation to clarify what sort of gauge condition is suitable for numerical analysis of coalescing compact binary neutron stars and gravitational waves from them. We adopt a kind of transverse gauge condition to determine the shift vector. On the other hand, for determination of the time slice, we adopt three slice conditions (conformal slice, maximal slice and harmonic slice) and discuss their properties. Using these conditions, the PN hydrodynamic equations are obtained up through the 2.5PN order including the quadrupole gravitational radiation reaction. In particular, we describe methods to solve the 2PN tensor potential which arises from the spatial 3-metric. It is found that the conformal slice seems appropriate for analysis of gravitational waves in the wave zone and the maximal slice will be useful for describing the equilibrium configurations. The PN approximation in the (3+1) formalism will be also useful to perform numerical simulations using various slice conditions and, as a result, to provide an initial data for the final merging phase of coalescing binary neutron stars which can be treated only by fully general relativistic simulations.