Post-Newtonian Hydrodynamic Equations Using the (3+1) Formalism in General Relativity

Using the (3+1) formalism in general relativity, we perform thepost-Newtonian(PN) approximation to clarify what sort of gauge condition issuitable for numerical analysis of coalescing compact binary neutron stars andgravitational waves from them. We adopt a kind of transverse gauge condition todetermine the shift vector. On the other hand, for determination of the timeslice, we adopt three slice conditions(conformal slice, maximal slice andharmonic slice) and discuss their properties. Using these conditions, the PNhydrodynamic equations are obtained up through the 2.5PN order including thequadrupole gravitational radiation reaction. In particular, we describe methodsto solve the 2PN tensor potential which arises from the spatial 3-metric. It isfound that the conformal slice seems appropriate for analysis of gravitationalwaves in the wave zone and the maximal slice will be useful for describing theequilibrium configurations. The PN approximation in the (3+1) formalism will bealso useful to perform numerical simulations using various slice conditionsand, as a result, to provide an initial data for the final merging phase ofcoalescing binary neutron stars which can be treated only by fully generalrelativistic simulations.Comment: 40 pages, TeX file using phyzzx, no figures, to appear in Prog. Theor. Phy