Comment on a relativistic model for coalescing neutron star binaries

The Wilson approximate dynamics and the Einstein dynamics are compared for binary systems. At the second post-Newtonian approximation, genuine two-body aspects are found to differ by up to 114%. In the regime of a formal innermost stable circular orbit (ISCO) the both dynamics differ by up to 7%. Recently Wilson and Mathews [1] proposed a truncated version of the Einstein field equations to treat the coalescence of binary neutron stars in a much simplified but still sufficiently precise manner. The main idea of this approximation is to neglect the independent (“true”) degrees of freedom of the gravitational field, i.e. in particular, the full gravitational radiation content. For spherically symmetric processes the proposed scheme is identical with the Einstein equations, in non-spherically symmetric dynamical situations, even stationary ones, the proposed scheme and the Einstein theory only coincide at the first post-Newtonian level of approximation. It is perhaps worth mentioning that, in contrast to the Einstein theory, the proposed scheme should allow post-Newtonian series expansions to all orders in (integer) powers of 1/c. Most recently, Wilson et al. [2] applied the Wilson scheme to the question of instabilities in close neutron star binaries and found the remarkable result that general relativity may cause otherwise stable stars to collapse prior to merging. In another recent paper Cook et al. [3] tested the Wilson scheme for isolated, rapidly rotating relativistic stars. They found a deviation from the Einstein theory of at most 5% which they interpreted as very encouraging for a better understanding of binary star evolution. In this paper we apply the Wilson scheme to point-like binary systems at the second postNewtonian approximation and calculate the periastron advance as well as the orbital period. For circular motion, also the dependence of the angular momentum on the orbital angular frequency is given. In addition, we use the Hamiltonian of the Wilson scheme to calculate a formal innermost stable circular orbit (ISCO) for binary systems. The obtained results are confronted with the corresponding results of the Einstein dynamics. In the Einstein theory the periastron advance and the orbital period have been calculated by Damour and Schäfer [4] starting from a second post-Newtonian Hamilton function. In isotropic coordinates and in the center of mass system (P1 = −P2 = P), in reduced varibales (p = P/μ, r = R/GM) the reduced Hamilton function Ĥ = H/μ reads Ĥ(r,p, ν) = 1 2 p − 1 r − 1 8c (1− 3ν)p − 1 2rc [(3 + ν)p + ν(n · p)]