Post-Newtonian Expansion of Gravitational Radiation

Abstract

The post-Newtonian expansion appears to be a relevant tool for predicting the gravitational waveforms generated by some astrophysical systems such as binaries. In particular, inspiralling compact binaries are well-modelled by a system of two point-particles moving on a quasi-circular orbit whose decay by emission of gravitational radiation is described by a post-Newtonian expansion. In this paper we summarize the basics of the computation by means of a series of multipole moments of the exterior field generated by an isolated source in the post-Newtonian approximation. This computation relies on an ansatz of matching the exterior multipolar field to the inner field of a slowly-moving source. The formalism can be applied to point-particles at the price of a further ansatz, that the infinite self-field of point-particles can be regularized in a certain way. As it turns out, the concept of point-particle requires a precise definition in high post-Newtonian approximations of general relativity.

Computation of gravitational waves from inspiraling binary neutron stars in quasiequilibrium circular orbits: Formulation and calibration

“…First, we calibrate our method by comparing the numerical results with post Newtonian formulas for gravitational waves from two point masses [2,8], adopting weakly gravitating binary neutron stars. We will demonstrate that our results agree well with post Newtonian analytic formulas [3]. Then, gravitational waves from more compact binaries are computed to point out the importance of tidal deformation and strong general relativistic effects on gravitational waves for close binaries.…”

“…As mentioned in Sec. I, the procedure for the computation of gravitational waves adopted here is quite similar to that for obtaining gravitational waves in the post Newtonian approximation [3,4]: In the post Newtonian work, one first determines an equilibrium circular orbit from post Newtonian equations of motion, neglecting the dissipation terms due to gravitational radiation. Then, one substitutes the spacetime metric and matter fields into the source term for a wave equation of gravitational waves.…”

“…For comparison, we also plot the energy and angular momentum for binaries of nonspinning stars derived in the 2PN approximation [3] as 13) where E b has to be added in the energy in comparison because in E t not only the binding energy between two stars but also the binding energy of individual stars is included. In [3], J 2PN is not shown but it is easily computed from the relation dE = ΩdJ for the point mass case. Figures 4 and 5 show that for distant orbits and for (M/R) ∞ = 0.14, the numerical results are fitted well with 2PN formulas except for a possible small systematic, numerical error.…”

“…For this reason, post Newtonian studies jointly using point particle approximations for compact objects have been carried out by several groups, producing a wide variety of successful results (e.g., [2][3][4][5][6][7]). However, for closer orbits such as for r o < ∼ 4R and v > ∼ c/3, the tidal effect is likely to become important, resulting in deformation of neutron stars and in the modification of the amplitude and luminosity of gravitational waves.…”

“…Our method is in a sense similar to the standard post Newtonian method for the computation of gravitational waves from binaries of two-point masses in circular orbits [3,4]. Thus, before proceeding, let us briefly review an outline of the post Newtonian method.…”